Special Functions
Special functions, which include the trigonometric functions, have been used for
centuries. Their role in the solution of differential equations was exploited by
Newton and Leibniz, and the subject of special functions has been in continuous
development ever since. In just the past thirty years several new special functions
and applications have been discovered.
This treatise presents an overview of the area of special functions, focusing pri-
primarily on the hypergeometric functions and the associated hypergeometric series.
It includes both important historical results and recent developments and shows
how these arise from several areas of mathematics and mathematical physics.
Particular emphasis is placed on formulas that can be used in computation.
The book begins with a thorough treatment of the gamma and beta functions,
which are essential to understanding hypergeometric functions. Later chapters
discuss Bessel functions, orthogonal polynomials and transformations, the Selberg
integral and its applications, spherical harmonics, g-series, partitions, and Bailey
chains.
This clear, authoritative work will be a lasting reference for students and re-
researchers in number theory, algebra, combinatorics, differential equations, math-
mathematical computing, and mathematical physics.
George E. Andrews is Evan Pugh Professor of Mathematics at The Pennsylvania
State University.
Richard Askey is Professor of Mathematics at the University of Wisconsin-
Madison.
Ranjan Roy is Professor of Mathematics at Beloit College in Wisconsin.

ENCYCLOPEDIA OF MATHEMATICS AND ITS APPLICATIONS
EDITED BY G.-C. ROTA
Editorial Board
B. Doran, M. Ismail, T.-Y. Lam, E. Lutwak
Volume 71
Special Functions
27 N. H. Bingham, С. М. Goldie, and J. L. Teugels Regular Variation
28 P. P. Petrushev and V. A. Popov Rational Approximation of Real Functions
29 N. White (ed.) Combinatorial Geometries
30 M. Pohst andH. Zassenhaus Algorithmic Algebraic Number Theory
31 J. Aczel and J. Dhombres Functional Equations in Several Variables
32 M.Kuczma, B.Choczewski,andR. Ger Iterative Functional Equations
33 R. V. Ambartzumian Factorization Calculus and Geometric Probability
34 G. Gripenberg, S.-O. Londen, and O. Staffans Volterra Integral and Functional Equations
35 G. Gasper and M.Rahman BasicHypergeometric Series
36 E. Torgersen Comparison of Statistical Experiments
37 A. Neumaier Interval Methods for Systems of Equations
38 N. Korneichuk Exact Constants in Approximation Theory
39 R. Brualdi and H. Ryser Combinatorial Matrix Theory
40 N. White (ed.) Matroid Applications
41 S.Sakai Operator Algebras in Dynamical Systems
42 W. Hodges Basic Model Theory
43 H.Stahland V. Totik General Orthogonal Polynomials
45 G. Da Prato and J. Zabczyk Stochastic Equations in Infinite Dimensions
46 A. Bjorner et al. OrientedMatroids
47 G. Edgar and L. Sucheston Stopping Times and Directed Processes
48 C.Sims Computation with Finitely Presented Groups
49 T. Palmer Banach Algebras and the General Theory of *-Algebras
50 F. Borceux Handbook of Categorical Algebra I
51 F. Borceux Handbook of Categorical Algebra II
52 F. Borceux Handbook of Categorical Algebra III
54 A. Katok and B. Hasselblatt Introduction to the Modern Theory of Dynamical Systems
55 V. N. Sachkov Combinatorial Methods in Discrete Mathematics
56 V. N. Sachkov Probabilistic Methods in Discrete Mathematics
57 P. M.Cohn Skew Fields
58 R. Gardner Geometric Topography
59 G. A. Baker Jr. and P. Graves-Morris Pade Approximants
60 J. Krajicek Bounded Arithmetic, Prepositional Logic, and Complexity Theory
61 H. Groemer Geometric Applications of Fourier Series and Spherical Harmonics
62 H. O. Fattorini Infinite Dimensional Optimization and Control Theory
63 A.C.Thompson Minkow ski Geometry
64 R. B. Bapat and Т. Е. S. Raghavan Nonnegative Matrices with Applications
65 K.Engel Sperner Theory
66 D. Cverkovic, P. Rowlinson, S. Simic Eigenspaces of Graphs
67 F. Bergeron, G. Labelle, and P. Leroux Combinatorial Species and Tree-Like Structures
68 R. Goodman and N. Wallach Representations and Invariants of the Classical Groups

ENCYCLOPEDIA OF MATHEMATICS AND ITS APPLICATIONS
Special Functions
GEORGE E.jANDREWS RICHARD ASKEY RANJAN ROY
CAMBRIDGE
UNIVERSITY PRESS

tier
PUBLISHED BY THE PRESS SYNDICATE OF THE UNIVERSITY OF CAMBRIDGE
The Pitt Building, Trumpington Street, Cambridge, United Kingdom
CAMBRIDGE UNIVERSITY PRESS
The Edinburgh Building, Cambridge CB2 2RU, UK
40 West 20th Street, New York, NY 10011-4211, USA
10 Stamford Road, Oakleigh, Melbourne 3166, Australia
Ruiz de Alarcon 13, 28014 Madrid, Spain
Dock House, The Waterfront, Cape Town 8001, South Africa
http://www.cambridge.org
© George E. Andrews, Richard Askey, and Ranjan Roy 1999
This book is in copyright. Subject to statutory exception and
to the provisions of relevant collective licensing agreements,
no reproduction of any part may take place without
the written permission of Cambridge University Press.
First published 1999
First paperback edition 2000
Printed in the United States of America
Typeset in 10/13 Times Roman in iftgX [ТВ]
A catalog record for this book is available from the British Library
Library of Congress Cataloging in Publication data is available
ISBN 0 521 62321 9 hardback
ISBN 0 521 78988 5 paperback

To Leonard Carlitz, От Prakash Juneja,
and Irwin Kra

Contents
Preface page xiii
1 The Gamma and Beta Functions 1
1.1 The Gamma and Beta Integrals and Functions 2
1.2 The Euler Reflection Formula 9
1.3 The Hurwitz and Riemann Zeta Functions 15
1.4 Stirling's Asymptotic Formula 18
1.5 Gauss's Multiplication Formula for Г(mx) 22
1.6 Integral Representations for Log Г(х) and тр-(x) 26
1.7 Kummer's Fourier Expansion of Log Г(х) 29
1.8 Integrals of Dirichlet and Volumes of Ellipsoids 32
1.9 The Bohr-Mollerup Theorem 34
1.10 Gauss and Jacobi Sums 36
1.11 A Probabilistic Evaluation of the Beta Function 43
1.12 The p-adic Gamma Function 44
Exercises 46
2 The Hypergeometric Functions 61
2.1 The Hypergeometric Series 61
2.2 Euler's Integral Representation 65
2.3 The Hypergeometric Equation 73
2.4 The Barnes Integral for the Hypergeometric Function 85
2.5 Contiguous Relations 94
2.6 Dilogarithms 102
2.7 Binomial Sums 107
2.8 DougalPs Bilateral Sum 109
2.9 Fractional Integration by Parts and Hypergeometric Integrals 111
Exercises 114

viii Contents
3 Hypergeometric Transformations and Identities 124
3.1 Quadratic Transformations 125
3.2 The Arithmetic-Geometric Mean and Elliptic Integrals 132
3.3 Transformations of Balanced Series 140
3.4 Whipple's Transformation 143
3.5 Dougall's Formula and Hypergeometric Identities 147
3.6 Integral Analogs of Hypergeometric Sums 150
3.7 Contiguous Relations 154
3.8 The Wilson Polynomials 157
3.9 Quadratic Transformations - Riemann's View 160
3.10 Indefinite Hypergeometric Summation 163
3.11 The W-Z Method 166
3.12 Contiguous Relations and Summation Methods 174
Exercises 176
4 Bessel Functions and Confluent Hypergeometric Functions 187
4.1 The Confluent Hypergeometric Equation 188
4.2 Barnes's Integral for 1F1 192
4.3 Whittaker Functions 195
4.4 Examples of i Fx and Whittaker Functions 196
4.5 Bessel's Equation and Bessel Functions 199
4.6 Recurrence Relations 202
4.7 Integral Representations of Bessel Functions 203
4.8 Asymptotic Expansions 209
4.9 Fourier Transforms and Bessel Functions 210
4.10 Addition Theorems 213
4.11 Integrals of Bessel Functions 216
4.12 The Modified Bessel Functions 222
4.13 Nicholson's Integral 223
4.14 Zeros of Bessel Functions 225
4.15 Monotonicity Properties of Bessel Functions 229
4.16 Zero-Free Regions for i Fi Functions 231
Exercises 234
5 Orthogonal Polynomials 240
5.1 Chebyshev Polynomials 240
5.2 Recurrence 244
5.3 Gauss Quadrature 248
5.4 Zeros of Orthogonal Polynomials 253
5.5 Continued Fractions 256

Contents ix
5.6 Kernel Polynomials 259
5.7 Parseval's Formula 263
5.8 The Moment-Generating Function 266
Exercises 269
6 Special Orthogonal Polynomials 277
6.1 Hermite Polynomials 278
6.2 Laguerre Polynomials 282
6.3 Jacobi Polynomials and Gram Determinants 293
6.4 Generating Functions for Jacobi Polynomials 297
6.5 Completeness of Orthogonal Polynomials 306
6.6 Asymptotic Behavior of Pff -^ (x) for Large n 310
6.7 Integral Representations of Jacobi Polynomials 313
6.8 Linearization of Products of Orthogonal Polynomials 316
6.9 Matching Polynomials 323
6.10 The Hypergeometric Orthogonal Polynomials 330
6.11 An Extension of the Ultraspherical Polynomials 334
Exercises 339
7 Topics in Orthogonal Polynomials 355
7.1 Connection Coefficients 356
7.2 Rational Functions with Positive Power Series Coefficients 363
7.3 Positive Polynomial Sums from Quadrature
and Vietoris's Inequality 371
7.4 Positive Polynomial Sums and the Bieberback Conjecture 381
7.5 A Theorem of Turan 384
7.6 Positive Summability of Ultraspherical Polynomials 388
7.7 The Irrationality of ?C) 391
Exercises 395
8 The Selberg Integral and Its Applications 401
8.1 Selberg's and Aomoto's Integrals 402
8.2 Aomoto's Proof of Selberg's Formula 402
8.3 Extensions of Aomoto's Integral Formula 407
8.4 Anderson's Proof of Selberg's Formula 411
8.5 A Problem of Stieltjes and the Discriminant of
a Jacobi Polynomial 415
8.6 Siegel's Inequality 419
8.7 The Stieltjes Problem on the Unit Circle 425
8.8 Constant-Term Identities 426
8.9 Nearly Poised 3F2 Identities 428
8.10 The Hasse-Davenport Relation 430

Contents
10
11
8.11
A Finite-Field Analog of Selberg s Integral
Exercises
Spherical Harmonics
9.1
9.2
9.3
9.4
9.5
9.6
9.7
9.8
9.9
9.10
9.11
9.12
9.13
9.14
9.15
9.16
Harmonic Polynomials
The Laplace Equation in Three Dimensions
Dimension of the Space of Harmonic Polynomials
of Degree к
Orthogonality of Harmonic Polynomials
Action of an Orthogonal Matrix
The Addition Theorem
The Funk-Hecke Formula
The Addition Theorem for Ultraspherical Polynomials
The Poisson Kernel and Dirichlet Problem
Fourier Transforms
Finite-Dimensional Representations of Compact Groups
The Group SUB)
Representations of SUB)
Jacobi Polynomials as Matrix Entries
An Addition Theorem
Relation of SUB) to the Rotation Group SOC)
Exercises
Introduction to ^-Series
10.1
10.2
10.3
10.4
10.5
10.6
10.7
10.8
10.9
10.10
10.11
10.12
The <?-Integral
The ^-Binomial Theorem
The q -Gamma Function
The Triple Product Identity
Ramanujan's Summation Formula
Representations of Numbers as Sums of Squares
Elliptic and Theta Functions
g-Beta Integrals
Basic Hypergeometric Series
Basic Hypergeometric Identities
<7-Ultraspherical Polynomials
Mellin Transforms
Exercises
Partitions
11.1
11.2
11.3
Background on Partitions
Partition Analysis
A Library for the Partition Analysis Algorithm
434
439
445
445
447
449
451
452
454
458
459
463
464
466
469
471
473
474
476
478
481
485
487
493
496
501
506
508
513
520
523
527
532
542
553
553
555
557

Contents xi
11.4 Generating Functions 559
11.5 Some Results on Partitions 563
11.6 Graphical Methods 565
11.7 Congruence Properties of Partitions 569
Exercises 573
12 Bailey Chains 577
12.1 Rogers's Second Proof of the Rogers-Ramanujan Identities 577
12.2 Bailey's Lemma 582
586
589
590
595
595
597
В Summabilitv and Fractional Integration 599
599
602
604
605
607
С Asymptotic Expansions 611
C.I Asymptotic Expansion 611
C.2 Properties of Asymptotic Expansions 612
C.3 Watson's Lemma 614
C.4 The Ratio of Two Gamma Functions 615
Exercises 616
D Euler-Maclaurin Summation Formula 617
D.I Introduction 617
D.2 The Euler-Maclaurin Formula 619
D.3 Applications 621
D.4 The Poisson Summation Formula 623
Exercises 627
E Lagrange Inversion Formula 629
E. 1 Reversion of Series 629
E.2 A Basic Lemma 630
E.3 Lambert's Identity • 631
E.4 Whipple's Transformation 632
Exercises 634
12.3
12.4
Watson's Transformation Formula
Other Applications
Exercises
Infinite Products
A.I
Infinite Products
Exercises
Summability and Fractional Integration
B.I
B.2
B.3
B.4
Abel and Cesaro Means
The Cesaro Means (C, a)
Fractional Integrals
Historical Remarks
Exercises

xii Contents
F Series Solutions of Differential Equations 637
F.I Ordinary Points 637
F.2 Singular Points 638
F.3 Regular Singular Points 639
Bibliography 641
Index 655
Subject Index 659
Symbol Index 661

Preface
Paul Turan once remarked that special functions would be more appropriately la-
labeled "useful functions." Because of their remarkable properties, special functions
have been used for centuries. For example, since they have numerous applications
in astronomy, trigonometric functions have been studied for over a thousand years.
Even the series expansions for sine and cosine (and probably the arc tangent) were
known to Madhava in the fourteenth century. These series were rediscovered by
Newton and Leibniz in the seventeenth century. Since then, the subject of spe-
special functions has been continuously developed, with contributions by a host of
mathematicians, including Euler, Legendre, Laplace, Gauss, Kummer, Eisenstein,
Riemann, and Ramanujan.
In the past thirty years, the discoveries of new special functions and of applica-
applications of special functions to new areas of mathematics have initiated a resurgence
of interest in this field. These discoveries include work in combinatorics initiated
by Schiitzenberger and Foata. Moreover, in recent years, particular cases of long
familiar special functions have been clearly defined and applied as orthogonal
polynomials.
As a result of this prolific activity and long history one is pulled different di-
directions when writing a book on special functions. First, there are important
results from the past that must be included because they are so useful. Second,
there are recent developments that should be brought to the attention of those
who could use them. One also would wish to help educate the new generation
of mathematicians and scientists so that they can further develop and apply this
subject. We have tried to do all this, and to include some of the older results that
seem to us to have been overlooked. However, we have slighted some of the very
important recent developments because a book that did them justice would have
to be much longer. Fortunately, specialized books dealing with some of these
developments have recently appeared: Petkovsek, Wilf, and Zeilberger [1996],
Macdonald [1995],Heckman and Schlicktkrull [1994], and Vilenkin and Klimyk

xiv Preface
[1992]. Additionally, I. G. Macdonald is writing a new book on his polynomials
in several variables and A. N. Kirillov is writing on R-matrix theory.
It is clear that the amount of knowledge about special functions is so great that
only a small fraction of it can be included in one book. We have decided to focus
primarily on the best understood class of functions, hypergeometric functions, and
the associated hypergeometric series. A hypergeometric series is a series Y,an with
an+i/an a rational function of n. Unfortunately, knowledge of these functions is
not as widespread as is warranted by their importance and usefulness. Most of the
power series treated in calculus are hypergeometric, so some facts about them are
well known. However, many mathematicians and scientists who encounter such
functions in their work are unaware of the general case that could simplify their
work. To them a Bessel function and a parabolic cylinder function are types of
functions different from the 3 — j or 6 — j symbols that arise in quantum angular
momentum theory. In fact these are all hypergeometric functions and many of
their elementary properties are best understood when considered as such.
Several important facts about hypergeometric series were first found by Euler
and an important identity was discovered by Pfaff, one of Gauss's teachers. How-
However, it was Gauss himself who fully recognized their significance and gave a
systematic account in two important papers, one of which was published posthu-
posthumously. One reason for his interest in these functions was that the elementary
functions and several other important functions in mathematics are expressible
in terms of hypergeometric functions. A half century after Gauss, Riemann de-
developed hypergeometric functions from a different point of view, which made
available the basic formulas with a minimum of computation. Another approach
to hypergeometric functions using contour integrals was presented by the English
mathematician E. W. Barnes in the first decade of this century. Each of these
different approaches has its advantages.
Hypergeometric functions have two very significant properties that add to their
usefulness: They satisfy certain identities for special values of the function and they
have transformation formulas. We present many applications of these properties.
For example, in combinatorial analysis hypergeometric identities classify single
sums of products of binomial coefficients. Further, quadratic transformations
of hypergeometric functions give insight into the relationship (known to Gauss)
of elliptic integrals to the arithmetic-geometric mean. The arithmetic-geometric
mean has recently been used to compute л to several million decimal places, and
earlier it played a pivotal role in Gauss's theory of elliptic functions.
The gamma function and beta integrals dealt with in the first chapter are essential
to understanding hypergeometric functions. The gamma function was introduced
into mathematics by Euler when he solved the problem of extending the factorial
function to all real or complex numbers. He could not have foreseen the extent of
its importance in mathematics. There are extensions of gamma and beta functions

Preface xv
that are also very important. The text contains a short treatment of Gauss and
Jacobi sums, which are finite field analogs of gamma and beta functions. Gauss
sums were encountered by Gauss in his work on the constructibility of regular
polygons where they arose as "Lagrange resolvents," a concept used by Lagrange
to study algebraic equations. Gauss understood the tremendous value of these
sums for number theory. We discuss the derivation of Fermat's theorem on primes
of the form An + 1 from a formula connecting Gauss and Jacobi sums, which is
analogous to Euler's famous formula relating beta integrals with gamma functions.
There are also multidimensional gamma and beta integrals. The first of these
was introduced by Dirichlet, though it is really an iterated version of the one-
dimensional integral. Genuine multidimensional gamma and beta functions were
introduced in the 1930s, by both statisticians and number theorists. In the early
1940s, Atle Selberg found a very important multidimensional beta integral in the
course of research in entire functions. However, owing to the Second World War
and the fact that the first statement and also the proof appeared in journals that were
not widely circulated, knowledge of this integral before the 1980s was restricted to
a few people around the world. We present two different evaluations of Selberg's
integral as well as some of its uses.
In addition to the above mentioned extensions, there are q-extensions of the
gamma function and beta integrals that are very fundamental because they lead to
basic hypergeometric functions and series. These are series Ес„ where cn+\/cn
is a rational function of q" for a fixed parameter q. Here the sum may run over
all integers, instead of only nonnegative ones. One important example is the
theta function Y^oo Q" x" ¦ This and other similar series were used by Gauss and
Jacobi to study elliptic and elliptic modular functions. Series of this sort are
very useful in many areas of combinatorial analysis, a fact already glimpsed by
Euler and Legendre, and they also arise in some branches of physics. For ex-
example, the work of the physicist R. J. Baxter on the Yang-Baxter equation led
a group in St. Petersburgh to the notion of a quantum group. Independently,
M. Jimbo in Japan was led by a study of Baxter's work to a related structure.
Many basic hypergeometric series (or g-hypergeometric series), both polyno-
polynomials and infinite series, can be studied using Hopf algebras, which make up
quantum groups. Unfortunately, we could not include this very important new
approach to basic series. It was also not possible to include results on the multidi-
multidimensional U(n) generalizations of theorems on basic series, which have been
studied extensively in recent years. For some of this work, the reader may
refer to Milne [1988] and Milne and Lilly [1995]. We briefly discuss the q-
gamma function and some important g-beta integrals; we show that series and
products that arise in this theory have applications in number theory, combi-
combinatorics, and partition theory. We highlight the method of partition analysis.

xvi Preface
P. A. MacMahon, who developed this powerful technique, devoted several chapters
to it in his monumental Сombinatory Analysis, but its significance was not realized
until recently.
The theory of special functions with its numerous beautiful formulas is very
well suited to an algorithmic approach to mathematics. In the nineteenth century,
it was the ideal of Eisenstein and Kronecker to express and develop mathematical
results by means of formulas. Before them, this attitude was common and best
exemplified in the works of Euler, Jacobi, and sometimes Gauss. In the twenti-
twentieth century, mathematics moved from this approach toward a more abstract and
existential method. In fact, agreeing with Hardy that Ramanujan came 100 years
too late, Littlewood once wrote that "the great day of formulae seem to be over"
(see Littlewood [1986, p. 95]). However, with the advent of computers and the
consequent reemergence of computational mathematics, formulas are now once
again playing a larger role in mathematics. We present this book against this
background, pointing out that beautiful, interesting, and important formulas have
been discovered since Ramanujan's time. These formulas are proving fertile and
fruitful; we suggest that the day of formulas may be experiencing a new dawn.
Finally, we hope that the reader finds as much pleasure studying the formulas in
this book as we have found in explaining them.
We thank the following people for reading and commenting on various chapters
during the writing of the book: Bruce Berndt, David and Gregory Chudnovsky,
George Gasper, Warren Johnson, and Mizan Rahman. Special thanks to Mourad
Ismail for encouragement and many detailed suggestions for the improvement of
this book. We are also grateful to Dee Frana and Diane Reppert for preparing the
manuscript with precision, humor, and patience.

The Gamma and Beta Functions
Euler discovered the gamma function, Г (x), when he extended the domain of the
factorial function. Thus Г(х) is a meromorphic function equal to (x — 1)! when x
is a positive integer. The gamma function has several representations, but the two
most important, found by Euler, represent it as an infinite integral and as a limit of
a finite product. We take the second as the definition.
Instead of viewing the beta function as a function, it is more illuminating to think
of it as a class of integrals - integrals that can be evaluated in terms of gamma
functions. We therefore often refer to beta functions as beta integrals.
In this chapter, we develop some elementary properties of the beta and gamma
functions. We give more than one proof for some results. Often, one proof gener-
generalizes and others do not. We briefly discuss the finite field analogs of the gamma
and beta functions. These are called Gauss and Jacobi sums and are important in
number theory. We show how they can be used to prove Fermat's theorem that a
prime of the form An + 1 is expressible as a sum of two squares. We also treat a
simple multidimensional extension of a beta integral, due to Dirichlet, from which
the volume of an n-dimensional ellipsoid can be deduced.
We present an elementary derivation of Stirling's asymptotic formula for n! but
give a complex analytic proof of Euler's beautiful reflection formula. However, two
real analytic proofs due to Dedekind and Herglotz are included in the exercises. The
reflection formula serves to connect the gamma function with the trigonometric
functions. The gamma function has simple poles at zero and at the negative inte-
integers, whereas esc nx has poles at all the integers. The partial fraction expansions
of the logarithmic derivatives of Г(х) motivate us to consider the Hurwitz and
Riemann zeta functions. The latter function is of fundamental importance in the
theory of distribution of primes. We have included a short discussion of the func-
functional equation satisfied by the Riemann zeta function since it involves the gamma
function.
In this chapter we also present Kummer's proof of his result on the Fourier
expansion of log Г(х). This formula is useful in number theory. The proof given
1

2 1 The Gamma and Beta Functions
uses Dirichlet's integral representations of log Г(х) and its derivative. Thus, we
have included these results of Dirichlet and the related theorems of Gauss.
1.1 The Gamma and Beta Integrals and Functions
The problem of finding a function of a continuous variable x that equals n! when
x = n, an integer, was investigated by Euler in the late 1720s. This problem was
apparently suggested by Daniel Bernoulli and Goldbach. Its solution is contained
in Euler's letter of October 13,1729, to Goldbach. See Fuss [1843, pp. 1-18]. To
arrive at Euler's generalization of the factorial, suppose that x > 0 and n > 0 are
integers. Write
*! = ?±^!, A.1.1)
( + 1)'
where (a)n denotes the shifted factorial defined by
(a)n=a(a + l)---(a+n-l) for n > 0, (aH = 1, A.1.2)
and a is any real or complex number. Rewrite A.1.1) as
!)* n]n* (" + *)*
Since
we conclude that
lim = 1,
n—>oo nx
n\nx
x!= lim . A.1.3)
П-+СО (x + 1)„
Observe that, as long as x is a complex number not equal to a negative integer, the
limit in A.1.3) exists, for
and

1.1 The Gamma and Beta Integrals and Functions
Therefore, the infinite product
... +3>7
converges and the limit A.1.3) exists. (Readers who are unfamiliar with infinite
products should consult Appendix A.) Thus we have a function
k\kx
П(х) = lim —'¦ A.1.4)
к-кх (x + l)k
defined for all complex x ф — 1, —2, —3,... and П(л) = n\.
Definition 1.1.1 For all complex numbers x ^= 0, — 1, —2, ..., the gamma func-
function Г(х) is defined by
к\кх~1
Г(х)=Ит——. A.1.5)
An immediate consequence of Definition 1.1.1 is
Г(х + 1)=хГ(х). A.1.6)
Also,
Г(л + 1) = л! A-1.7)
follows immediately from the above argument or from iteration of A.1.6) and use
of
ГA) = 1. A.1.8)
From A.1.5) it follows that the gamma function has poles at zero and the negative
integers, but 1/ Г(х) is an entire function with zeros at these points. Every entire
function has a product representation; the product representation of 1/Г(х) is
particularly nice.
Theorem 1.1.2
1
A.1.9)
I \ П I I
n=\
where у is Euler 's constant given by
у = lim > --logn . A.1.10)

4 1 The Gamma and Beta Functions
Proof.
1 = Um x(x+ !)¦•• (*+«-!)
X
n
The infinite product in A.1.9) exists because
and the factor e~x/n was introduced to make this possible. The limit in A.1.10)
exists because the other limits exist, or its existence can be shown directly. One
way to do this is to show that the difference between adjacent expressions under
the limit sign decay in a way similar to 1/n2. ¦
One may take A.1.9) as a definition of Г(х) as Weierstrass did, though the
formula had been found earlier by Schlomilch and Newman. See Nielsen [1906,
p. 10].
Over seventy years before Euler, Wallis [ 1656] attempted to compute the integral
/o Vl - x2dx = \ J^(l-xI/2(l+xI/2dx. Since thisintegral gives thearea of
a quarter circle, Wallis's aim was to obtain an expression for jr. The only integral
he could actually evaluate was /0 xp(l — x)qdx, where p and q are integers or
q = 0 and p is rational. He used the value of this integral and some audacious
guesswork to suggest that
1 .. Г 2-4-6---2И 1 I2 /3
xldx = - hm • —pz = ГI -
A [l - 3 -5-- -B/г-1) VJ \2
A.1.11)
Of course, he did not write it as a limit or use the gamma function. Still, this
result may have led Euler to consider the relation between the gamma function
and integrals of the form Jo xp(\ — x)qdx where p and q are not necessarily
integers.
Definition 1.1.3 The beta integral is defined for Rex > 0, Re у > 0 by
B(x,y)= f tx-l{\-t)y~ldt. A.1.12)

1.1 The Gamma and Beta Integrals and Functions 5
One may also speak of the beta function B(x,y), which is obtained from the
integral by analytic continuation.
The integral A.1.12) is symmetric in x and у as may be seen by the change of
variables и — \ — t.
Theorem 1.1.4
Г(х)ГСу)
B^y)=nx-^7y AЛЛЗ)
Remark 1.1.1 The essential idea of the proof given below goes back to Euler
[1730, 1739] and consists of first setting up a functional relation for the beta
function and then iterating the relation. An integral representation for Г(х) is
obtained as a byproduct. The functional equation technique is useful for evaluating
certain integrals and infinite series; we shall see some of its power in subsequent
chapters.
Proof. The functional relation we need is
B(x,y) = X-^B(x,y + \). A.1.14)
First note that for Rex > 0 and Re у > 0,
/¦i
B(x,y + 1) = tx-\\-t){\-t)y-ldt
Jo
= B(x,y)-B(x + \,y). A.1.15)
However, integration by parts gives
\-tx(l - ty] + У- f
= У-В(х + \,у). A.1.16)
X
Combine A.1.15) and A.1.16) to get the functional relation A.1.14). Other proofs
of A.1.14) are given in problems at the end of this chapter. Now iterate A.1.14) to
obtain
m ^ (x + y)(x+y + l) (x+y)nDt , .
B(x, y) = B(x, у + 2) = • • • = —— B(x, у + n).
Я? +1) (y)n
Rewrite this relation as
B(x, y + l)= \-tx(l - ty] + У- f tx(\ -
[ J J
(У)п Jo \n
,y-i
= ——^jl—— / tx~x[\ — ) dt.

6 1 The Gamma and Beta Functions
As n -» oo, the integral tends to /0°° tx~le~'dt. This may be justified by the
Lebesgue dominated convergence theorem. Thus
B(x, y) = ^ / t^e-'dt. A.1.17)
Г(х + у) J
Set у = 1 in A.1.12) and A.1.17) to get
-= f tx~xdt = B(x, 1) = ГA) Г
x Jo Г(х + 1) 7o
Then A.1.6) and A.1.8) imply that /0°° tx~le~'dt = Г(х) for Rex > 0. Now use
this in A.1.17) to prove the theorem for Rex > 0 and Key > 0. The analytic
continuation is immediate from the value of this integral, since the gamma function
can be analytically continued. ¦
Remark 1.1.2 Euler's argument in [1739] for A.1.13) used a recurrence relation
in x rather than in y. This leads to divergent infinite products and an integral that
is zero. He took two such integrals, with у and у = m, divided them, and argued
that the resulting "vanishing" integrals were the same. These canceled each other
when he took the quotient of the two integrals with у and у = m. The result was an
infinite product that converges and gives the correct answer. Euler's extraordinary
intuition guided him to correct results, even when his arguments were as bold as
this one.
Earlier, in 1730, Euler had evaluated A.1.13) by a different method. He expanded
A — t)y~l in a series and integrated term by term. When у = n + 1, he stated the
value of this sum in product form.
An important consequence of the proof is the following corollary:
Corollary 1.1.5 For Rex > 0
Г(х) = / tx-'e-'dt. A.1.18)
Jo
The above integral for Г(х) is sometimes called the Eulerian integral of the
second kind. It is often taken as the definition of Г(х) for Rex > 0. The Eulerian
integral of the first kind is A.1.12). Legendre introduced this notation. Legendre's
Г(х) is preferred over Gauss's function П(х) given by A.1.4), because Theorem
1.1.4 does not have as nice a form in terms of П (x). For another reason, see Section
1.10.
The gamma function has poles at zero and at the negative integers. It is easy
to use the integral representation A.1.18) to explicitly represent the poles and the

1.1 The Gamma and Beta Integrals and Functions
analytic continuation of Г(х):
T(x) = /
Jo J\
/•OO
-ie-dt + l t^e-'dt
—11" Г°°
—^+ / r'-^-'dr. A.1.19)
+ x)n\ Ji
The second function on the right-hand side is an entire function, and the first shows
that the poles are as claimed, with (—1)"/и! being the residue at x = —n, n =
0.1,....
The beta integral has several useful forms that can be obtained by a change of
variables. For example, set t — s/{s + 1) in A.1.12) to obtain the beta integral on
a half line,
Then again, take t = sin2 в to get
Г/2 sin2-1 в cos2? Ode = Г(Х)Г°° . A.1.21)
Jo 2Г(х+у)
Put x = у = 1/2. The result is
_
2ГA) 2'
or
ГA/2) = V5F. A-1.22)
Since this implies [Г(|)]2 = я/4, we have a proof of Wallis's formula A.1.11).
We also have the value of the normal integral
= ^. A.1.23)
-oo JO Jo
Finally, the substitution t = (u — a)/(b — a) in A.1.12) gives
A.1.24)
The special case a — — l,b = Us worth noting as it is often used:

8 1 The Gamma and Beta Functions
A useful representation of the analytically continued beta function is
Г(х)Г(у) (x+y)^ A + ^)
B(x,y)= = TT^—w \ч- A.1.26)
T{x+y) xy l=l(l + ^)(l + z)
This follows immediately from Theorem 1.1.2. Observe that B(x,y) has poles at
x and у equal to zero or negative integers, and it is analytic elsewhere.
As mentioned before, the integral formula for Г(х) is often taken as the defini-
definition of the gamma function. One reason is that the gamma function very frequently
appears in this form. Moreover, the basic properties of the function can be devel-
developed easily from the integral. We have the powerful tools of integration by parts
and change of variables that can be applied to integrals. As an example, we give
another derivation of Theorem 1.1.4. This proof is also important because it can
be applied to obtain the finite field analog of Theorem 1.1.4. In that situation one
works with a finite sum instead of an integral.
Poisson [1823] and independently Jacobi [1834] had the idea of starting with
an appropriate double integral and evaluating it in two different ways. Thus, since
the integrals involved are absolutely convergent,
/CO ЛОО ЛОО ЛОО
/ tx-]sy-]e-{s+l)dsdt = / t^e-'dt / s^e^ds = Г(х)Г(у).
Jo Jo Jo
Apply the change of variables s = uv and t = u(l — v) to the double integral,
and observe that 0 < и < oo and 0 < v < 1 when 0 < s, t < oo. This change
of variables is suggested by first setting s + t = u. Computation of the Jacobian
gives dsdt = ududv and the double integral is transformed to
A comparison of two evaluations of the double integral gives the necessary result.
This is Jacobi's proof. Poisson's proof is similar except that he applies the change
of variables t = r and s = ur to the double integral. In this case the beta integral
obtained is on the interval @, oo) as in A.1.20). See Exercise 1.
To complete this section we show how the limit formula for Г(х) can be derived
from an integral representation of Г(х). We first prove that when n is an integer
> 0 and Rex > 0,
j;
tx-\\-t)ndt = —: -. A.1.27)
x(x + \){x+n)
This is actually a special case of Theorem 1.1.4 but we give a direct proof by
induction, in order to avoid circularity in reasoning. Clearly A.1.27) is true for

1.2 The Euler Reflection Formula
n = 0,and
1 И
tx-\\-t)n+ldt= I tx-\\-t){\-t)ndt
о Jo
(x)n+1 (x + l)n+i
= (и + 1)!
(x)n+2
This proves A.1.27) inductively. Now set t = u/n and let n —> oo. By the Lebesgue
dominated convergence theorem it follows that
tx-le~'dt = lim for Rex > 0.
n^oo (x)n
Thus, if we begin with the integral definition for Г(х) then the above formula can
be used to extend it to other values of x (i.e., those not equal to 0,-1,-2,...).
Remark 1.1.3 It is traditional to call the integral A.1.12) the beta function. A
better terminology might call this Euler's first beta integral and call A.1.20) the
second beta integral. We call the integral in Exercise 13 Cauchy's beta integral.
We shall study other beta integrals in later chapters, but the common form of these
three is jc[l\{t)]p[l2(t)\4dt, where l\(t) and I2(t) are linear functions of t, and
С is an appropriate curve. For Euler's first beta integral, the curve consists of a
line segment connecting the two zeros; for the second beta integral, it is a half line
joining one zero with infinity such that the other zero is not on this line; and for
Cauchy's beta integral, it is a line with zeros on opposite sides. See Whittaker and
Watson [1940, §12.43] for some examples of beta integrals that contain curves of
integration different from those mentioned above. An important one is given in
Exercise 54.
1.2 The Euler Reflection Formula
Among the many beautiful formulas involving the gamma function, the Euler
reflection formula is particularly significant, as it connects the gamma function
with the sine function. In this section, we derive this formula and briefly describe
how product and partial fraction expansions for the trigonometric functions can be
obtained from it. Euler's formula given in Theorem 1.2.1 shows that, in a sense,
the function \/T(x) is half of the sine function.
Theorem 1.2.1 Euler's reflection formula:
^ A.2.1)

10 1 The Gamma and Beta Functions
Remark The proof given here uses contour integration. Since the gamma function
is a real variable function in the sense that many of its important characterizations
occur within that theory, three real variable proofs are outlined in the Exercises.
See Exercises 15, 16, and 26-27.
Since we shall show how some of the theory of trigonometric functions can be
derived from A.2.1), we now state that sinx is here defined by the series
The cosine function is defined similarly. It is easy to show from this definition that
sine and cosine have period 2л and that e1" = — 1. See Rudin[ 1976, pp. 182-184].
Proof. Set у = 1 - x, 0 < x < 1 in A.1.20) to obtain
Г(х)ГA-х)= / 1—dt. A.2.2)
Jo J +'
To compute the integral in A.2.2), consider the integral
-dz.
I
)c 1 -1
where С consists of two circles about the origin of radii R and € respectively,
which are joined along the negative real axis from — R to — e. Move along the
outer circle in the counterclockwise direction, and along the inner circle in the
clockwise direction. By the residue theorem
L
z
dz = -2л i, A.2.3)
с 1-z
when zx has its principal value. Thus
fn iRxeix9
1€е t?
dt+ -^d0+ dt.
^J l'e J \
IR l+t }n l-ee'" Л l+t
Let R —> oo and e —> 0 so that the first and third integrals tend to zero and the
second and fourth combine to give A.2.1) for 0 < x < 1. The full result follows
by analytic continuation. One could also argue as follows: Equality of A.2.1) for
0 < x < 1 implies equality in 0 < Rex < 1 by analyticity; for Rex = 0, x ф 0
by continuity; and then for x shifted by integers using Г(х + 1) — хГ(х) and
sin(x + л) = — sinx. ¦
The next theorem is an immediate consequence of Theorem 1.2.1.
Theorem 1.2.2
™( x2
Sin7TX = ЛХ \\ 1 г

1.2 The Euler Reflection Formula
11
71 COt7TX
=l+Ё (— + -1-) = lim E -4'
Sin7TX
n=l
= lim
7isecnx= lim
: + i-x
A.2.6)
A.2.7)
A.2.8)
4У ^
Sin27TX ^^ (Х+ИJ
/I——OO
Proof. Formula A.2.4) follows from the product formula
r<*>
»=.
proved in the previous section and from Theorem 1.2.1 in the form Г(х)ГA— х) =
—хГ(х)Г(—x) = л I sin7rx.
Formula A.2.5) is the logarithmic derivative of A.2.4), and A.2.6) follows from
A.2.5) since esc x = cot |—cotx. The two formulasA.2.7)andA.2.8)are merely
variations of A.2.5) and A.2.6). Formula A.2.9) is the derivative of A.2.5). ¦
It is worth noting that A.2.6) follows directly from A.2.1) without the product
formula. We have
/00 fX— 1 />1 ^Л—1 /0 ^X — 1
df = / df + / -—
l+t Jo l+t Jl 1+t
dt
\n+\tn+\
dt
where
k=—n
f\tn"
JO
t"-x+l
)dt
n+x + 1 ' n —x +2
Thus A.2.6) has been derived from A.2.1).
Before going back to the study of the gamma function we note an important
consequence of A.2.5).
Mathernati:?; V :¦

12 1 The Gamma and Beta Functions
Definition 1.2.3 The Bernoulli numbers Bn are defined by the power series ex-
expansion
n °° 2k
4Ч1
^4Ч
It is easy to check that -?y + | is an even function. The first few Bernoulli
numbers are Bx = -1/2, B2 = 1/6, B4 = -1/30, B6 = 1/42.
Theorem 1.2.4 For each positive integer к,
- 2
f^2k Bk)\
Proof. By A.2.10)
eix+e~ix 2ix ^ ... 2lkxlk
xcotx = ix- =ix + —. = 1 - > (-\)k+yB2k
and A.2.5) gives the expansion
x cot x =
n=\ n=\ k=\
Лк
Now equate the coefficients of x in the two series for x cot л to complete the
proof. ¦
Eisenstein [ 1847] showed that a theory of trigonometric functions could be sys-
systematically developed from the partial fractions expansion of cot x, taking A.2.5)
as a starting point. According to Weil [1976, p. 6] this method provides the sim-
simplest proofs of a series of important results on trigonometric functions orginally
due to Euler. Eisenstein's actual aim was to provide a theory of elliptic functions
along similar lines. A very accessible account of this work and its relation to mod-
modern number theory is contained in Weil's book. Weil refers to Пт,,^^ Yl"-n ak as
Eisenstein summation.
Theorem 1.2.2 shows that series of the form
where к is an integer, are related to trigonometric functions. As we shall see next,
the "half series"
,
0

1.2 The Euler Reflection Formula 13
bears a similar relationship to the gamma function. In fact, one may start the study
of the gamma function with these half series.
Theorem 1.2.5
Г'A) = -У, A.2.12)
A.2.13)
Г(х) ^
^fL A-2.14)
froo/. Take the logarithmic derivative of the product for 1 / Г (x). This gives
Г(х) ' ' f^\x+k-l k,
The case x = 1 gives A.2.12). The other two formulas follow immediately. ¦
Corollary 1.2.6 Log Г(х) is a convex function of x for x > 0.
Proof. The right side of A.2.14) is obviously positive. ¦
Remark The functional equation A.1.6) and logarithmic convexity can be used
to derive the basic results about the gamma function. See Section 1.9.
We denote Г" (x) / Г (x) by xfr (x). This is sometimes called the digamma function.
Gauss proved that is(x) can be evaluated by elementary functions when x is a
rational number. This result is contained in the next theorem.
Theorem 1.2.7
++ +т
x x+1 x+n—1
A.2.15)
p\ n np 4^' 2лпр / nn
)Y cot log^+ 2> coslog 2 sin
¦ f p\ n np 4^' 2лпр /
t[-)=-Y- ^cot— -log^+ 2> cos log 2 sin
A.2.16)
where 0 < p < q; ^' means that when q is even the term with index n = q/2 is
divided by 2. Here lq/2\ denotes the greatest integer in q/2.
Proof. The first formula is the logarithmic derivative of
Г(х +и) = (x+n- l)(x+n -2)---

14 1 The Gamma and Beta Functions
We derive Gauss's formula A.2.16) by an argument of Jensen [1915-1916] using
roots of unity. Begin with Simpson's dissection [1759]:
k-l
n=0 j=0
where w = el7Z'lk is a primitive fcth root of unity. This is a consequence of
Y^ wjm = 0, m ф O(modfc). Now by A.2.13)
00 / 1
= У) (—-r -
j^ p + nq
= lim У" ( ?—\tp+nq =: lim s(t)
by Abel's continuity theorem for power series. From the series — log(l — t) =
YlT=i tn/n>an^ Simpson's dissection with со = e2nilq, we get
q-\
s(t) = -tp~q log(l - tq) + ^оо~пр log(l - wnt)
\-tq VX
= -tp~glog (tp~g - l)log(l -t) + 22o~nplog(l -oft).
Let? ->¦ 1~ to get
q-\
if(p/q) = -y - log? + ]Tornp iog(i - a,").
n=\
Replace p by q — p and add the two expressions to obtain
The left side is real, so it is equal to the real part of the right side. Thus
(p\ (q — p\ %-^ 2irnp ( 2ттп\
- }+ir[ ?—!- =-2y-21og^+> cos -log 2-2cos .
q) \ q ; ^ q \ q )
A.2.17)
But
f(x)-\lr(l -x) = —logr(x)r(l -x) = -
dx

1.3 The Hurwitz and Riemann Zeta Functions 15
So
i,(p/q) - YK1 - (p/q)) = -л cotnp/q. A.2.18)
Add this identity to A.2.17) to get
(p\ л лр 1 t-^ 2лnp, / 2лп\
- = -у cot — - logo + - > cos log 2 - 2cos .
4J 2 4 2flt 4 \ 4 )
A.2.19)
But cos2jr(^ — n)/q = cos2jrn/^, so the sum can be cut in half, going from 1
to lq/2], where |_*J denotes the greatest integer in x. Thus
р\ л лр L^J, 2л tip , ( 2лп\
— = — у cot log о + > cos log 2 — 2 cos
ч) 2 ч 7^ ч \ q )
л лр 4~~*' 2л tip ( лп\
= — у cot log о + 2 у cos log 2 sin — . щ
2 ч t=i ч \ ч J
1.3 The Hurwitz and Riemann Zeta Functions
The half series
00
for*>0, A.3.1)
called the Hurwitz zeta function, is of great interest. We have seen its connection
with the gamma function for positive integer values of s in the previous section.
Here we view the series essentially as a function of* and give a very brief discussion
of how the gamma function comes into the picture.
The case x = 1 is called the Riemann zeta function and is denoted by ?(.?). It
plays a very important role in the theory of the distribution of primes. The series
converges for Re s > 1 and defines an analytic function in that region. It has a
continuation to the whole complex plane with a simple pole at 5 — 1. The analytic
continuation of ?(s) up to Re s > 0 is not difficult to obtain. Write the series for
as a Stieltjes integral involving [jcJ . Thus for Re s > 1
c Ixjdx
n=\
= 1 + —-+s I ^—^dx.
s -
s+l
The last integral converges absolutely for Re s > 0 and we have the required

16 1 The Gamma and Beta Functions
continuation. The pole ats = l has residue 1 and, moreover,
lim
f 1 ) f°° UJ
im r{s) = 1 + / V-
1- / i , / L*J -x
= lim 1 + / z—dx
/1 \
= lim V log« = y.
A.3.2'
The best way to obtain analytic continuation to the rest of the plane is from the
functional relation for the zeta function. We state the result here, since the gamma
function is also involved. There are several different proofs of this result and we
give a nice one due to Hardy [1922], as well as some others, in the exercises. In
Chapter 10 we give yet another proof.
Theorem 1.3.1 For all complex s,
(°)/2) - s). A.3.3)
If s < 0, then 1 — s > 1 and the right side provides the value of f (s). This
relation was demonstrated by Euler for integer values of s as well as for 5 = 1/2 and
i = 3/2. He had proofs for integer values of 5, using Abel means. An interesting
historical discussion is contained in Hardy [1949, pp. 23-26]. The importance of
%(s) as a function of a complex variable in studying the distribution of primes was
first recognized by Riemann [1859].
The last section contained the result
The following corollary is then easy to prove.
Corollary 1.3.2
<A -2k) = —-B2k, ?@) = — and f(-2/t)=O /or Jk = 1, 2, 3,....
A.3.4)
Corollary 1.3.3
С '@) = —1ог2л\ A.3.5)
2
Proo/. From the functional equation and the fact that

1.3 The Hurwitz and Riemann Zeta Functions 17
we have
-?A - s) = *-s+l'z ^^ (s - l)f (s). A.3.6)
sv 2F(C-5)/2)v
Now A.3.2) implies that (s - \)S(s) = 1 + y(s - 1) + A(s - IJ -\ .So take
the logarithmic derivative of A.3.6) to get
_,,__, y+2A(s-\)
= l°g,r--^-,j--,
Set 5 = 1 and use Gauss's result in Theorem 1.2.7 with p = 1 and q = 2. This
proves the corollary. ¦
There is a generalization of the last corollary to the Hurwitz zeta function S(x,s).
A functional equation for this function exists, which would define it for all complex
s, but we need only the continuation up to some point to the left of Re s = 0. This
can be done by using the function l;(s). Start with the identity
S(x, s) - «E) - sxS(s + 1)) = x~s + 2^n~'[(l +x/n)-s - A - sx/n)}.
n=\
The sum on the right converges for Re s > — 1, and because Sis) is defined for all
s, we have the continuation of Six, s) to Re 5 > — 1.
The following theorem is due to Lerch.
Theorem 1.3.4
=,„81« „3.7,
V2
s=0 V ?Л
Proof. The derivative of the equation Six + 1, s) = Six, s) — x's with respect
to 5 at s = 0 gives
/dS(x+l,s)\ fdSix,s)\
- =logA\ A.3.8)
For Re s > 1,
«=o
so
A39)

18 1 The Gamma and Beta Functions
Now A.3.8) and A.3.9) together with A.2.14) of Theorem 1.2.5 imply that
To determine that the constant С = — \ log 2тт, set x = 1 and use Corollary 1.3.3
This completes the proof of Lerch's theorem. ¦
For a reference to Lerch's paper and also for a slightly different proof of Theoren
1.3.4, see Weil [1976, p. 60].
1.4 Stirling's Asymptotic Formula
De Moivre [1730] found that и! behaves like Cnn+]/2e~n for large n, where С is
some constant. Stirling [1730] determined С to be -J2n; de Moivre then used г
result of Stirling to give a proof of this claim. See Tweddle [1988, pp. 9-19]. This
formula is extremely useful and it is very likely that the reader has seen applications
of it. In this section we give an asymptotic formula for Г(х) for Rex large, wher
Imx is fixed. First note that log T{x + n + 1) = J21=\ l°g(& + дг) + log Г(* + 1)
We then employ the idea that an integral often gives the dominant part of the surr
of a series so that if the integral is subtracted from the series the resulting quantity
is of a lower order of magnitude than the original series. (We have already used
this idea in Equation A.3.2) of the preceding section.) In Appendix D we prove the
Euler-Maclaurin summation formula, a very precise form of this idea when the
function being integrated is smooth. Two fuller accounts of the Euler-Maclaurir
summation formula are given by Hardy [1949, pp. 318-348] and by Olver [1974.
pp. 279-289].
Theorem 1.4.1
Г(х) ~ *j2nxx~'42e-x as Re* -> oo.
Proof. Denote the right side of the equation
n-\
log T(jc + n) = ]T log(* + jc) + log Г(х + 1)
*=]
by с„, so that
cn+i -с„ =log(x+n).
By the analogy between the derivative and the finite difference we consider cn to
be approximately the integral of log(jc + n) and set
cn = (n+ x) log(« + x) - (n + x) + dn.

1.4 Stirling's Asymptotic Formula 19
Substitute this in the previous equation to obtain
log(* +n) = (n+\+x)log(n + 1 + jc) - (n + x)log(n + x) + dn+x -dn-\.
Thus
dn+i - dn = 1 - (и + х + 1) к
\ « -г л /
1 1 1
1 1
2(n+jt) 6(n + xJ
Proceeding as before, take
dn=en-- log(n
and substitute in the previous equation to get
e», ~ «. = '- log(l
{п+х)г)'
Now
en-e0 = l>+i ~ek) = ^ |- щк + xJ + O[ -^—^ ) \; A.4.1)
therefore, limn^.oo(en — eo) = A"] (jc) exists. Set
1
c"-"^12(n+jt)^V(«+*)V'
where ^T(jc) = ^Ti(jc) + e0. The term (и + jc) comes from completing the sum
in A.4.1) to infinity and approximating the added sum by an integral. So we can
write
1
cn = (n + x) log(n +x)-(n + x) - - log(n + *)
+ logC(*)
+o( ),
I2(n+x) \(n+xJJ
where ^T(jc) = log C(jc). This implies that
\ .+O[ ^ )}.
Щп+х) \(n+xJJ]
A.4.2)

20 1 The Gamma and Beta Functions
We claim C(x) is independent of x. By the definition of the gamma function
.. Г(л+л:) Г(лО (х)п T(x) T(y)
lim ny = lim ny x = • = 1. A.4.3)
п-+осГ(п + у) Г (у) п^оо (у)„ Г (У) Г(х)
Now, from A.4.2) and A.4.3) we can conclude that
, ,. -хГ(п + х) C(x) ( x\n _x C(x)
1 = hm n x^—— = ——- hm 1 + - } e =
Г (и) C@)»-»»\ n) C@)
Thus C(x) is a constant and
T(jc) ~ Cxx~l/2e~x as Re* -> oo.
To find C, use Wallis's formula:
22«(»!J 1
= lim —
n-*oo Bи)! Vй
2
= lim
"^°
С
This gives С = л/2тт and proves the theorem. Observe that the proof gives the
first term of an error estimate. ¦
We next state a more general result and deduce some interesting consequences.
A proof is given in Appendix D. For this we need a definition. The Bernoulli
polynomials Bn(x) are defined by
7
e' — 1
n=0
The Bernoulli numbers are given by Bn @) = Bn for n > 1.
Theorem 1.4.2 For a complex number x not equal to zero or a negative real
number,
„,, ,,.4.5)
2m 7 (x + 02ffl
Г/ге va/ие о/log jc is //ге branch with logx rea/ w/геи jc is rea/ and positive.
The expansion of log F(jc) in A.4.5) is an asymptotic series since the integral is
easily seen to be O(x~2m+l) for |argx\ < л - 8, S > 0.

1.4 Stirling's Asymptotic Formula 21
From this theorem the following corollary is immediately obtained.
Corollary 1.4.3 For S > 0 and |argx\ < ж - S,
Г(х) ~ -/2nxx~y2e-x as \x\ -+ oo.
Corollary 1.4.4 When x = a + ib,a\ < a < a2 and \b\ —> oo, then
\Y(a + ib)\ = V2^\b\a-^2e^w/2[l + O(l/\b\)],
where the constant implied by О depends only on a\ and a2.
Proof. Take \b\ > \,a > 0. It is easy to check that the Bernoulli polynomial
\\B2 - B2(t)\ < \\
B2 - B2(t) =t -t2. Thus \\B2 - B2(t)\ < \\t(\ -01 < | forO < t < l.So
A.4.5) with m = 1 is
log Г (a + ib) = (a + ib--\ log(a + ib) - (a + ib) + - login + R(x),
and
dt 1 __, \b\ L , n
Now
Re ( a + ib - - ) log(a + ib)\ = (a - -) log(a2 + b2I'2 - barctan-.
V 2 / V 2 / a
Also,
log(a2 +b2)l<2 =l-\ogb2 + ilog^l + °^\ =\og\b\ + o(~
Moreover,
This gives
b a \t,
arctan - + arctan - = < 2Л
a b [ -f, if b < 0.
b , Г , л- а \
-b arctan - = —b\± \- O\ —
a [2b \fc2
Putting all this together gives
log \Г(а + ib)\ = (a - ^ log \b\ - ||ft| + 11оё2я + О l^~ ).

22
1 The Gamma and Beta Functions
The condition a > 0 is removed by a finite number of uses of the functional
equation A.1.6) and the corollary follows. Observe that the proof only uses a =
o{\b\) rather than a bounded. ¦
Corollary 1.4.5 For |argjc| < л - 8, 8 > 0,
Corollary 1.4.4 shows that Г (a + ib) decays exponentially in the imaginary
direction. This can be anticipated from the reflection formula, for
or
or
Similarly,
and
\2
г(
[it
>)
\+ib)
2я
7TD _|_ я— Л"О
-VS.—
—ibsmnbi
e "'*
as
/><7
as
2я
±O0
V27T\b\'l/2e~nlbl/2 as fe -> ±c».
Since Г(д:) increases rapidly on the positive real axis and decreases rapidly in the
imaginary direction, there should be curves going to infinity on which a normalized
version of P(jc) has a nondegenerate limit. Indeed, there are. See Exercise 18.
1.5 Gauss's Multiplication Formula for Г(отдг)
The factorization
together with the definition of the gamma function leads immediately to Legendre's
duplication formula contained in the next theorem.
Theorem 1.5.1
ГBа)Г( l-] =
A.5.1)

1.5 Gauss's Multiplication Formula for Г(тх) 23
This proof suggests that one should consider the more general case: the factor-
factorization of (a)mn, where m is a positive integer. This gives Guass's formula.
Theorem 1.5.2
Г(та)Bл)(т-])/2 = тта/2Г(а)Г (а+-)---г(а+ ^—!Л. A.5.2)
V m) V m )
Proof. The same argument almost gives A.5.2). What it gives is A.5.2) but with
»-ii /IV (m-\\
Bл)—т-1 replacedby Г - ...Г =: P. A.5.3)
\ m I \ m I
To show that A.5.3) is true, we show that
P2 =
2 Bя)"-]
m
By the reflection formula
sin^'
m
So it is enough to prove
, л 2л (т — Х)л
2 sin — sin — • • • sin = m.
mm m
Start with the factorization
Г
x — 1
Let x ->¦ 1 to obtain
m-I
m = JJA
k=\
, л 2л (т — \)л
= 2м sin — sin sin —.
mm m
This proves A.5.3). ¦
Remark 1.5.1 A different proof of A.5.1) or A.5.2) that uses the asymptotic
formula for Г(х) and the elementary property Г(х + 1) = хГ(х) is also possible.
In fact it is easily verifed that
S
ГA/2)ГBд:)
satisfies the relation g(x + 1) = g(x). Stirling's formula implies that g(x) ~ 1 as
jc —>¦ oo so that ЦгПп-юо g(x + n) = 1 when n is an integer. Since g(x+n) = g(x)
we can conclude that g(x) = 1. A similar proof may be given for Gauss's formula.
This is left to the reader.

24 1 The Gamma and Beta Functions
An elegant proof of the multiplication formula using the integral definition of
the gamma function is due to Liouville [1855]. We reproduce it here.
The product of the gamma functions on the right side of A.5.2) is
xm vtf+((m— l)/m)~l//v
Xm UXm
rOQ rOQ r
/ e-'^r'Aci / e^xa2+(l/m)-ldx2 ¦¦•
Jo Jo Jo
CO /-CO /.CO
/ ¦¦•/ ^
о Jo Jo
Introduce a change of variables:
_ z _ _
X\ — , X2 — X2, ... i Xm — Xm.
X2 ' ' ' xm
The Jacobian is easily seen to be
mzm'1
х2хъ ¦ ¦ ¦ xm
and the integral can be written
roc /-oo
exp | — ( X2 + хъ + ¦ ¦ ¦ + х„
о Jo
a+(l/m)—1 a-\-((m~l)/m)—l z j j j
n ил™.
Set t = X2 + хт, + ¦ ¦ ¦ + xm + zm/(x2X3 ¦ ¦ ¦ xm), and rewrite the integral as
/•CO /"CO /"CO
m / ¦•¦/ e-tzma-lx{2l"n)-lxf"n)~X---x«m-l)lm)-'dzdx2---dxm.
Jo Jo Jo
A.5.4)
First compute
POO POO m~l
I = • ¦ ¦ / e~* TT Xj_li~ dx2dxT, ¦ ¦ -dxm.
Jo Jo J=l
Clearly,
dl m_j f™ f -t T[ (j/m)-\dx2 ¦ ¦ -dxm
dz Jo Jo jJi J+i x2---xm
Now introduce a change of variables,
x2 =zm/(xiX3---xm),x3 =x3,...,xm =xm,

1.5 Gauss's Multiplication Formula for Г(тх) 25
and
h = x3 + x4 H h xm + xi + zm/(x3 ¦ ¦ ¦ xmxi).
The Jacobian is
and j- is given by
dl m_, Г00
dz ~ mZ Jo
m-\
•п^и)-
= -m ¦¦¦
Jo
= -ml.
Therefore,
x2x3
Г "'¦ (
Jo e V
\dx\dxj, ¦ ¦ dxm
Г/хг
/•со m—l
Jo )}2]+l
¦Xm-l
m \(l/m)-l
)
C1X3 ¦¦¦Xm)
n)—\ ((m—l)/m)—1 ,
X1 tZX
/ = Ce~mz.
To find C, set z = 0 in the integral for / as well as in the above equation and
equate to get
m
By A.5.3), С = Bn){m-l)l2m-ll2 and / = {2n)(m-^l2m-xl2e'mz. Substitution
in A.5.4) gives
РОС
Г(а)Г(а + 1/m) ¦¦¦Г(а + (т- l)/m) = т1/2Bя)(т-1)/2 / e-mzzma~ldz,
Jo
which is Gauss's formula.
Remark 1.5.2 We pointed out earlier that 1 / Г (x) is a half of sin n x. In this sense
the duplication formula is the analog of the double angle formula
( 1\
sin 2nx = 2 sin nx sin n I x + - I.
This is usually written as sin2^x = 2 sin л* cos л* and so is thought of as a
special case of the addition for sin(x + y). The gamma function does not have an
addition formula.

26 1 The Gamma and Beta Functions
1.6 Integral Representations for Log T(x) and ip(x)
In A.2.13), we obtained
(^)
from the product for 1/ Г(x). We start this section by rederiving it from the beta
integral. Note that, for x > 1,
-(x - 1) /
Jo
-t)dt = T{x-l)(l-€)X+k
'Ьо (*+*)(*+*)
by term-by-term integration, which is valid because of uniform convergence in
[0, 1-е]. Now let € -> 0. By Abel's continuity theorem for power series,
r-2 V^ (X ~ О
t log(l — t)dt = y^ •
We can introduce log(l — t) in the beta integral /0' tx~2(l — t)ydt by taking the
derivative with respect to y. In that case,
Г(*+у) '
or
/•I
(x-l) tx~2(\ - t)y log(l - t)dt
Jo
_ Г(х)Г'(у + \)Г(х + у)~ Г(х)Г(у + 1)Г'(* + у)
~ Г(х + уJ '
The case у — 0 gives the necessary result. The differentiation is justified since
the integrands involved are continuous. Some care should also be taken of the fact
that the integrals are improper. The details are easy and left to the reader. The next
theorem gives the integral representations of f(x) due to Dirichlet and Gauss.
Theorem 1.6.1 For Re x > 0,
Jo z\ A+.
(ii) f(x) = J (:y-y:—^)rfz (Gauss).
(i) fix) = -[e~z- )dz (Dirichlet),
J \ A + zYJ

1.6 Integral Representations for Log Г(х) and \jr(x) 27
Proof, (i) Evaluate the integral /0°° /* e~ndtdz in two different ways by changing
the order of integration to get the formula
e-z _ g-sz
- - A.6.1)
Similarly, the double integral
POO pO
Jo Jo
p —-Ui-t-^./
: dsdz
z
when first integrated with respect to z yields (by A.6.1))
d f°° _s .
logs ds = —— / e s ds = Г (х).
о dx Jo
If we integrate the double integral with respect to s we get
ri/ i \
Г(х) / -(e-t-—-—)dz.
Jo z\ (l+z)xJ
Equate the last two expressions to get Dirichlet's formula.
(ii) Gauss's formula is obtained from Dirichlet's by a change of variables:
= lim / — dz- / -dt
+ \JS z J 1 *
—dz+ (—-t^ 7)dt
since
1
g
dz
z
-dz = log ^ ^ ^0 as S -> 0.
г log(l + S)
/log(l+«) Z ' lOg(l + S)
This proves (ii). ¦
The integrated form of the last theorem is given in the next result.
Theorem 1.6.2 For Re x > 0,
(l) --o- V-/ I ! V" -/- jQg^

28 1 The Gamma and Beta Functions
and
(ii) log Г(*) = / (*-!)«"'-
-L
00 < _t
1-е-'
Proof. The integrals in Theorem 1.6.1 are uniformly convergent for Re x > S >
0, so we can integrate from 1 to x under the sign of integration. The integrals in
Theorem 1.6.2 are the corresponding integrated forms.
A change of variables и = e~' in (ii) gives
\ du
-x+l — • d-6-2)
1 — и ) log и
There are two other integrals for log Г (х) due to Binet that are of interest. These
are given in the next theorem. A proof of one of them is sketched and the other is
left as an exercise. See Exercise 43. ¦
Theorem 1.6.3 ForRex>0,
_ / 1 \ 1 f°° /1 1 1 \ e~tx
and
1\ , 1 „ , „ f°° arctan(r/x)
(ii) \ogT(x)lx
Proof. Gauss's formula in Theorem 1.6.1 together with Equation A.6.1) give
1A(x + l) ^logr(x + l) +logx f (+
dx 2x Jo \2 t e' - \
Integrate from 1 to x, changing the order of integration to get
Use log Г(х + 1) = log Г(х) + logx to rewrite the above formula as
'tx
where
Stirling's formula applied above gives / = 1 — A/2) log 2л.
The second Binet formula can be used to derive the asymptotic expansion for
log Г(х) contained in Corollary 1.4.5.

1.7 Kummer's Fourier Expansion of Log Г(х) 29
Expand \/(e2nt — 1) by the geometric series and integrate term by term to see
that
(-1) it- (L6-4)
The last equality comes from Theorem 1.2.4. Now,
gives, after integration,
t W3 , ] t5 , (-1)" t2n~l , (-1)" f z2ndz
x 5x* 5 xJ In - 1 xLn~l xLn~
Substitute this in Binet's formula (ii) and use A.6.4) to arrive at
ii _ x ^_ _ j.— o_ , \
) log, , + iog&r + g 2 .B. ;)х2;_1
For |argx| < | — €, € > 0, it can be seen that |^~i I < esc ze for all г > 0.
This implies that the last term involving the integral is 0(,l+l). So we have
the asymptotic series but only for |argx| < | - € instead of |argx| < л — e.
Whittaker and Watson [1940, §13.6] show how to extend the range of validity.
It is also possible to derive an asymptotic formula for log Г(х) from Binet's first
formula. See Wang and Guo [1989, §3.12]. For references to the works of Gauss,
Dirichlet, Binet, and others, see Whittaker and Watson [1940, pp. 235-259]. ¦
1.7 Kummer's Fourier Expansion of Log T(x)
Kummer [1847] discovered the following theorem:
Theorem 1.7.1 For 0 < x < 1
logloBsin) + ^]
logBsin^x) + (y+log2^)(l2x) + ^]
\1tz 2 2 л
where у is Euler's constant.
Proof. Start with the identity
„Aitix (mix
-log(l - e2™) = e2™ + ^- + — + ..., 0<

30 1 The Gamma and Beta Functions
The real and imaginary parts are
Ecos 2knx
7 A-7.1)
*=i k
and
л
?^ d.7.2)
Since log Г(х) is differentiable in 0 < x < 1, it has a Fourier expansion
logГ(х) = C0 + 2^2ckcos2knx + 2^ Dk si
<t=l k=\
where
Ck= I XogV(x)cos2knxdx and Dk = \ogT(x)sin2knxdx. A.7.3)
We use Kummer's method to compute Ck and Ць The Ck are easy to find. Take
the logarithm of Euler's reflection formula A.2.1):
logr(x)+logr(l -x) = log 2л -IogBsin7rx)
= Iog2^ + cos 2лx H— cos4^x + • • • ¦
The Fourier series of log Г(х) gives
logr(x)+logT(l -x) = 2C0 + 4C{ cos2лx +4C2cos4^x -\ .
Equating the last two relations gives
Co = - log 2л and Ck = — for к > 1.
2 4k
Now use integral A.6.2) for log Г(х) in A.7.3) so that
1 fl-ux'] \sin2knxdudx
A
io Jo \ l-« /
fl -n /'
Уо ' Jo
log и
But
2кл'
and
sin 2knxdx =
The first two integrals are easy to solve and the third is the imaginary part of
ex(logu+2kn,)dx _ 1 M ~
l

1.7 Kummer's Fourier Expansion of Log Г(х) 31
Therefore,
f1 f -2kn 1 \ du
°k~ Jo \u((loguJ +4k2n2) + 2/br,
or, with и = e~2knt,
Take к = 1 and we have
1 -2яг
Moreover, x = 1 in Dirichlet's formula (Theorem 1.6.1) gives
у 1 f°°
~2n ~ 2n Jo
where у is Euler's constant. Therefore,
00 e-t - e~27" 1
+2^
By A.6.1), the first integral is log In and a change of variables from t to 1 /t shows
that the second integral is 0. Thus
To find Dk, observe that
„-2л7 _ e-2knt
1 "' In
where the integral is once again evaluated by A.6.1). Thus
1 f°° e~2wt — e 1
kDk-Dl = — dt = — log*,
In Jo t In
= —(y+log2*jr), *= 1,2,3,....
2А:я
The Fourier expansion is then
logr(*) = - log In + V C0S f X + - (y + log In) V^
2/fc ' /'T '^ 2*
1 7
log A:
si
H>
Apply A.7.1) and A.7.2) to get the result. ¦
Kummer's expansion for log (T(x)/V2n) and Theorem 1.3.4 have applications
in number theory. Usually they give different ways of deriving the same result.
This suggests that the Hurwitz zeta function itself has a Fourier expansion from

32 1 The Gamma and Beta Functions
which Kummer's result can be obtained. Such a result exists and is simply the
functional equation for the Hurwitz function:
2r(l-s)| 1 *^cos2m7rx 1 ^sin2m^x|
Ц^<sin^> tcosTrs) ¦ }. A.7.4)
BлI-* 2 ^ m1-* 2 ^-j m1"
V. m=l m=l /
The functional equation for the Riemann zeta function is a particular case of this
when x = 1. See Exercises 24 and 25 for a proof of A.7.4) and another derivation
of Kummer's formula.
1.8 Integrals of Dirichlet and Volumes of Ellipsoids
Dirichlet found a multidimensional extension of the beta integral which is use-
useful in computing volumes. We follow Liouville's exposition of Dirichlet's work.
Liouville's [1839] presentation was inspired by the double integral evaluation of
the beta function by Jacobi and Poisson.
Theorem 1.8.1 If"V is a region definedby xt > 0, / = 1,2,.. .,n,andJ2xi 5 1.
then for Re a, > 0,
f
J
xa.-i
Proof. The proof is by induction. The formula is clearly true for n = 1. Assume
it is true for и = к. Then for &(k+ 1)-dimensional V
J-.J x«>-lx?
/1 /-1-х, f
/ •¦¦/
Jo Jo
¦ dxk+l
1 /-1-х, fl-x,~x2 xt
/ / ll
-* 1 ""' X/c — 1
«1-1 at-l
Xl ' ' ' Xk
0 ./0
¦ A — xi — • • • — Xk)at+'dxkdxk-i ¦ ¦ ¦ dx\
Now set Xk = A — xi — ¦ ¦ • — Xk-i)t to get
ч'Г-
о Jo
A — xi — ¦ ¦ ¦ — xk-i)ak+ak*[tak~ A — t)ak+ldtdxk-i ¦ ¦ ¦ dxi
Г(ак)Г(ак+] +1)
ак+1Г(ак + ak+l
II
Jo Jo

1.8 Integrals of Dirichlet and Volumes of Ellipsoids 33
Compare this with the integral to which the change of variables was applied and
use induction to get
( f Г(«))Г(at + ak+l)
ак+1Г(ак + ak+l + 1) ГA + Y^i «/)
This reduces to the expression in the theorem. ¦
Corollary 1.8.2 If V is the region enclosed by x; > 0 and J2 (•*; M Y' < 1.
2
dx
Proof. Apply the change of variables, y, = (x,¦ /щ)р', i = ],... ,n. Then
Эх, 1 x,
Эу; Pi yi
and the Jacobian is
1 X1X2 ¦ ¦ • Xn
P\P2---Pn У\Уг---Уп
The integral becomes
PlP2'-Pn
¦ dyn,
where V is defined by yt > 0 and J2 У' 5 1 ¦ The corollary now follows from the
theorem. ¦
Corollary 1.8.3 ThevolumeenclosedbyJ2(xi/ai)Pi < 1, x, >0is Пм^
/и particular the volume of the n-dimensional ellipsoid ^(x,/a,J < 1 is
n/2 -an
ГО+и/2)
Proof. For the first part of the corollary take a, = 1. For the particular case take
(|) = \
Pi — 2 and use the fact that Г(|) = \yfn.
Corollary 1.8.4 IfV is given by x, > 0 andJ2(^~)Pi < ^ ги Dirichlet's integral,
then its value is
Liouville also gave the following extension of Dirichlet's result, which can be
proven in the same way.

34 1 The Gamma and Beta Functions
Theorem 1.8.5 If V consists of xt >0, t\ <J2(xi/ai)Pl < t2 and f is a continuous
function on (?ь h), then
f ¦¦ f
+¦¦¦ + (xnlanY"\dxx ---dx»
Па"'Г (aj/pi)/pi
A related integral is given next.
Theorem 1.8.6 IfV is the set xt > 0, YTi=\ xi = 1. then
This is a surface integral rather than a volume integral, but it can be evaluated
directly by induction or from Corollary 1.8.2. It is also a special case of Theorem
1.8.5 when f(u) is taken to be the delta function at и = 1. This function is not
continuous, but it can be approximated by continuous functions.
1.9 The Bohr-MoIIerup Theorem
The problem posed by Euler was to find a continuous function of x > 0 that equaled
n! at x = n, an integer. Clearly, the gamma function is not the unique solution to
this problem. The condition of convexity (defined below) is not enough, but the
fact that the gamma function occurs so frequently gives some indication that it
must be unique in some sense. The correct conditions for uniqueness were found
by Bohr and Mollerup [1922]. In fact, the notion of logarithmic convexity was
extracted from their work by Artin [1964] (the original German edition appeared
in 1931) whose treatment we follow here.
Definition 1.9.1 A real valued function f on (a, b) is convex if
f(Xx + A - k)y) < kf{x) + A - k)f(y)
for x, у € (a, b) andO < X < 1.
Definition 1.9.2 A positivefunction f on (a, b) is logarithmically convex if log /
is convex on (a, b).
It is easy to verify that if / is convex in (a, b) and a<x<y<z<b, then
/(У) ~ fix) < f(z) - f(x) < f(z) - f(y)
у — x ~~ z — x ~ z - у
With these definitions we can state the Bohr-Mollerup theorem:

1.9 The Bohr-Mollerup Theorem 35
Theorem 1.9.3 If f is a positive function on x > 0 and (i) /A) = 1, (ii)
f(x + 1) = x/(x), and (Hi) f is logarithmically convex, then f(x) = Г(х) for
x > 0.
Proof. Suppose и is a positive integer and 0 < x < 1. By conditions (i) and (ii)
it is sufficient to prove the theorem for such x. Consider the intervals [n, n + 1],
[n + 1, n + 1 + x], and [n + 1, n + 2]. Apply A.9.1) to see that the difference
quotient of log /(x) on these intervals is increasing. Thus
lo /(" + !) , l lQ /(n + l+x) /(и+ 2)
Simplify this by conditions (i) and (ii) to get
\(x+n)(x+n-l)---xf(x)]
xlogn <log <xlog(n + 1).
L «! J
Rearrange the inequalities as follows:
Therefore,
n\nx
fix) = lim = Г(х),
"->oo x(x + 1) • ¦ • (X +П)
and the theorem is proved. ¦
This theorem can be made the basis for the development of the theory of the
gamma and beta functions. As examples, we show how to derive the formulas
/•OO
Г(х) = / e~'tx~ldt, x > 0,
= / e-'t
Jo
and
tx~\\ - ty~ldt = Г(Х)Г(:У), x > 0 and у > 0. A.9.2)
/о Г(х + у)
We require Holder's inequality, a proof of which is sketched in Exercise 6. We
state the inequality here for the reader's convenience. If / and g are measurable
nonnegative functions on (a, b), so that the integrals on the right in A.9.3) are
finite, and p and q positive real numbers such that \/p + \/q = 1, then
fgdx<^J fPdxj i^J g<dx) . A.9.3)
It is clear that we need to check only condition (iii) for log Г(х). This condition
can be written as
Г(ах + Ру)<Г(х)аГ(у)/}, a>0,p>0 and a + p = I. A.9.4)

36 1 The Gamma and Beta Functions
Now observe that
and apply Holder's inequality with a = \/p and fi = \/q to get A.9.4).
To prove A.9.2) consider the function
Г(х + у)В(х,у)
fix) = =-— •
г Су)
Once again we require the functional relation A.1.14) for B(x, y). This is needed
to prove that / (x +1) = xf (x). It is evident that / A) = 1 and we need only check
the convexity of log/(x). The proof again uses Holder's inequality in exactly the
same way as for the gamma function.
We state another uniqueness theorem, the proof of which is left to the reader.
Theorem 1.9.4 If f{x) is defined for x > 0 and satisfies (i) /A) = 1,
(ii) f{x + 1) = xf(x), and (шДт^со f(x + n)/[nx/(«)] = 1, then f(x) =
Fix).
For other uniqueness theorems the reader may consult Artin [1964] or
Anastassiadis [1964]. See Exercises 26-30 at the end of the chapter. Finally, we
note that Ahern and Rudin [1996] have shown that log |Г(дг + iy)\ is a convex
function of x in Rex > 1/2. See Exercise 55.
1.10 Gauss and Jacobi Sums
The integral representation of the gamma function is
Г(х)
Here dt/t should be regarded as the invariant measure on the multiplicative group
@, oo), since
d(ct) dt
ct t
To find the finite field analog one should, therefore, look at the integrand e~cttx.
The functions e~ct and tx can be viewed as solutions of certain functional relations.
This point of view suggests the following analogs.
Theorem 1.10.1 Suppose f is a homomorphism from the additive group of real
numbers R to the multiplicative group of nonzero complex numbers C*, that is,

1.10 Gauss and Jacobi Sums 37
and
f(x + y) = fix)fiy). A.10.1)
/// is differentiable with /'@) = с ф 0, then fix) = ecx.
Remark 1.10.1 We have assumed that /(x) фО for any x but, in fact, the relation
g(x + y) — gix)g(y), where g : R —>• C, implies that if g is zero at one point it
vanishes everywhere.
Proof. First observe that /@ + 0) = /@J by A.10.1). So /@) = 1, since /@)
cannot be 0. Now, by the definition of the derivative,
,,, , .. fix + 0 - fjx) fix)fjt) - fjx)
/ (x) = lim = lim
,, ,,. /@-/@)
= fix) lim
= сf{x).
So fix) = e". Ш
Remark 1.10.2 In the above theorem it is enough to assume that / is continuous
or just integrable. To see this, choose а у e R such that /oy fit)dt / 0. Then
fix) H fit)dt = H fix + t)dt = Jxx+y fit)dt. So
This equation implies that if / is integrable, then it must be continuous and hence
differentiable.
Corollary 1.10.2 Suppose g is a homomorphism from the multiplicative group
of positive reals R+ to C*, that is,
gixy) = gix)giy). A.10.2)
Then gix) — xc for some с
Proof. Consider the map / = g о exp : R ->• C*, where exp(x) = ex. Then /
satisfies A.10.1) and giex) = ecx. This implies the result. ¦
A finite field has p" elements, where p is prime and и is a positive integer. For
simplicity we take n = 1, so the field is isomorphic to Ъ(р), the integers modulo
p. The analog of / in A.10.1) is a homomorphism
TJf.Zip) -+ C*.
Since "Lip) is a cyclic group of order p generated by 1 we need only specify i^ (!)•
Also, VK1K = VKO) = 1 and we can choose any of the p\h roots of unity as the

38 1 The Gamma and Beta Functions
value of фA). We therefore have p different homomorphisms
ij,j(x) = ei*iixlp, j = O,l,...,p-l. A.10.3)
These are called the additive characters of the field. In a similar way the multi-
multiplicative characters are the p — 1 characters defined by the homomorphisms from
Щр)* to C*. Here Щр)* = Z(p) - {0}. Since Щр)* is a cyclic group of order
p — 1, we have an isomorphism Щр)* = Щр — 1). The p — \ characters on
Щр)* can be defined by means of this isomorphism and A.10.3). We denote a
multiplicative character by either x or r\, unless otherwise stated.
It is now clear how to define the "gamma" function for a finite field.
Definition 1.10.3 For an additive character т/л,- and multiplicative character Xt
we define the Gauss sums gj (xd, j = 0, 1,..., p — 1 by the formula
x=o
where we extend the domain of Xi by setting x,- @) = 0.
It is sufficient to consider g(x) := g\ (x)> for when j ф 0,
8j(x) = Y, X(x)fj(x)
= X(j)g(x)- A.10.5)
This formula corresponds to /0°° e~jxxs~ldx = F(s)/js, where j is a nonzero
complex number with positive real part. When j = 0 in A.10.4) the sum is
Ylx X(x), which can be shown to be zero when х{х)ф 1 for at least one value of x.
Theorem 1.10.4 For a character x,
Remark 1.10.3 The identity character is the one that takes the value 1 at each
point in Щр)*.
Proof. The result is obvious for x = id. If x ф id, there is а у е Щр)* such
thatx(j) ф l.Then

1.10 Gauss and Jacobi Sums 39
which implies the theorem. There is a dual to A.10.6) given by the following
theorem: ¦
Theorem 1.10.5 For the sum over all characters we have
Proof. It is sufficient to observe that if x ф 1, then there is a character x such
that x (x) Ф 1 • The theorem may now be proved as before. ¦
We now define the analog of the beta function.
Definition 1.10.6 For two multiplicative characters x and r\ the Jacobi sum is
defined by
J(X,V)= Yl Х(хШУ)- A-10.8)
x+y=\
The following theorem gives some elementary properties of the Jacobi sum. We
denote the trivial or identity character by e. The reader should notice that the last
result is the analog of the formula B(x, у) = Г(х)Г(у)/[Г(х + у)].
Theorem 1.10.7 For nontrivial characters x and i), the following properties
hold:
J(e,X)=0. A.10.9)
J(e,e) = p-2. A.10.10)
,X~1) = -X(-D- A.10.11)
then J(x,r1) = 8ix)8(f. A.10.12)
Remark 1.10.4 From the definition of characters it is clear that the product of
two characters is itself a character and so the set of characters forms a group.
The additive characters form a cyclic group of order p and the multiplicative
characters a cyclic group of order p — 1. Also, x(x) = x(x~') = l/x(*) and
since |x(*)l = 1 it follows that x~lix) =
Proof. The first part of the theorem is a restatement of Theorem 1.10.3 and the
second part is obvious. To prove A.10.11), begin with the definition
¦/(X, X) =
Now note that as x runs through 2,..., p—l, then x(l - x) runs through 1,...,
p — 2. The value у — p— 1 = — l(mod;>) is not assumed because x = уA+у)~1 ¦
Therefore,
J(x,x~l)=
уф-\

40
1 The Gamma and Beta Functions
by Theorem 1.10.4. This proves the third part. The proof of the fourth part is
very similar to Poisson's or Jacobi's proofs of the analogous formula for the beta
function. Here one multiplies two Gauss sums and by a change of variables arrives
at a product of a Jacobi sum and a Gauss sum. Thus, for x Ц ф е,
8(X)g(ri)
Х(*ЖH+ Y, X(x)r](t-x)e
-x)e2*it/p
x+y=0
The first sum is ^ X(x)i(-x) — rj(—l) YjX Xi(x) = 0 since xi Ф id. The
second sum with x — st is
-s)e2nitlp =
= g(X4)J(X,4).
This proves the fourth part of the theorem. ¦
We were able to evaluate Г (s) in a nice form for positive integer values and half-
integer values of s. Evaluations of special cases of Gauss sums are also possible and
important, but in any case the magnitude of the Gauss sum can always be found.
Theorem 1.10.8 For nontrivial multiplicative and additive characters x and \j/,
Proof. By A.10.5) it is enough to prove that |gi(x)l2 — l#(x)l2 = P-
\8(X)\2 =
Set x = ty. Then
\8ix)\2 =
r.v^O
1
tjtO or 1

1.10 Gauss and Jacobi Sums 41
The first sum is p — 1 and the inner sum in the second term is — 1. Thus
\g(x)\2 = p-i-J2x{t) = P-l + l = P>
and the result is proved. ¦
Corollary 1.10.9 If x, Ц and xi] are nontrivial characters, then
\J(X,ri)\=VP- A.10.13)
Proof. This follows from Theorems 1.10.7 and 1.10.8. ¦
As an interesting consequence we have:
Corollary 1.10.10 If p = An + 1 is a prime, then there exist integers a and b
such that p = a2 + b2.
Proof. The group 2>{p)* is of order p — 1 = An, which is also isomorphic to
the group of multiplicative characters on Z(p)*. Since the latter group is cyclic
there exists a character x of order 4 that takes the value ±1, ±i. It follows that
J(X, x) = a + bi for integers a and b. Since x2 Ф id, apply Corollary 1.10.9 to
obtain the desired result. ¦
Corollary 1.10.10 is a theorem of Fermat, though Euler was the first to publish a
proof. See Weil [1983, pp. 66-69]. Later we shall prove a more refined result that
gives the number of representations of a positive integer as a sum of two squares.
This will come from a formula that involves yet another analog of the beta integral.
We have seen that characters can be defined for cyclic groups. Since any abelian
group is a direct product of cyclic groups, it is not difficult to find all the characters
of an abelian group and their structure. The following observation may be sufficient
here:
If xi is a character of a abelian group G\, and xi of G2, then we can define a
character / : Gx x G2 -> C* by x(x, y) = Xi(x)Xi(y)-
We thus obtain n additive characters of Z(n) and ф(п) multiplicative characters
of Z(«)*. The Gauss and Jacobi sums for these more general characters can be
defined in the same way as before. Gauss [1808] found one derivation of the law
of quadratic reciprocity by evaluating the Gauss sum arising from the quadratic
character. (A character x ф id is a quadratic character when x2 = id.) Details of
this connection are in Exercise 37 at the end of the chapter. One problem that arises
here, and which Gauss dealt with, is evaluating the sum G = Ylx=o e27lix2/N. As
in Theorem 1.10.8 one can show that G2 = ±N depending on whether N = 1
D) or 3 D). The problem is to determine the appropriate square root for obtaining
G. According to Gauss, it took him four years to settle this question. Dirichlet's
evaluation of Ylx=o е2т"х ^N by means of Fourier series is given in Exercise 32.

42 1 The Gamma and Beta Functions
Jacobi and Eisenstein also considered the more general Jacobi sum
J(Xi,X2,...,)U)= J2 Xi(ti)X2(t2)---Xdti). A.10.14)
This is the analog of the general beta integral in Theorem 1.8.6. Eisenstein's result,
corresponding to the formula in Theorem 1.8.6, follows.
Theorem 1.10.11 If xi, X2, ¦ ¦ ¦ > Xi are nontrivial characters and X1X2 ¦ • ¦ Xi ™
nontrivial, then
j, ч g(Xi)g(X2) ¦ ¦ ¦ g(Xe) /11Л1СЧ
J(Xi,X2,---,Xt) = : : • A.10.15)
g(XiX2---Xe)
The proof of this is similar to that of Theorem 1.10.7, and the reader should fill in
the details.
In Section 1.8 the volume of «-dimensional objects of the form ayx\[ + aix5^ +
h a^xf < b was determined by means of the gamma function. In the same way,
for finite fields, the number of points satisfying а\х\* +a2x2 H 1-a**** = b can
be found in terms of Gauss sums. Gauss himself first found the number of points
on such (but simpler) hypersurfaces and used this to evaluate some specific Gauss
sums. Weil [1949] observed that it is easier to reverse the process and obtain the
number of points in terms of Gauss sums. For an account of this the reader should
see Weil [1974]. It may be mentioned that Weil's famous conjectures concerning
the zeta function of algebraic varieties over finite fields are contained in his 1949
paper. It also contains the references to Gauss's works. One may also consult
Ireland and Rosen [1991] for more on Jacobi and Gauss sums and for references
to the papers of Jacobi and Eisenstein.
The form of the Gauss sums also suggests that they are connected with Fourier
transforms. Let T denote the vector space of all complex valued functions on
Z(N), the integers modulo N. Let F be the Fourier transform on T defined by
1 N-\
(Ff)(n) = -= V f{x)^l"'N. A.10.16)
It can be shown that the trace of this Fourier transform with respect to the basis
{So, Si,... ,S/v}, where
'. x Ф У,
, x=y,
is the quadratic Gauss sum Ylx=o e2l"x ^' ¦ Schur [1921] gave another evaluation
of this sum from this fact. The details are given in Exercise 47. One first proves
that the fourth power of F is the identity so that the eigenvalues are ± 1, ±i and
the essential problem is to find the multiplicity of these eigenvalues.

1.11 A Probabilistic Evaluation of the Beta Function 43
Discrete or finite Fourier analysis was not applied extensively before 1965 be-
because of the difficulty of numerical computation. This changed when Cooley and
Tukey [1965] introduced an algorithm they called the Fast Fourier Transform
(FFT) to reduce the computation by several orders of magnitude. The reader may
wish to consult the paper of Auslander and Tolimieri [1979] for an introduction
to FFT, which emphasizes the connection with group theory. Some of the earlier
instances of an FFT algorithm are mentioned here. Computational aspects are also
interesting. See de Boor [1980] and Van Loan [1992, §1.3].
1.11 A Probabilistic Evaluation of the Beta Function
When a and fi are positive integers,
I
It seems that it should be possible to arrive at this result by a combinatorial argu-
argument. But working with only a finite number of objects could not give an integral.
Here is a combinatorial-cum-probabilistic argument that evaluates the integral.
Choose points at random from the unit interval [0, 1]. Assume that the probability
that a point lies in a subinterval (a, b) is b — a. Fix an integer n and let P(xk < t)
denote the probability that, of n points chosen at random, exactly к of them have
values less than t. The probability density function for P(xk < t) is
,.\ v P{Xk < t + Af) ~ P{Xk < r)
p(t) = hm
hm .
д/->о At
Now
P(xk <t + At)- P(xk < t)
— the probability that one point lies in (t, t + At),
к — 1 points less than t and n — к points greater than t + At,
+ the probability that two points lie in (t, t + At),
к — 2 points less than t and n — к points greater than t + At,
Since there are n points, the number of ways that one point is in (t, t + At), к — 1
points are less than t, and n — k, are greater than t + At is
n-l\fn-k\ (n-\
-i L-J="U-i

44 1 The Gamma and Beta Functions
The probability of each such event is Attk~l A —t — At)n~k, and since the events
are mutally exclusive, we get
P(xk <t + At)- P(xk < t)
n~1 )At tk~l(\ -t- At)n~k
к — I I
+ ¦••
= "(?"! V~'d - t - At)n~kAt + O((AtJ).
\ к - 1 /
Therefore,
Since
s:
we obtain
(n-k+l)
/'
Jo
n\ Г(« + 1)
We made use of probability theory here to indicate its relationship with the beta
function. Though we do not use it elsewhere, probability theory can be used to
derive formulas involving some extensions of the beta function.
1.12 Thep-adic Gamma Function
In number theory there are completions of the rationals other than the reals that
are of great importance. These are the p-adic completions of the rationals. There
is an analog of the gamma function defined on the p-adic numbers that is useful.
The following is a very brief account of the p-adic gamma function. The interested
reader should consult the references given later.
Suppose a is an integer and p a prime. Define ord^a to be the highest power of
p that divides a. Let Q be the set of rational numbers. For x = alb e Q, where a
and b are integers, define ord^x = ord^a — oxdpb. The p-adic norm | 1^ on Q is
defined by
l/pordp\ x ф 0,
0, x = 0.

1.12 The /?-adic Gamma Function 45
Thus in the p-adic norm, p" gets small as n gets large. In contrast, for negative
values ofn,p" becomes big. So it is reasonable to write numbers in powers of p.
An integer would have an expansion of the form
a0 + a\p + агР2 Л +anp",
where a,¦ e {0, 1, 2,..., p — 1}. For rational numbers negative powers of p will
also be involved. The p-adic norm is non-Archimedean, that is,
and so the triangle inequality holds. This gives a metric on Q.
We can then obtain a completion of Q with this metric in the same way as we
get the real numbers by taking the ordinary metric on Q. This involves taking
the Cauchy sequence of rationals. The p-adic completion is denoted by Qp. The
p-adic numbers can be represented by the series
b-m Ь-m+l b_i 2 ,
1 r -\ 1 h b0 + hx p + Ъгр Л •
pm pm-1 р
The subset of Qp which contains all numbers with nonnegative powers of p forms
a ring denoted by Zp. This is the ring of p-adic integers. The positive integers
Z+ form a dense subset of Zp. This makes sense because a member of Zp can be
represented as an infinite series
ao + a\P + агр1 + ¦ ¦ ¦, a, e {0, 1,... p - 1},
and the partial sums are integers that converge to the p-adic number. So, if there
is a function / defined on the positive integers and the values of / at two integers
that are p-adically close are close to each other, then / has a unique continuous
extension to Zp.
Define a function / on the positive integers n by the formula
рЧк
It is not difficult to show that /(« + pmt) = f{n)modpm, where n,m, and
I are positive integers. Now n + p"? and n are p-adically close to each other
and the values of / at these points are also p-adically close. Consequently, /
has an extension to Zp. This extension gives the p-adic gamma function due to
Morita [1975]. The p-adic gamma function is defined by ГДх) = —f(x — 1).
This function also has a functional relation and other useful properties. There is
a formula of Gross and Koblitz [1979] that gives the Gauss sum as a product of
values of the p-adic gamma function.

46 1 The Gamma and Beta Functions
A good treatment of the p-adic numbers and functions is given in Koblitz
[1977]. An account of the p-adic gamma function and the Gross-Koblitz formula
is available in Lang [ 1980], including a reference to a paper by "Boyarsky" [ 1980].
In fact, p-adic extensions of the beta function, and more generally, the Mellin
transform, are also available.
Exercises
1. Use the change of variables s = ut to show that
/•00 pOG
Jo Jo
is Г(х + y)B(x, y). (Poisson)
2. Let / = /0°° e-*2dx. Observe that /2 = /0°° /0°° e-^'+>'2)dxdy. Evaluate this
double integral by converting to polar coordinates and show that / = ф:/2.
3. A proof of Wallis's formula is sketched below:
(a) Show that
/•00
Jo
dx ж
/o x2+a
(b) Take the derivative of both sides n times with respect to the parameter a
to conclude that
roc dx 1 • 3 • 5 • • • In - 1 л 1
f
Jo
o (x2+a)"+! 2-4-6---2n 2
(c) Set x = y/s/n, a = 1, let n —>- oo, and use Exercise 2 to obtain Wallis's
formula.
4. Evaluate j\{i — t2)x~ldt in two different ways to prove the duplication
formula given in Theorem 151To get another proof evaluate
formula given in Theorem 1.5.1. To get another proof evaluate
/-Л-/2
/ sin2* 2<Ы6>
Jo
in two ways.
5. Suppose that / is twice differentiable. Show that /" > 0 is equivalent to
/(ax + fiy) < a/(x) + fif{y) for a and /9 nonnegative and a + /9 = 1.
6. Convexity can be used to prove some important inequalities, for example,
Holder's inequality:
b 11//p Г гь
\ 0
/ fga
Ja
where / and g are integrable functions and | + - = 1. We sketch a proof

Exercises 47
(a) Note that ex is a convex function; use this and the result of Exercise 5 to
show that if и and v are nonegative real numbers then
uv < 1 .
p q
Equality holds if and only if up = vq.
(b) Deduce Holder's inequality from (a).
It might be appropriate to call this the Rogers-Holder inequality since Rogers
[1888] had the result before Holder [1889]. Other important results of
L. J. Rogers are discussed later in the book.
7. Here is another proof of the functional relation
)
Write
Jo
B(x,y+l)= I tx+y~M — 1 dt
and perform an integration by parts to show that
B(x,y + l) = -?—B{x,
x + y
8. Show that
B{x,y) f00 tx~ldt
¦y Jo (c
+ t)x+y
Take the derivative with respect to с and derive the functional equation
Give a similar argument using
t*-\c-tydt.
Jo
9. Write Gauss's formula as
. =
Bл-)(п-1)/2„A/2)-х •
Show that the right side satisfies all the conditions of the Bohr-Mollerup
theorem. This proves the formula.
10. Give a proof of Gauss's formula by using the definition of Г(х).
11. Prove Gauss's formula by the method given in the remark after Theorem 1.5.2.
12. It is clear from Г(х + 1) = хГ(х) that //+1 logT{t)dt = xlogx -x + C.
Show that С = j log27T. Stirling's formula will work, but there is a more
elegant argument using Gauss's multiplication formula first.

48 1 The Gamma and Beta Functions
13. There is another beta integral due to Cauchy defined by
dt п21-х-УТ{х+у-\)
(a) To prove this, show:
(i) Integration by parts gives C(x,y + 1) — *,C(x + l,y).
(ii) Write
. . (-l-it
/-00
= 2C(x,y+l)-C(x-l,
This together with (i) gives
C(x, y) = —-C(x, v + 1).
x + у - 1
(iii) Iteration gives
22" (*)„(>%
- lJn
= /
— 00
¦00
-oo A+?2)"A+/0л"A -ity'
Set f -> 11yfn in the second integral and let n —> со.
(b) The substitution t = tan в leads to an important integral. Find it.
14. Use the method for obtaining Stirling's formula to show that
11 1 ,- 1 / 1
~r + ~F + • ¦ ¦ + —F = 1^n + C °
where
Sum
n n *
cn = у (Ck — Ck-i) with cn = > —= —
and use some algebra to change cn — cn_i to an expression that goes to zero
like и/2 to show that
See Ramanujan [1927, papers 9 and 13] for further results of this type.

Exercises 49
15. Here is an outline of a real variable proof of A.2.1). Let
N .
Ж v-^ 1
g(x) = lim > .
tan7TX N^oc*-^1 n + X
— N
oo
(a) g'(x) = -7T2/sm27TX + Y,l/(n+xJ-
—00
(b) g'(x) is continuous forO < x < 1 if g'@) = g'(l) = 0.
(c) g'(x/2) + g'((x + l)/2)=4g'(x).
(d) Let M = maxo<x<i \g'(x)\. Then M < M/2 so M = 0.
00
(e) g(x/2) - g((x + l)/2) = 27r/(sinjrx) - 2^ (-!)"/(« + x).
—00
(f) g(x + l) = g(x).
(g) g(x) = constant.
(h) / tx-1/(i+t)dt = Y/(-iy/(n+x)+Y/(-iT/(n + i-x)
J° n=0 n=0
so A.2.1) holds. This proof, due to Herglotz, was published by Caratheodory
[1954, pp. 269-270]. Bochner's [1979] review of the collected works of
Herglotz also includes this proof.
16. The following is Dedekind's [1853] proof of Г(х)ГA -x)=n/ sin7rx. Set
(a) Show that
/•OO fX — l
Jo st + 1 ~
and
(b) Deduce that
00 fX — 1
Л = (/>(x)^.
-1 (.~*"\ Z100 fx
s - 1 Jo (*?
(c) Use the second formula in (a) to get
,00 j / ,00 fX-| X
= J0 VTTkL 7T~sdt)ds'y

50 1 The Gamma and Beta Functions
then change the order of integration to obtain
[ф(х)]2 = / f-dt.
Jo t - 1
(d) Deduce
y />oo *v—1
/
/y />oo *v—1 f — v
№(x)fdx= / —dt.
-y Jo I ~ 1
(e) Integrate (b) with respect to s over @, oo) and use (d) to derive
/x f00 tx~l log?
m)?dt = 2 / f dt = 2ф'(х).
J l +
(f) Show that ф(х) = ф(\-х) implies ф'ф = 0 and
= 2 /
l-л: Jl/2
(g) Deduce that
ф(х) f
Jl/2
(h) Show that ф satisfies the differential equation фф" — (ф1J = ф4.
(i) Solve the differential equation with initial condition ф(^) = ж and
ф'(\) = 0 to get ф(х) = ж свсях.
17. Show that
Г *-xQ-*y-1* = W) ,Rex>0,Re>,>0,fl>0,
Jo [at + b(l -t)]x+y ахЪУТ(х+у)
18. Show that
hm
sin ax 1 h ,cscGrb/2)
—— dx = -жаь~1—
b 2
19. Prove that for a > 0,
X *b "" 2"" Г(Ь)
and
f°° cos ax 1 , _, 8ес(яЬ/2)
/ ;—dx = -жа , С
Jo xb 2 Гф)
20. For к > 0, x > 0 and —я/2 < a < я/2, prove that
f tx-le-ktcosaCQS(kt ^na)dt = X-x
Jo

Exercises 51
and
~v~ isinax.
f
Jo
/o
21. Prove that n-sl2Y(s/2)l;(s) = л--A-5)/2Г(A - s)/2)f(l - s) as follows:
(a) Observe that
, ¦ = (— l)mW4 for mn < x < (m + lOr, m = 0, 1,
^ 2n + l
(b) Multiply the equation by xs~l@ < s < 1) and integrate over @, oo).
Show that the left side is V(s) sin(sjr/2)(l - 2-s-x)t;(s + 1) and that
the right represents an analytic function for Res < 1 and is equal to
2A - 2i+1)f A - s) for Res < 0.
(c) Deduce the functional equation for the zeta function. (Hardy)
22. Let С be a contour that starts at infinity on the negative real axis, encircles
the origin once in the positive direction, and returns to negative infinity. Prove
that
-L = -L / л-*.
i(s) 2ni Jc
This formula holds for all complex s.
(a) Note that the integral represents an analytic function of s.
(b) С may be taken to be a line from -oo to —8, then a circle of radius 8 in
the positive direction, and finally a line from —8 to —oo. Show that
/ e'rsdt = 2i sinns / e~"u~sdu + I,
Jc Js
where / is the integral on the circle \t\ =8.
This representation of the gamma function is due to Hankel; see Whittaker
and Watson [1940, p. 244].
23. Prove that
e"ijrsr(l -s) Г ts-ye~xl
Г ts-ye
/
Jc 1 -
Tdt,
t(x,s) Ц
2ni
where С starts at infinity on the positive real axis, encircles the origin once
in the positive direction, excluding the points ±2nni, n > 1 an integer, and
returns to positive infinity.
Hint: First prove that
and then apply the ideas of the previous exercise. Note also that t(x,s) is now

52 1 The Gamma and Beta Functions
defined as a meromorphic function by the contour integral with a simple pole
at s — 1.
24. Prove the functional equation
2Y{\—s) Г
I. m=\ m=\ )
Hint. Let С denote the line along the positive real axis from oo to B« + 1)ж,
then a square with corners Bn + 1)jt(±1 ± /), and then the line from (In + 1)
to oo. Show that
f ts-xe~xl Г f-xe'"
/ "i Tdt = / i Tdt
Jc 1 - e-< JCn 1 - e~<
— the sum of the residues at ± 2mni, m = 1,..., n,
where С is the curve in the previous exercise.
Note that the sum of the residues at ±2mni is
У'" sinBm7rx + ns/2).
Now let n -*¦ oo and show that /c ->•().
25. Show that the functional equation for f (x, s) easily implies
(a) the functional equation for t, (s),
(b) Kummer's Fourier expansion for log Т{х)/^/Ът.
The next five problems are taken from Artin [1964].
26. For 0 < x < oo, let ф(х) be positive and continuously twice differentiable
satisfying (а) ф(х + 1) = ф(х), (b) ффф(^) = dф{x), where d is a
constant. Prove that ф is a constant.
Hint:Letg{x) = ^ log ф(х). Observe that g(x + l) = g(x) and i(g(f ) +
g(^r))=g(x).
27. Show that ф(х) = r(x)r(l-x)sin7rx satisfies the conditions of the previous
problem. Deduce Euler's reflection formula.
28. Prove that a twice continuously differentiable function / that is positive in
0 < x < oo and satisfies (a) f(x +1) = xf(x) and (b) 22x~1f(x)f(x + \) =
л/л fBx) is identical to Г(х).
29. It is enough to assume that / is continuously differentiable in the previous
problem. This is implied by the following: If g is continuously differentiable,
g(x + 1) = g{x), and g(f) + g(*±!) = g{x), then g = 0.
Hint: Observe that
k=0
-I
The left side tends to /J g'{x)dx = g{\) - g@) = 0 as n -> oo.

Exercises 53
30. Prove that the example g{x) = X^li ^ sinBn7rx) shows that just continuity
is insufficient in the previous problem.
31. Suppose / and g are differentiable functions such that f(x +y) = f(x)f(y) —
g (x) g(y) and g(x+y) = f(x)g(y)+g(x)f{y). Prove that/(x) = eaxcosbx
and g(x) = eax sin их, unless f(x) — g{x) = 0.
32. Prove that Yfx^ elni%1lN = Ц^-VN, where i = v^l.
(a) Set f(t) = J2^e2nKx+lJ/N,0 < t < 1. Note that /@) = /(l) and
extend /(?) as a periodic function to the whole real line.
(b) Note that fit) = ?-oc ane2*inl, where an = ft f{t)e-2*imdt.
Conclude that /@) = Et"o *W/" = E-oo ««•
(c) Show that an = e-™N*l* J%f} e
(d) Show that
-nN/2 rN-nN/2
-E/
(e) Use Exercise 19 to evaulate the integral. Another way is to take N = 1 in
(d). (Dirichlet)
33. If p is an odd prime, then there is exactly one character /2 that maps Z(p)*
onto {±1}. Recall that Z(p)* is the integers modulo p without 0. Prove that
X2(a) = 1 if and only if x2 —a mod p is solvable, that is, a is a square in
Z(p)*. Usually one writes x2 (a) = (-), which is called the Legendre symbol.
34. Prove that if a is a positive integer prime to p, then ap~x/2 = (-) (mod p).
Here p is an odd prime. (Use the fact that Ъ{р)* is a cyclic group.)
35. For pan odd prime, use the previous problem to prove that (—) = (—1)^p~1^2
and i2-) = (-l)^2-"/8. (Use 2p/2 = {еп11А + e~ni/4)P = (epni/4 + e-pni/4)
(modp). Consider the two cases p = ±1 (mod 8) and p = ±3 (mod 8)
separately.)
36. Prove the law of quadratic reciprocity: For odd primes p and q, (?)(?) =
(a) For S = Y?x=\ (f )e2nix/p, show that S2 = {-~)p- (The proof is similar
to that of Theorem 1.10.8.)
(b) Use (a) and Exercise 34 to prove that Sq~l = (-l)^1?1 (?) (mod^).
(c) Show that Sq = T,xZl(pe2nillx/p = (pS (modq).
(d) Deduce the reciprocity theorem from (b) and (c). (Gauss)
37. For integers a and N with N > 0, define G(a, N) = ^^o e2niax2/N.
(a) For p prime, show that G(l, p) = ??=i(f У2*1'*7*"-
(b) For p prime show that G(a, p) = (-)G(l, p).

54 1 The Gamma and Beta Functions
(c) Prove that G(q, p)G(p, q) = G(l, pq) when p and q are odd primes.
(d) Now use the result of Exercise 32 to deduce the reciprocity law. (Gauss)
For a discussion of Exercises 32-37 and for references, see Scharlau and
Opolka [1985, Chapters 6 and 8].
38. Prove Theorems 1.8.5 and 1.8.6.
39. Prove Theorem 1.9.4.
40. Weierstrass's approximation theorem: Suppose / is a continuous function
on a closed and bounded interval, which we can choose to be [0, 1] without
any loss of generality. The following exercise shows that / can be uniformly
approximated by polynomials on [0, 1].
(a) Show that it is enough to prove the result for /@) = /A) = 0. Now
extend / continuously to the whole real line by taking / = 0 on x < 0
andx > 1.
(b) Observe that
is a polynomial such that
Qn(t)dt=l.
Show that Pn(x) = J_{ f{x + t)Qn{t)dt is a polynomial in x for x e
[0, 1].
(c) Use Stirling's formula to show that for S > 0and<5 < \t\ < 1, Qn{t) -*¦ 0
uniformly as n —>¦ oo.
(d) Note that forO < x < 1, Pn(x) - f(x) = J^lfix + t) - f(x)]Qn(t)dt.
To show that Р„(х) ->¦ f(x) uniformly on [0, 1], break up the integral
into three parts, J~{ + J_s + Js, and use (c).
41. Prove Plana's formula (see Whittaker and Watson [1940, p. 145] for references
to Plana): For positive integers m and n
k=m
ф(т)+ф(п)
/ фШх - i /
Jm JO
ф(п + iy) - ф(т + ty) - ф(п - ty) + ф(т - ly)
X z : flV,
e2ny _l •y'
where ф(х + iy) is a bounded analytic function in m < x < n.
Hint:
(a) Consider the integral /c ф{г)/(е~2пп — \)dz where С is a suitable in-
indented rectangle with vertices ifc,ifc+l,ifc+l + L/, and к + Li. Then let
L —>¦ oo.
(b) Now replace i with —i in the contour С and repeat the process in (a).
(c) Add the results in (a) and (b) and sum over k.

Exercises 55
42. (i) In Plana's formula let m = 0, n -> oo, and suppose that ф(п) -> О,
ф(п ± iy) -> 0, to get
v>(*) = W- Г\,._... Г*<оО-*(-оО
?? 2 Jo Л
(ii) Deduce Hermite's formula (for reference, see Whittaker and Watson,
[1940, p. 269])
x~s xl~s f°° (x2 + t2)~s/2 sin(s arctanf/x)
(iii) Conclude that ?(x, 2) = —5Ч h , / , —^ ,
2x2 x Jo (x2 + t2J(e2711 - I)
43. (a) For f{x) = T'ix)/ Г(х), note that fix) = ?(x, 2).
(b) Deduce that
ltdt
fix) =lnx-
(Use part (iii) of the previous exercise.)
(c) Deduce Binet's second formula
( x\ !
1пГ(х)^^ I x — — 1 in x — x -\—
\ 2) 2
where x is complex and Re x > 0.
(d) Use Hermite's formula in the previous problem to obtain Lerch's formula
A.3.7) for ij-Jix,s))s=0.
44. Prove the following properties of Bernoulli polynomials:
(a) Bqix + \) - Bq{x) = qxq~l.
(b)
1
n=M
(c) Bn(jt) = 2j ,_ )BkXn-
k=0
4=0
45. Prove that
(a) B2q-i(x -[x]) = 2(

56 1 The Gamma and Beta Functions
and
(b) Deduce
CBq) = (—X)q~ , q > 1,
B#)! 2
and
46. Prove that
B2n=G2n- p
(p-\)\2n F
where dn is some integer and p is a prime such that p — 1 divides 2и.
(Clausen-von Staudt)
Hint: Define ]>X0 %xn = ?X0 If-*" (mod k)if к divides an - bn for all
и > 0. Show that
^ Z- + ¦ ¦ ¦ ) (mod4).
(b) For prime p,
(c) For composite m > 4
(ez-l)m-1=O (modm).
«0
4,^ + ^ + .
ez — 1 2 3 4
Deduce the result on Bernoulli numbers. (See Polya and Szego [1972, Vol. II,
p. 339].
47. Let C(Z(n)), where Z(n) is the integers modulo n, be the set of all complex
functions on Z(n), where n is an odd positive integer. Define F: C(Z(n)) —>
C(Z(/i)) by
1 ""'
(F/)(*) = -= V /(t)^"" for x e Z(n).
(a) Show that Trace F = -^ YXX e2nik2/n.
Hint: Use the functions Sx,x e Z(n), where 5^(y) = 0, x ф у, and
^ (x) = 1, as a basis for Z(n).

Exercises 57
(b) Prove that (F2f)(x) = /(-*). Conclude that F4 = id and hence that
±1, ±i are the eigenvalues of F. Let m\, rm, m^, m4 be the multiplicities
of 1, i, — l.and —i respectively. Thus mi + гп2 + тз +П14 = 1.
(c) Show that Trace F2 — 1 and conclude that m\ — ni2 + ni-$ — тц = 1.
(d) Show that 1;^ ЕЙ e27Iik2/n\2 = 1. Use (a) to get (m, - miJ + (w3 -
(e) Prove that
f l " «з)гA-")/2, И = 1D),
F - ;(«l-«4)+(»i-«3)-(»+D/2 _
"U-m^10'2, n = 3D),
= det f-Le2nixy/n
and also
0<дс,у<я-1
where Л" is a positive number.
(f) Show that nti = a + 1, and»?2 = m^ = m.4 — a when n — Aa + 1 and
7щ = ni2 = mi = a and m.4 = a — 1 when и = Aa — 1.
(g) Obtain the value of -^ ЕЙ e2nik^n for и odd. (Schur)
Let mbea positive integer and let / be a character on the group Z(m)*.
The function / can then be defined on all the integers by setting / (k) =0
when gcd(?, m) > 1. Clearly / has period m. We call / primitive if it
does not have a smaller period. Also, / is even if / (— 1) = 1 and odd if
Also, define
m-l
n=0
48. For a e Z(p), let N(xn = a) denote the number of solutions of the equation
xn = a. If n I p — 1, then prove that
N(xn = a) = 1 -f
where the sum is over all nontrivial characters of order dividing n.
Let a be a nonzero integer. Consider the elliptic curve E defined by
xqxI — x\ — cixq = 0, which in affine coordinates is y2 = x3 + a. Suppose
p ф 2 or 3 is a prime that does not divide a. Then у2 = хъ + a is an elliptic
curve over Z(p) with a point at infinity. If Np denotes the number of Z(p)
points on the curve, then Np = 1 + N(y2 — x3 + a).
(a) Show that if p = 2 (mod 3), then Np = p + 1.
(b) Let p = 1 (mod3) and let /3 and /2 denote the cubic and quadratic
characters ofZ(p)*. Note that N(y2 = x3 + a) = EB+V=a N(y2 = u)

58 1 The Gamma and Beta Functions
N(x3 = -v). Deduce that
Np=p+l+ X2Xi(a)J(Xi, Хз) + Х2Хз(а) ДХг, Хз)-
(с) Show that if Np = p + 1 - ap then \ap\ < 2^/p.
49. By the method used in the previous problem, show that
\N(x3+y3 = \)-р + 2\<2^/р.
For Exercises 48 and 49, see Ireland and Rosen [1991, Chapters 8 and 18].
50. Prove that if x is primitive, then
( ) = fx(k)g(x) when gcd(k,m) = l,
8kKX) \0 when gcd(?, m) > 1.
Define the Dirichlet L-function by
oo
X(n)
n=\
The series converges for Re s > 0, when % is a nontrivial character, that is,
X(n) ф 1 for at least one n e Z(m)*.
51. (a) Prove that when / is nontrivial
. m— 1
L(X, 1) = —
m
(b) Show that if / is primitive
ix) V
> X(k) log sin — H .
k?Z(m)* Ч 7
(c) Prove that when / is even, 53 X (k)k = 0. and also, when x is odd,
Ex(*)l°g
(d) Prove that
f ^r E k€Z(mf X №) log sin ^, when x is even,
1^?2) h x is odd.
52. Prove that
(a) l-| + 5-^ + --- = f (Madhava-Leibniz)
(b) l + 5-5-^ + 5 + TT = i7! (Newton)
(c) 1-1 + 1-1 + 1-1 + ... = ^= (Euler)

Exercises 59
(d) 1 + 1-1 + 1-1-1 + 1 + ... = ^ (Euler)
V4/ * 2 3 "г 4 ' 6 7 8 ' 9 '" 11 ¦¦'-"V5l0S 2
The series for n/4, usually called Leibniz's formula, was known to Madhava
in the fourteenth century. See Roy [1990]. Newton [1960, p. 156] produced
his series in response to Leibniz's formula by evaluating the integral
' 1+x2 ,
/
Jo
in two different ways. Series (c) and (d) are attributed to Euler by Scharlau
and Opolka [1985, pp. 30 and 83].
Define the generalized Bernoulli numbers by the formula
a=l n=0
53. (a) Prove the following functional equation for L(x,s), x primitive:
2is \r
where 8 = 0 or 1 according as / is even or odd.
Hint: Consider the integral
pat
—dt,
L
Ic emt - 1
where С is as in problems 23 and 24. Follow the procedure given in those
problems,
(b) For any integer n > 1, show that
n
(c) For n > 1 and n = 8 (mod 2) (8 as defined in (a)), prove that
54. Let P be any point between 0 and 1. Show that
A+,o+,i_.o-) ^_^ ^ ^_^ ^
Jp ГA—
The notation implies that the integration is over a contour that starts at P,
encircles the point 1 in the positive (counterclockwise) direction, returns to
P, then encircles the origin in the positive direction, and returns to P. The
1 —, 0— indicates that now the path of integration is in the clockwise direction,
first around 1 and then 0. See Whittaker and Watson [1940, pp. 256-257].

60
1 The Gamma and Beta Functions
55. Let G(z) = log Г (г). Show that
(a) If x > 1/2, then Re G"(x + iy) > 0 for all real y.
(b) If x < 1/2, then Re G"{x + iy) < 0 for all sufficiently large y.
(c) If 1/2 < a <b, then
is an increasing function of у on (—oo, oo).
(d) The conclusion in (c) also holds if 0 < a < 1/2 and b > I — a.
(Ahern and Rudin)
56. Show that
This problem was given without the value by Amend [1996]. FOXTROT
© 1996 Bill Amend. Reprinted with permission of Universal Press Syndicate.
All rights reserved.

The Hypergeometric Functions
Almost all of the elementary functions of mathematics are either hypergeometric
or ratios of hypergeometric functions. A series Ес„ is hypergeometric if the ratio
cn+i/cn is a rational function of n. Many of the nonelementary functions that arise
in mathematics and physics also have representations as hypergeometric series.
In this chapter, we introduce three important approaches to hypergeometric func-
functions. First, Euler's fractional integral representation leads easily to the derivation
of essential identities and transformations of hypergeometric functions. A second-
order linear differential equation satisfied by a hypergeometric function provides a
second method. This equation was also found by Euler and then studied by Gauss.
Still later, Riemann observed that a characterization of second-order equations
with three regular singularities gives a powerful technique, involving minimal
calculation, for obtaining formulas for hypergeometric functions. Third, Barnes
expressed a hypergeometric function as a contour integral, which can be seen as
a Mellin inversion formula. Some integrals that arise here are really extensions
of beta integrals. They also appear in the orthogonality relations for some special
orthogonal polynomials.
Perceiving their significance, Gauss gave a complete list of contiguous relations
for 2Fi functions. These have numerous applications. We show how they imply
some continued fraction expansions for hypergeometric functions and also contain
three-term recurrence relations for hypergeometric orthogonal polynomials. We
discuss one case of the latter in this chapter, namely, Jacobi polynomials.
2.1 The Hypergeometric Series
A hypergeometric series is a series Y^, cn such that cn+\/cn is a rational function
of n. On factorizing the polynomials in и, we obtain
cn+i _ (n + a\)(n + a2) • ¦ • (n + ap)x
61

62 2 The Hypergeometric Functions
The x occurs because the polynomial may not be monic. The factor (n + 1)
may result from the factorization, or it may not. If not, add it along with the
compensating factor (n + 1) in the numerator. At present, a reason for inserting
this factor is to introduce n! in the hypergeometric series J2 cn • This is a convenient
factor to have in a hypergeometric series, since it often occurs naturally for many
cases that are significant enough to have been given names. Later in this chapter
we shall give a more intrinsic reason.
From B.1.1) we have
> с» = Cn ) =: en п га I \ х ). B.1.2)
to 0t0(b0n---(bq)nnl Ч*ь-Л )
Here the ft, are not negative integers or zero, as that would make the denominator
zero. For typographical reasons, we shall sometimes denote the sum on the right
side of B.1.2) by pFq{a\,..., ap; ftb ..., bq; x) or by pFq. It is natural to apply
the ratio test to determine the convergence of the series B.1.2). Thus,
cn+\
<
\x\nP-i-\\ + \a\\/n) ¦ ¦ ¦ (\ + \ap\/n)
An immediate consequence of this is the following:
Theorem 2.1.1 The series pFq(a\, ...,ap;b\, ...,bq;x) converges absolutely
for all x if p < q and for \x\ < 1 if p = q + 1, and it diverges for all x^O if
p > q + 1 and the series does not terminate.
Proof. It is clear that \cn+\/cn\ —>¦ 0 as n —>¦ oo if p < q. For p = q + 1,
linWoo \cn+\/cn\ — \x\, and for p > q + 1, \cn+\/cn\ -> oo as n -> oo. This
proves the theorem. ¦
The case \x \ = 1 when p = q + 1 is of great interest. The next result gives the
conditions for convergence in this case.
Theorem 2.1.2 The series q+\Fq(a\,..., aq+l; b\, ..., bq; x) with \x\ = 1 con-
converges absolutely /fRe(^fc, — Y2ai) > 0- ^e series converges condition-
conditionally if x = е'в ф 1 and 0 > Re(^fc, — ^a,) > -1 and the series diverges if
Proof. The coefficient of nth term in q+l Fq is
(fli)n • ¦

2.1 The Hypergeometric Series 63
and the definition of the gamma function implies that this term is
ПГ(а,-)
as n —>¦ oo. Usually one invokes Stirling's formula to obtain this, but that is not
necessary. See Formula A.4.3). The statements about absolute convergence and
divergence follow immediately. The part of the theorem concerning conditional
convergence can be proved by summation by parts. ¦
This chapter will focus on a study of the special case 2F\(а, b; с; x), though
more general series will be considered in a few places. The 2Ft series was
studied extensively by Euler, Pfaff, Gauss, Kummer, and Riemann and most
of the present chapter and the next one is a discussion of their fundamental
ideas.
We saw that 2^1 (a, b; c; x)diverges in general forx = landRe(c — a—b) < 0.
The next theorem due to Gauss describes the behavior of the series as x —>¦ 1~. A
proof is given later in the text, where it arises naturally.
Theorem 2.1.3 //Re(c - a - b) < 0, then
,. 2Fi(a,b;c;x) Г(с)Г(а+Ь-с)
hm = '
*-*i- (l-x)c-a-b Г(а)Г(Ь)
and for с = a +b,
2Fl(a,b;a+b;x) Г(а+Ь)
*->i- log(l/(l-*)) Г(а)Г(Ь)
The next result about partial sums of 2Fi(a,b; c; 1) is due to Hill [1908]. It can
be stated more generally for p+\Fp. The proof is left as an exercise.
Theorem 2.1.4 Let sn denote the nth partial sum of 2F1(a,b;c; 1). For
Re(c - a - b) < 0,
T{c)na+b-c
S" ~ Г(а)Г(Ь)(а+Ь-сУ
and for с = a + b,
^ Г (с) log n
Sn~ Г(а)Гф)'
The theorem is easily believable when we note that the nth term is
Г(с) па+ь-с-\
The necessary result would now follow if we replace the sum with an integral.

64 2 The Hypergeometric Functions
Many of the elementary functions have representations as hypergeometric series.
Here are some examples:
tan x = x 2FX ( ^ 2 l; -xl ) ; B.1.4)
U/2-"- B.1.5)
B.1.6)
This last relation is merely the binomial theorem. We also have
sin x=x0Fl( ~ ¦ -x2/4 J ; B.1.7)
cosx=0F^i/2;—J; B.1.8)
() B.1.9)
The next set of examples uses limits:
(hbl) B.1.10)
coshx = lim г/7]! ; —r 1 ; B.1.11)
,*(%)= lim 2*(в'*;^; B.1.12)
Vc ) b-oc \ с bj
0Fi(~;jc) = lim 2Fj (а'Й; 4) . B.1.13)
\ с J a,b-><x> \ с ab)
The example of log(l — x) = —x 2/^i A, 1; 2; x) shows that though the series con-
converges for \x\ < 1, it has a continuation as a single-valued function in the complex
plane from which a line joining 1 to oo is deleted. This describes the general sit-
situation; a 2Fj function has a continuation to the complex plane with branch points
at 1 and oo.
Definition 2.1.5 The hypergeometric function 2F\ (a, b; c; x) is defined by the
series
^ (a)n(b)n „
/ \
fl=0
for \x\ < I, and by continuation elsewhere.

2.2 Euler's Integral Representation 65
When the words "hypergeometric function" are used, they usually refer to the
function 2F1 (a, b\ c; x). We will usually follow this tradition, but when referring
to a hypergeometric series it will not necessarily mean just iF\. Hypergeometric
series will be the series defined in B.1.2).
2.2 Euler's Integral Representation
Contained in the following theorem is an important integral representation of the
2F\ function due to Euler [1769, Vol. 12, pp. 221-230]. This integral also has an
interpretation as a fractional integral as discussed in Section 2.9.
Theorem 2.2.1 If Re с >Reb>0, then
Г(Ь)Г(с-Ь) Jo
in the x plane cut along the real axis from 1 to 00. Here it is understood that
arg? = arg(l — t) = 0 and A — xt)~a has its principal value.
Proof. Suppose at first that \x\ < 1. Expand A — xt)~a by the binomial theorem
given in B.1.6) so that the right side of the formula becomes
x
n\
This is a beta integral, which in terms of the gamma function is
Ь)Г(с-Ь)
T(n +
Substitute this in the last expression to get
„ „ (a,b
xn = 2FA ;x
^ л!Г(л + с) \ с
n~\)
This proves the result for \x\ < 1. Since the integral is analytic in the cut plane,
the theorem holds for x in this region as well. ¦
The integral in Theorem 2.2.1 may be viewed as the analytic continuation of
the jF\ series, but only when Re с > Reb > 0. The function A — xt)~a in the
integrand is in general multivalued and one may study the multivalued nature of
2Fi(a,b;c;x) using this integral. To discuss analytic continuation more deeply
would require some ideas from the theory of Riemann surfaces, which goes beyond
the scope of this book. See Klein [1894].
It is also important to note that we view 2^1 (a. b; c; x) as a function of four
complex variables a,b,c, and x instead of just x. It is easy to see that ^ 2 F\ (a, b;
c; x) is an entire function of a, b, с if x is fixed and \x\ < 1, for in this case the

66 2 The Hypergeometric Functions
series converges uniformly in every compact domain of the a, b, с space. Analytic
continuation may be applied to the parameters a,b,c. The results may at first be
obtained under some restrictions, and then extended. For example:
Theorem 2.2.2 (Gauss [1812]) For Re(c - a - b) > 0, we have
^ (g)n(b)n _ р /a,b \ = Г(с)Г(с-а-Ь)
^ n\{c)n \ с ' ) T{c-a)T{c-b)
Proof. Let x —>¦ 1 ~ in Euler's integral for 2 F\. The result is, by Abel's continuity
theorem,
b; Л = Г(с)
r(b)r(c-b)J0
_ Г(с)Г(с-а-Ь)
~ Г(с-а)Г(с-Ь)'
when Re с > Re ? > 0 and Re(c — a - b) > 0. The condition Re с > Re b > 0
may be removed by continuation. It is, however, instructive to give a proof that
does not appeal to the principle of analytic continuation.
Our first goal is to prove the relationship
c(c — a — b)
If
, (a)n(b)n (a)nib)n
An = and Bn = ,
л!(с)„ л!(с+1)„'
then
<2,л)
1
c-a-b-
и!(с+1)„_1
and
с(иА„ — (и + l)An+i) = — я
и!(с+1)„_! L c-
So, since the right sides in the last two expresions are equal,
c(c —a — b)An = (c - a)(c — b)Bn + cnAn — c(n
and
N N
c(c — a — b)y An = (c — a)(c — b)y Bn — c(N
о о

2.2 Euler's Integral Representation 67
Now let N -* oo and observe that (N + l)AN+i ~ \/Nc~a'b -> 0, because
Re(c — a — b) > 0. This proves B.2.1). Iterate this relation n times to get
Г(с-а)Г(с-Ь) -.(а,Ь \ Г(с+ n - а)Г(с+ n - b) ( a,b
Г(с)Г(с-а-Ь) \ с J Г(с + п)Г(с + п-а-Ь) \c + n
It is an easy verification that the right side —>• 1 as n —*¦ oo. This proves the theorem
for Re(c — a — b) > 0. The theorem is called Gauss's summation formula. ¦
The case where one of the upper parameters is a negative integer, thereby making
the 2F1 a finite sum, is worthy of note. This result was essentially known to the
thirteenth century Chinese mathematician Chu and rediscovered later. See Askey
[1975, Chapter 7].
Corollary 2.2.3 (Chu-Vandermonde)
-n,a \ _ (c-a)n
с ) (с)„
Euler's integral for 2F\ can be generalized to pFq. Rewrite it as
Thus, integrating a i Fo with respect to the beta distribution tb~l A — t)c~b~l gives
a 2F1, that is, a parameter i> is added in the numerator and с in the denominator of
the original \Fo(a; t).
More generally, we have
/fli a,,yi \_
1 9+1\bu...,bq,bq+l'X)
f и---,aP
4\bu...,bq
J
when Rei>9+i > Reap+i > 0. This condition is needed for the convergence of
the integral. By a change of variables the expression on the right of B.2.2) also
equals
B.2.3)
Note also that B.2.2) can be used to change the value of a denominator or numerator
parameter in pFq(a\, ...,ap;bi,...,bq;x). For example, take ap+\ — bq in

68 2 The Hypergeometric Functions
B.2.2) to get
, U1, „ ,
" «v*i.---.Vi.Vi'"y r(bq)r(bq+i-bq)
¦ [ tb<-l(l-t)b'+1-b<-1pFq(a,u"-'a,p;xt)dt. B.2.4)
Jo \bu---,bq J
It should be remarked that when x is a complex variable in B.2.2) to B.2.4),
then the pFq is in general a multivalued function. Thus, the variable x has to be
restricted to a domain where the pFq in the integrand is single valued. One must
take care to state the conditions for single-valuedness. We note a special case of
B.2.4).
Theorem 2.2.4 For Rec > Red > 0, x ф 1, and |arg(l — x)\ < я,
r{d)T(c-d)J0 \ d '
One pecularity of Euler's integral for 2Fi is that the 2F\ is obviously symmetric
in the upper parameters a and b, whereas it is not evident that the integral remains
the same when a and b are interchanged. Erdelyi [1937] has presented a double
integral from which the two representations can be obtained:
[Г(
^ /' /Vv-'o-o'-*-1
-a)F(c-b)Jo Jo
Г(а)Г(Ь)Г(с
• A - s)c-"-1 A - tsx)~cdtds. B.2.5)
The next theorem gives an important application of Euler's integral to the deriva-
derivation of two transformation formulas of hypergeometric functions.
Theorem 2.2.5
b) ^C;b ^) (Pfaff), B.2.6)
в;^)=A-*)~-*2*(С~в;С~*:*) (Euler). B.2.7)
Proof. Replace t with 1 — s in Euler's integral (Theorem 2.2.1) to obtain
r
Г(Ь)Г(с-Ь) Jo

2.2 Euler's Integral Representation 69
This proves Pfaff's [1797] transformation for Re с > Reb > 0. The complete
result follows by continuation of с and b.
The hypergeometric function is symmetric in the parameters a and й, so we
apply Pfaff's transformation to itself:
This is Euler's [1794] formula and the theorem is proved. ¦
The right-hand series in Pfaff's transformation converges for \x/(x — 1)| < 1.
This condition is implied by Rex < 1/2; so we have a continuation of the series
2F1 (a, b\ c; x) to this region by Pfaff's formula.
The following two examples give an indication of the power of the transforma-
transformation formulas. By Pfaff's transformation,
1 + x2
In Chapter 1 we showed how the gamma function could be used to develop some
aspects of trigonometric functions, starting with the series definitions of the sine
and cosine functions. The above relation can now be the basis for the connection
between the trigonometric functions and a right triangle.
For the second example, write Euler's transformation as
Equate the coefficient of x" on both sides to get
a)j(b)j(c-a-b)n-j (c-a)n(c-b)n
j^ j\(c)j(n-j)l n\(c)n
Rewrite this as:
Theorem 2.2.6 (Pfaff-Saalschutz)
3F2( n'a'b Л = (са)АсЬ)п
г\с,1+а + Ь-с-п' ) (c)n(c-a-b)n
Gauss's 2F1 sum (Theorem 2.2.2) follows from this by letting n -> 00. The
limiting procedure may be justified by Tannery's theorem, which is a discrete
form of Lebesgue's dominated convergence theorem. Theorem 2.2.6 was first
discoverd by Pfaff [1797a] and rediscovered by Saalschlitz [1890]. It is often
called Saalschiitz's theorem but this nomenclature does not give due credit to

70 2 The Hypergeometric Functions
Pfaff. Surprisingly, a special case seems to have been found by Chu. See Takacs
[1973].
Remark 2.2.1 The Chu-Vandermonde identity (Corollary 2.2.3) gives the sum
of a terminating 2^1- The Pfaff-Saalschiitz identity involves a special type of
terminating 3F2. The sum of denominator parameters is one more than the sum
of the numerator parameters. Such a series is called balanced. This identity
was obtained by a factorization of a jF\ and it is worth noting that the Chu-
Vandermonde identity can be derived from a factorization of a 1 Fq. Thus, one may
equate the coefficients of x" in
A - xTa{\ - x)'b = A - x)-{a+b)
to get an equivalent identity:
{а)кф)п-к (a+b)n
The right side of Pfaff's transformation formula when expanded as a series
equals
(аЖе - b)k k (a+k)jxj
X ,
Note that (a)k(a + k)j = (a)j+k; then write j + к = п to see that the sum is
—x — > x ,
where the inner sum was evaluated by the Chu-Vandermonde identity. This gives
another proof of Pfaff's transformation.
The following definition is suggested by the Pfaff-Saalschiitz formula.
Definition 2.2.7 A series
/au...,ap+i
p+\tp I ,X
\ 0\, .. . ,0p
is called balanced ifx — 1, one of the numerator parameters is a negative integer,
and a\-\ h ap+\ + 1 = b\ H \-bp.
Remark 2.2.2 The Pfaff-Saalschiitz identity can be written as
-n, -a, -b
(c)n(c + a + b)n3F2 [ ;
\c, l—a—b — n—c

2.2 Euler's Integral Representation 71
This is a polynomial identity in a, b, с Dougall [1907] took the view that both
sides of this equation are polynomials of degree и in a. Therefore, the identity is
true if both sides are equal for n + 1 distinct values of a. Clearly the result is true
when n = 0. Assume the result true for n = 0, 1,..., к — 1. Now set n — k. By
symmetry in a and n, it follows that the identity is true for a = 0, 1,..., к — 1.
These are к values, so if we can find one more value of a for which the identity
holds, then it is proved. Note that
(l-a-b-n-c)j v ' v
So, if a = —b — с then both sides of the identity are equal to (—a)n(—b)n. This
proves the identity. Dougall showed that a more general identity could be proved
by this method. The identity is
a, 1 + ^a, —b, —c, —d, —e, —n
lFe у \a, 1 + a + b, 1 + a + c, 1 + a + d, 1 + d
= A + a)n(l +a+b + c)»(l + a + b + d)n(l +a + c + d)n
~ {l+a+b)n{l+a+c)n{l+a+d)n{l+a+b + c + d)n'
where l+2a+b + c + d + e + n — 0 and и is a positive integer. This condi-
condition means that the series terminates and the sum of the denominator parameters
is 2 more than the sum of the numerator parameter. Such a series is called 2-
balanced. (A 1-balanced series is balanced as in Pfaff-Saalschiitz.) Note that the
sum of the parameters in a column in this tF6 add up to the same quantity. Thus,
\ + a = 1 + \a + \a = 1 + a + b — b = I + a + с — с and so on. This type
of series is called well poised. Dougall's identity thus gives the sum of a class of
well-poised 2-balanced 7F6. These series are called very well poised because the
series contains the factor
(f + 1),. a + 2k
This identity was also discovered by Ramanujan at around the same time as
Dougall's discovery. See Hardy [1940, p. 102]. A proof of the sum B.2.9) is
given later. To obtain another important identity from Dougall, let n —>¦ oo to
get
/ a,a/2+ \,-b,-c,-d
5 4 \a/2,a+b + l,a + c+ l,a+d + l'
T(a+b + 1)Г(д +с+ 1)Г(а + d + 1)Г(а +b + c + d+l)
+ + d + l)

72 2 The Hypergeometric Functions
when Re(a + b + c + d + l)>0. Then take d = -a/2 to get
f ( a' ~b' ~c ¦
3 2 \a+b+l,a+c+l'
b + с + I)
Г (а + 1)Г(§ + b + 1)Г(| + с + 1)Г(а + Ъ
This gives the sum of a general well-poised 3F2 series. It is due to Dixon [1903].
The limiting procedure above may be justified by Tannery's theorem. A more
general result than B.2.10) had been found by Rogers [1895]. This will be given
later.
Remark 2.2.3 We have seen that the Chu-Vandermonde identity can be obtained
from the Euler's integral in Theorem 2.2.1, which is an immediate consequence of
the value of the beta function. A type of converse holds. The Chu-Vandermonde
identity is a discrete form of the beta integral formula,
.e-i,, .,*-, ,. Па)Т{Ъ)
! t"-l(l-
Jo
By Remark 2.2.1, we have
n! ^ ia)k{b)n-k _
(a + Ь)„ f^
We briefly sketch the argument showing that a limiting form of this identity is the
beta integral formula. Rewrite the identity as
(a+b)n n + l f-? k\{n-k)\
or
A [(* + l)l-a(a)k/k\][(n + 1 - k)'-\b)n-k/(n - *)!]
Recall that by definition
k^o jfei = r^)"

2.3 The Hypergeometric Equation 73
If we break up the sum as
logn n—logn n
?+? + ?'
k=0 logn n—logn
the first and third sums go to zero and the second tends to
This expression equals 1 and we have the result. The reader should try to find the
beta integral that corresponds to Gauss's formula for 2^1 at x = 1.
2.3 The Hypergeometric Equation
The hypergeometric function satisfies a second-order differential equation with
three regular singular points. This equation was found by Euler [1769] and was
extensively studied by Gauss [1812] and Kummer [1836]. Riemann [1857] in-
introduced a more abstract approach, which is very important. Our treatment will
basically follow Riemann, in a more explicit form given by Papperitz [1889]. The
reader who has never seen series solutions of differential equations with regular
singular points might find it helpful to read Appendix F first.
Let p(x) and q(x) be meromorphic functions. Suppose that the equation
ii+P(x)^+q(x)y=0 B.3.1)
has regular singularities at the finite points a, ft, у and that the indicial equations
at these points have solutions a\, a2; bu bj; and с\,сг respectively. Assume that
a\ — аг,Ъ\ —Ъг, and c\ — сг are not integers. Set x = \/t so that the differential
equation is transformed to
Since 00 is an ordinary point, 2x —x2p(x) and x4q (x) are analytic at 00. Moreover,
since a, fi, у are regular singular points,
ABC
p(x) = + + + Ui(x)
x — a x — p x — у
and
(x - a)(x - p){x - y)q(x) = + + + U2(x),
x — a x — p x - у
where u\{x) and ui{x) are analytic functions.

74 2 The Hypergeometric Functions
The last two relations together with the analyticity of 2x —x2p(x) and xAq{x) at
infinity imply that A + B + C = 2 and u\ (x) = u2(x) = 0. Suppose a solution has
the form YlT=o а"(х ~ a)n+X> where the exponent X satisfies the indicial equation
X(X - 1) + XA + -? = 0.
(a - P)(a - y)
Since a\ and a2 are roots of this equation,
fll + U2 = 1 — A
and
D
axa2 =
{a - /J)(a - y)
Therefore,
A = 1 — a\ - аг and D = (a — f})(a — y)a\a2.
Similarly,
B = l-bi-b2 and E = (P - a)(P - у)Ьф2,
and
C=\—C\-C2 and F = (y - a)(y - P)c\c2.
Since Л + В + С = 2, it follows that the exponents of the differential equation
are related by the equation
fCl+c2 = l. B.3.3)
We summarize the results in the following theorem due to Papperitz [1889].
Theorem 2.3.1 A differential equation with three singular points a, p, у and
exponents a\, a2; b\, b2; and Ci, c2 respectively has the form
d2y \\-a\-a2 \-b\-b2 1 - C\ - c2 \ dy
d2y (I — a\— a2 1 —b\—b2
~dx^+ \ x-a + x-p
{x - a){x - p){x - y)
x — у ) dx
(a - P)(a - y)axa2
| (p - a)(p - у)Ьф2 | (y - a)(y - P)ac2\
x-p x-y /
and the exponents satisfy B.3.3).

2.3 The Hypergeometric Equation 75
It is customary to take the regular singularities at 0, 1, and oo. So let a = 0, /3 =
1, and у -> oo in the above differential equation to obtain
x2(x - \)ld-\ + {A - fl, - аг)х{х - IJ + A - bx - b2)x\x -
dxl
+ {axa2{l-x)+b\b2x+clc2x{x- \)}y = 0. B.3.4)
The hypergeometric equation is obtained from this one by another simplification.
Write this equation in the form B.3.1). If у satisfies B.3.1) and у = xxf, then /
satisfies
d2f ( 2X\ df ( Xp(x) X(X-\)\
-4 + p{x) + — )-L+( q{x) + J^-L + / = 0.
dx2 \ x ) dx \ x x1 )
This equation also has 0, 1, oo as singular points, but the exponents are different.
Equation B.3.4) has exponents ax and a2 at 0; the new equation has exponents
a\ — X and a2 — X at zero. The exponents at oo, however, are a + X and a + X.
By this procedure we can arrange that one exponent at 0 and one exponent at 1
be equal to 0. (At x = 1, we set у = A — x)x f{x).) Thus the new equation
has exponents 0, a2 — a\; 0, b2 — b\\ c\ + a\ + b\, c2 + a\ + b\. This brings
considerable simplification in B.3.4) since the terms a\a2 and b\b2 vanish. It is
traditional to write a — c\ + a\ + b\, b = c2 + a\ + b\, and с = 1 + a\ — a2.
After simplification, the equation becomes
x{\ - xy-?- + [c - (a + b + l)x]~ - aby = 0. B.3.5)
dx2 dx
This is Euler's hypergeometric differential equation. It has regular singularities
at 0, 1, and oo with exponents 0, 1 — c; 0, с — a — b; and a, b respectively. Unless
specifically stated, we assume that c, a — b, and с — a — b are not integers.
Riemann [1857] denoted the set of all solutions of the equation in Theorem
2.3.1 by
B.3.6)
In particular, the set of solutions of B.3.5) is denoted by

76 2 The Hypergeometric Functions
Our earlier discussion implies that
xx(\-xYP lax ex bx x
B.3.7)
Every conformal mapping of the Riemann sphere С U {oo} is of the form
Xx + [i
t =
Sx + v '
where X v — /л8 = 1. Such a mapping takes any set of three distinct points {a, f), у}
to another set of three distinct points {a\, fix> Y\ }¦ In this case
[a P Y ) («i А И )
P lax bx cx x \ = P I ax bx cx t\. B.3.8)
[a2 b2 c2 J [ a2 b2 c2 J
This is easily checked. Moreover, there are six linear fractional transformations
that will map a set of three points to a permutation of the three points. For example,
the set {0, 1, oo} will be mapped to itself by the mappings
11 1 x-1 1 x
x-+x,l-x,-, , 1 - - = , —- = -. B.3.9)
X I — X X X 1 — l/X X — 1
We note a few particular cases of B.3.7)-B.3.9):
0 oo 1
= P< 0 a 0 1-х
с — a — b b I — с
B.3.10)
(A)
(B)
л — i I
c-b b-a )
(Note: A ^y)" = A — x)~a.)

2.3 The Hypergeometric Equation 77
= Р{а О О -) (С)
(D)
(Е)
О оо 1
= {\-х)с-а-ьР \ 0 с-а О х) (F)
1-е с — Ъ а+Ь — с
О оо 1
= х1'ср{ 01+а-с 0 х). (G)
кс—1 l+b — c c — a — b
Now from the full set of solutions P { } of B.3.5) we choose two solutions about
x = 0 which form a basis. For a solution of the form xx Y^ anxn, X is either 0 or
1-е. When X = 0, the coefficients an satisfy
So one solution is 2Fi(a,b;c;x). If с is not an integer, then the hypergeometric
equation has only one independent solution analytic at x = 0. In particular,
2F\{a,b;c;x) is the only solution analytic at x = 0 and with value 1 at x =
0. The other solution is of the form W = xl~cg, where g is analytic at x =
0. It follows from B.3.10G) that g = k2Fx(a + 1 - c, b + 1 - c; 2 - c; x),
where к is a constant. Therefore, the independent solutions are 2F\(a,b;c;x) and
x1~c2F\(a + 1 — c, b + 1 — c; 2 — c; x). In a similar way, B.3.10) immediately
implies that two independent solutions at x — 1 are
2FX (a, b\a +b+ 1 - c; 1 - x)
and
A - x)c~a~b2Fi{c - а, с - b; с + \-a-b\l-x)
and at oo are
(-xya2Fi(a, a + l-c;a+l-b; l/x)
and
b + l-c\b + l-a; l/x).

78 2 The Hypergeometric Functions
The powers of — 1 have been introduced for convenience in expressing some later
formulas.
It is important to note that some of the hypergeometric transformation formu-
formulas can also be obtained from B.3.10). Pfaff's transformation 2F\{a, b; c; x) =
A — x)~a2F\(a,c — b; c; x/(x — 1)) is an immediate consequence of B.3.10B),
combined with the fact that there is only one solution analytic and equal to 1 at x =
0. Euler's transformation 2F\_(a, b; c; x) = A — x)c~a~b2F\{c — а, с — b;c; x)
follows from B.3.10F).
A number of relations involving hypergeometric functions arise from the fact
that the hypergeometric equation has two independent solutions, so that any three
solutions must be linearly related.
Theorem 2.3.2
v( a'b
iF\ ; 1 — x
\a+b+l -c
(a,b \ , /1 +a - c, 1 +b - с \
= A2Fll Ус -xj+Bx'-^FA 2c ;x\, B.3.11)
where
T
=
(а + 1-с)Гф+1-с)
. T{a+b+l-c)T{l-c) , n Г(с-1)Г(а
A = and В —
Г( 1)Гф1)
/a,b \ (a,a-c+\ 1
2F1[ ;x)= C{-x)-a2Fx ^ ; -
; / \ a — b + I x
b-a+l
where
С = Г(с)Г(Ь - a) and D =
~ T{)T{b) пП ~
and D
Proof. When x = 0 and Re с < 1, Gauss's summation formula gives
Г(а+1-с)Г(Ь+1-с)'
Observe that Re с < 1 was used twice. First it was used to make the second term
on the right vanish; it is also the condition for the series on the left to converge at
x = 0. Then x = 1 gives (for Re(c - a - b) > 0)
l = лГ(с)Г(с-а-Ь) вГB-с)Г(с-а-Ь)
~ Г(с-а)Г(с-Ь) ГA-

2.3 The Hypergeometric Equation 79
After some tedious trigonometric calculation, which comes in after applying
Euler's reflection formula to the second term on the right, we arrive at the value of
В required by the theorem. This proves B.3.11).
Suppose Reb > Rea. The right side of B.3.12) as x —> oo is ~C(—x)~". To
see the behavior of the left side, apply Pfaff's transformation. Then
5 1
x - I
Г(с-а)Г(Ь)'
The assumption that Reb > Re a was used in the last step to evaluate the 2FX by
Gauss's formula. It follows that
Г(с)Г(Ь-а)
Г(с-а)Гф)'
The value of D follows from the symmetry in a and b. ¦
Corollary 2.3.3
'a,b \ T(c)T(c-a-b)
Г(а)Г(й) ( Х) 2^{l+c-a-b'1 *) '
B.3.13)
7 ;) (Pfaff) B.3.14)
+l-n-c )
Proof. In B.3.11), replace x by 1 - x and с by a + b + 1 - с Then B.3.14)
l
Г(-л)
follows from B.3.13). Just take a = —и and recall that r ' , = 0 when и is a
nonnegative integer. ¦
The first part of Theorem 2.1.3 also follows from B.3.13) above. It should also
be noted that, since Pfaff's formula
2F\{a,b;c\x) = A - x)~a2F\{a, с - b; c; x/{x - 1))
gives a continuation of 2F\ from |jc| < 1 to Rex < \, then B.3.13) gives the
continuation to Rex > 1/2 cut along the real axis from x = 1 to x — oo. The cut
comes from the branch points of A — x)c~a~b, and once this function is defined
on a Riemann surface, 2F\ {a, b; c; x) is also defined there.

80 2 The Hypergeometric Functions
Now consider the function
f0° dt
S(x)
4°
Jo
We show how Theorem 2.3.2 can be employed to find the asymptotic expansion of
the function given above. Wong [1989, p. 18] used this function to demonstrate that
a certain amount of care should be taken when finding the asymptotic expansion
of a function. In this instance, when the method of integration by parts is applied,
one gets the expansion
z.r.^^-i)^" as x' <)
which is obviously incorrect, since the integral is positive and every term of the
expansion is negative. However, for t > 1 we have
«=o
If this series is substituted in the integral, then term-by-term integration produces
the divergent integrals
/•oo ,-«-1/3
/
JO
x+t
-dt.
If these are interpreted in a "distributional" sense the value of the above integral
can be set equal to
2л (-1)"
With this interpretation, S(x) has the expansion (after termwise integration)
S(x) ~%Y Ш«х-«-1/з as x -+ oo. B.3.16)
The correct result is, however, the sum of the two expansions in B.3.15) and
B.3.16). We obtain this from Theorem 2.3.2. First note that forRe(a + 1-е) > 0
and Re b > 0,
/•00
\C-b-\.
/ tb~l{\+t)c~b~l{l+xt)~adt
Jo
Г(а + 1-с)Г(Ь) ( a,b \
= — — -2FA ;l-x). B.3.17)
F(a + b+l— c) \a+b+l—c )
This follows from Euler's integral representation of a iF\ given in Theorem 2.2.1.
To reduce this integral to Euler's form, set r = и/A — и). From B.3.17) it follows

2.3 The Hypergeometric Equation 81
that
1 f°
= - /
X Jo
ц /i,i i
3 /1,1 1
Apply B.3.11) to get
3 [ГD/3)Г(-2/3) /1,1 1
ГB/3)ГD/3) 23 /1/3,1/3 1
* 2f4 1/3 ;x
FJ + X 4 1/3 '
3 f/^1'1 l\ л.27Г : F /1/3,1/3 1
2x \5/3 x/ V3* / V 1/3 ¦«
This is equivalent to the sum of the series in B.3.15) and B.3.16).
Remark 2.3.1 We have seen that 2F1 {a, b; c; x) is one solution of the hyperge-
hypergeometric equation. This fact was used to show that another independent solution
is
xl~c2Fx(a + 1 - c, b+ 1 - c; 2 - c; x).
Here we show how the other solutions can be obtained formally from the series
for 2F1. We should write the hypergeometric series as a bilateral series
Since ГA + x) has poles at x = —1, —2,... , the series has no negative powers
of x. It is clear that a change of variables n -> n + m, where m is an integer, does
not change the bilateral series. Consider the transformation n -> n + a, where a
is a noninteger. Now the series takes the form
(a)n+a(b)n+axn+a _ (a)a(b)a a ^ (a+a)n(b
^ (a)n+a(b)n+axn+a _ (a)a(b)a a ^ (a+a)n(b + u) n
fl (c)n+a(\)n+a (c)e(l)a ,f-L(c+ «)„(!+а)„
«= — 00

82 2 The Hypergeometric Functions
In the latter expression, the terms with negative values of n vanish if we set
c+a = lorl+a = l. The last condition gives back the original iF\ series. The
first case, where a = 1 — c, gives
(a + 1 - c)n{b + 1 - c)n j_ (a + l-c,b+l-c
=X 2^1 ^ >x
^ A)вB-с)„ 2 Ч 2-е
which is the second independent solution. The solutions at 00 are obtained in a
similar manner by changing и to —и. In this case (A) becomes
kxa
X
where к is a constant. Since (a + «)_„ = (—1)"/A — a — a)n, write the last
series as
kxa V (l-c-a)nX~n(-<x)n -n
Once again, to eliminate that portion of the sum involving negative values of и, take
either a = —aora = — b. In the first case we get ex ~a 2^1 (я + 1-е, a; a + l—b;
l/x) and in the second case cx~b2F\(b + 1 — c, b; b + \-a\ \/x).
Remark2.3.2 In Section 2.1 we proved that the series 2F\(a, b; c; x) has radius
of convergence 1. We now reverse the point of view taken in Remark 2.3.1 and
see how to obtain convergence from the theory of differential equations. Since the
singularities of the equation are at 0, 1, and 00, the radius of convergence is at least
1. If it is more than 1, then the series is an entire function. Moreover, by B.3.12)
it is a linear combination of x~a f\(x) and x~bf2(x), which are solutions at 00.
This is possible only if either a or b is an integer. Otherwise, both solutions at 00
are multivalued. (Both a and b cannot be integers since a — b is not an integer.)
Liouville's theorem shows us that the integer must be negative and that the 2F1 is
a polynomial. Hence, if the 2^1 is an infinite series then the radius of convergence
must be 1.
We now consider the case where с is an integer. Suppose с is a positive integer;
then
Г(а)Г(Ь) (a,b
к!Г(с + к)
and
Г(а+\-с)Гф + \-с) x_ (a + c- l.fc+1 -c
F2:= ^—; x 2F>( 2_c ;*
Г (a + 1 - с + к)Г(Ь + l - с + fc)
к\ГB-с + к)

2.3 The Hypergeometric Equation 83
are equal. To find the second solution in this case, suppose a and b are not negative
integers. Consider the limit
hm = — (F, - F2) \c=n
с-m С — П ОС
Г(а+к)Г(Ь + к)Г'(п+к) к
к\Г(п+к)Г(п + к)
logx
k\T{2-n+k)
f^Q к\ГB-п+к)
'(a + l-n+Jfc) Г'(Ь + 1 - n + к)
— hmx
Г'B-с + к) к
The second series is
Г»Г(?), ^ (a,b
Г (и)
;x
and the first n — 1 terms in the third series are zero because TAln+K) = 0
for к = 0,1,... ,n — 2. The same is not true in the fourth series, because
Г'B— с + к)I ГB — с + к) has poles at these points. By Euler's reflection for-
formula,
+ cos лхТ{х).
[ГA-дс)р Г(х)ГA-х)
Set x — n — к — 1 to get
The fourth series can now be written as
n — \ ,,
Г (a + 1 - n + k)V{b + 1 - n + к) к

84 2 The Hypergeometric Functions
So, when a and b are not negative integers, the second solution is
; x log*
n )
E°° (a)k(b)k k
{yf(a + k) + yf(b + k) — i/f(l + k) — Ф(п + k)\x
(*-1)!Г(«-*)Г(Ь-*)
(лЛ1)! V 7
where f{x) = Г'(х)/Г(х).
If a is a negative integer, say —m, then -ф-(а + к) is undefined for some values of
к. Consequently, the solution given above does not work. To resolve this difficulty
observe that
lim {ir(a + k)-if(a)} = ir(l+m-к)-if A+m) fotk<m. B.3.19)
a> m
a—>— m
Now, if \p-(a) iF\(a, b; c; x) is subtracted from the second term in B.3.18), then
the resulting series is again a solution of the hypergeometric equation except that
now we may let a tend to the negative integer —m. The reader should verify
B.3.19) and also that in this case the second solution is
2F1I ;x 1 logx
( \ (h\
7\ №Q +mk) + ir(b + k) ф(п + к) фA + к)}хк
к=о
Г(Ь) (^ (п - к - 1I(т + 1)к
The case where both a and b are negative integers may be treated in the same way.
When с — 0, — 1, —2,..., then the indicial equation shows that the first solution
is
a -c + l,b -c + 1
The second solution in this case can be obtained from B.3.18) by replacing a and
b with a — с + 1 and b — с + 1 respectively.
Theorem 2.3.2 and its corollary must be modified when c, a — b, or с - a — b
is an integer. The reader should work out the necessary changes.
An interesting history of the hypergeometric equation is contained in Gray
[1986].

2.4 The Barnes Integral for the Hypergeometric Function 85
2.4 The Barnes Integral for the Hypergeometric Function
In a sequence of papers published in the period 1904-1910, Barnes developed an
alternative method of treating the hypergeometric function 2 F\. A cornerstone
of this structure is a contour integral representation of jF\(a, b; c; x). An under-
understanding of this representation can be obtained through the concept of a Mellin
transform. We begin with a simple and familiar example: T(s) = /0°° xs~xe~xdx.
It turns out that it is possible to recover the integrated function e~x in terms of a
complex integral involving Г (s). This inversion formula is given by
-* = -L Г
2ni Jc_i
¦C+iOO
—s
x~sr(s)ds, о 0. B.4.1)
This can be proved by Cauchy's residue theorem. Take a rectangular contour L
with vertices с ± iR, с — (N + |) ± iR, where N is a positive integer. The poles
of F(s) inside this contour are at 0, —1,..., —N and the residues are {—\УЦ\ at
j = 0, 1,..., N. Cauchy's theorem gives ^ JL x~sT(s)ds = ^=0 (-l)jxj/jl.
Now let R and N tend to infinity and use Theorem 1.4.1 and Corollary 1.4.4 to
show that the integral on L minus the line joining с — iR to с + iR tends to zero.
This proves B.4.1).
The Mellin transform of a function /(jc) is defined by the integral F(s) =
/0°° xs~l f(x)dx. We have studied other examples of Mellin transforms in Chap-
Chapter 1. The integral /J jcs-1A - x)'~ldx is the transform of
I — jc)'-1, 0<jc<1,
and
poo rs~X
-dx
A+JC)'"
is the transform of f(x) = 1/A + x)'. Once again, one can prove that
A - jc)' = —- / x~s -!— ds, Re t > 0 and c> 0, B.4.2)
)+ 2ni Jc_ioo r(s + r)
and
1 I pc+ioo
(l+хУ 2niT{t)
/C+l
x~sr(s)T(t - s)ds, 0 < с < Re t. B.4.3)
—ioo
The phenomenon exhibited by B.4.1) through B.4.3) continues to hold for a fairly
large class of functions /(jc). Thus, if F(x) = f™xs~lf(x)dx then /(jc) =
2S7 Sc-ioo X~S F(s)ds is true for a class of functions. We do not develop this
theory as a whole but prove a few interesting cases. The Mellin transform is
further discussed in Chapter 10 with a different motivation.

86 2 The Hypergeometric Functions
The above discussion shows that if we want a complex integral representation
for the hypergeometric function we should find its Mellin transform. Now,
'00
xs~l2F\ ( ; — x \dx
с
T{b)(c-b)Jo Jo (\+xt)a
T(s)T(a-s)Tic) f\b-s-4l_tr-b-,dt
Г(а)Г(Ь)Г(с-Ь)Л ( '
r(s)r(a - s)F(c) Г(Ь - s)V(c - b)
T(a)T(b)T(c - b) T(c-s)
T(c) T(s)T(a-s)T(b-s)
B.4.4)
Г(а)Г(Ь) T(c-s)
These formal steps can be justified by assuming min(Re a, Re b) > Res > 0. The
auxiliary condition Re с > Re b can be removed by analytic continuation or via
contiguous relations, which are treated in Section 2.5. Note that we integrated 2F\
at — x because, in general, 2F\(a, b; c; x) has branch points at x = 1 and x = oo
and in B.4.4) the integral is over the positive real axis. There is another proof of
B.4.4) in Exercise 35.
We expect, by inversion, that
T(a)T(b) (a,b \ 1 [k+i°° Г(х)Г(а - s)F(b - s)
K\ = JL fk+ic
) 2xi Jk-ioo
T{c) "'\ с  2л-/ Л-,-оо Пс-s) v ' '
B.4.5)
where min(Re a, Reb) > к > 0 and с ф 0, — 1, -2, This is Barnes's formula
and it is the basis for an alternative development of the theory of hypergeometric
functions. It should be clear that we can represent a pFq by a similar integral. The
precise form of Barnes's [1908] theorem is given next.
Theorem 2.4.1
T(a)T(b) „ (a,b \ 1 fi0° Г (a + s)T(b + s)V(-s)
¦x)=t- „, , ч (-x)sds,
Г(с) 'V с J 2niJ_i0
|arg(—jc)| < n. The path of integration is curved, if necessary, to separate the
poles s = —a — n,s = —b — n, from the poles s = n, where n is an integer > 0.
(Such a contour can always be drawn if a and b are not negative integers.)
Proof. Let L be the closed contour formed by a part of the curve used in the
theorem from —(N+ i)i to (N + |)i together with the semicircle of radius N +^

2.4 The Barnes Integral for the Hypergeometric Function 87
drawn to the right with 0 as center. We first show that the above integral defines
an analytic function in |arg(— jc)| < n — <5, 8 > 0. By Euler's reflection formula,
write the integrand as
By Corollary 1.4.3, this expression is asymptotic to
a
л
SHlSTT
Set* = it to get
„i»(log|jt|+iarg(-jt))
-{it)a+b-c~l2ni =
p—7X1 pTlt
for |arg(— x) | < n — 8. This estimate shows that the integral represents an analytic
function in |arg(—jc)| < л — 8 for every 8 > 0, and hence it is analytic in
|arg(—jc)| < n. We now show that the integral represents the series
xn for \x < 1.
This will prove the theorem by continuation. (Note that, if we start with the series
for iF\(a, b; c; x), then Barnes's integral gives the continuation to the cut region
|arg(-x)| < л.)
On the semicircular part of the contour L the integrand is
sin sn
for large N. For s = (N + \)ew and \x\ < 1,
sinra
Since —л + 8 < arg(—x) < n — S, the last expression is
In 0 < \в\ < |,cos6» > ^ and in I < \в\ < \, |sin6»| > ^. So, since
log|jcI < 0, the integrand is O(Na+b-c-leT2iN+^bglxl) for 0 < |6»| < f and
O(Na+b-c-\e-j-24N+{)^ for n_ < щ < | This implies that the integral on the
semicircle -^ 0 as N ->¦ oo. Since the pole s = n of the integrand has residue
n\T(c
the theorem is proved. ¦
Г(а+п)Г(Ь + п) п

88 2 The Hypergeometric Functions
We can recover the asymptotic expansion contained in B.3.12) from Barnes's
integral quite easily. Suppose a — bis not an integer. Move the line of integration
to the left by m units and collect the residues at s = —a — n and s = —b — n. The
residue at s — —a — n is
„•-а-п)Г(а + п)(-хГа-"
п\Т(с-а-п)
Г(с-а) ' n\(l+a-b)n '
after a little simplification. Thus,
Г(а)Г(Ь)
(a,b \ 1 Гт+
; x = /
Чс У 2ni J_m_io
Г (с) 1\ с J 2л i J_m_ioo Г (с
+ (~хГа
Г (с - а)
а - b) ^ (b)n(l + b - с)п
+ ( X) T(c-b) ^ n\(l+b-a)n X '
B.4.6)
where w(a) is the largest integer n suchthata+л < m. We define m(b) similarly.
The integral is equal to
Х—х-т [' —^—Ш ' "/- v~ ' ''" {-x)'ds.
2ni J-ioo Г (с — /и + *)ГA — w + s) sin л*
For |arg(— jc)I < n — S, S > 0, the last integral is a bounded function of m and
x. This implies that the expression is O(l/:cm), so that we have an asymptotic
expansion for 2F1 (a, b; c; x) in B.4.6). If a — b is an integer, then some of the
poles of Г(а + s)F(b + s) are double poles and logarithmic terms are involved.
The reader should work out this case as an exercise.
To gain insight into the next result of Barnes [1910], suppose F(s) andG(s)are
the Mellin transforms of /(jc) and g(jc) respectively. The problem is to determine
how the Mellin transform of f(x)g{x) is related to F(s) and G(s). Formally, it is
easily seen that
/00 1 /-oo /-c+zoo
*s-7(jc)g(;c)rfjc - — / x'-lg(x) / F(t)x-'dtdx
Ini Jo Jc-ioo
1 pc~\~ioo poo
= ;r-7 / FW / xs-'~lg(x)dxdt
2ni Jc-ioo Jo
1 pc+ioo
/ F(t)G(s - t)dt. B.4.7)
2ТГI J^^

2.4 The Barnes Integral for the Hypergeometric Function 89
The case of interest to us is where s = 1. Then
/00 1 /-C+/00
f(x)g(x)dx = — / F{t)G{\ - t)dt. B.4.8)
ШI Jc-ioo
Apply this to the Mellin pairs
tt л х" cv л r(b + s)r(a-b-s)
f(x) = ?ГТ7?' F(s) =
and
xd nr^ r(d + s)T(c-d-s)
to obtain
1 fk+i0° Г(Ь + s)T(a -b- s)T(d + 1 - s)V(c - d - 1 + s)
iTii Л-,-» Г(а)Г(с)
-L
00 xb+d T{b+d+ \)T{a +c-b-d-l)
-dx =
Г(а+с)
for a suitable k. By renaming the parameters, this can be written as
1 rlOQ
I 1 id
2Л7 J-ioo
B.4.9)
This formula is due to Barnes and the above proof is due to Titchmarsh [1937]. It
is correct when Re(a, b, c, d) > 0. We give another proof, because we have not
developed the general theory of Mellin transforms in a rigorous way here. But
first note that, if we take f(x) = jc?A - jc)^ and g(x) = xc-\\ ~ x)d+c~x
in B.4.8), we get
2лi Л-ioo r(b + s)T(d-s) T(b-a)T(d-c)T(b + d-Y)
max(-a, -b) < к < min(c, d). B.4.10)
Theorem 2.4.2 If the path of integration is curved to separate the poles of Г
{a + s)F(b + s)from the poles ofV(c - s)F(d - s), then
1 f'°°
I := : Г(а+ s)T(b + s)T(c - s)T(d - s)ds
Г (a + с)Г(а + d)T(b + с)Г(Ь + d)
Note that a+c, a + d,b + c, b + d cannot be 0 or a negative integer.

90 2 The Hypergeometric Functions
Proof. As in the proof of the previous theorem, use Euler's reflection formula to
write the integrand as
ГA — с + s){\ — d + s) sinn(c -:
Also, let L be a closed contour formed by a part of the curve in the theorem
together with a semicircle of radius R to the right of the imaginary axis. By
Stirling's theorem (Corollary 1.4.3), the integrand is o(sa+b+c+d~2e'2j']lms]) as
\s\ —>¦ oo on L. So the integral in Theorem 2.4.2 converges, but for JL we see
that Im s can be arbitrarily small when \s\ is large. Thus, we have to assume that
Re(a + b + с + d — 1) < Oto ensure that JL on the semicircle tends to 0 as
R ->¦ oo. By Cauchy's residue theorem,
_ ^ Г (а + с + п)Г(Ь + с + n)V(d - с - и)(-1)и
^ Г(а + d + п)Г(Ь + d + п)Г(с - d - п)(-\)"
= Г(а + с)Г(Ь + c)T(d - cJFi i ~ •--¦-. х
1 +с -d
Г(а + d)T(b + */)Г(с - dJFx (, ^
V 1 +J-
The 2^iS can be summed by Gauss's formula. After some simplification using
Euler's reflection formula and trigonometry, the right side of the theorem is ob-
obtained. This is under the condition Re(a + b + с + d — 1) < 0. The complete
result follows by analytic continuation of the parameters a,b,c,d. ¦
Theorem 2.4.2 is the integral analog of Gauss's summation of the jF\ at x = 1.
Moreover, if we let b — e — it, d = f — it, and s = itx in the theorem and let
t —> oo, we get, after some reduction employing Stirling's formula,
i:
о i ya-t с + е + j )
Thus, Barnes's integral formula is an extension of the beta integral on @, 1) and
so will be called Barnes's beta integral. It is also called Barnes's first lemma.
Theorem 2.4.2 can also be used to prove that
(a,b \ T(c)T(c-a-b) ( a,b
Г(с)Г(а + Ь-с) c^a_b ( c-a,c-b
+ Г(а)Г(Ь) ( ~X) lF

2.4 The Barnes Integral for the Hypergeometric Function 91
с — a — b ф integer. The proof is an exercise. This result was derived from the
hypergeometric differential equation in the previous section.
The next theorem gives an integral analog of the Pfaff-Saalschiitz identity.
Theorem 2.4.3 For a suitably curved line of integration, so that the decreasing
sequences of poles lie to the left and the increasing sequence of poles lies to the
right of the contour:
2л- i J_
100 Г(а + s)T(b + s)r(c + s)T(l -d- s)r(-s)
ds
—ioo ^ \^ ' S)
Г(д)Г(Ь)Г(с)ГA -d + а)ГA -d + Ь)ГA - d + с)
Г(е-а)Г(е-Ь)Г(е-с)
where d-\-e = a-\-b-\-c-\-\.
Proof. Start with the following special case of Theorem 2.4.2:
Г(с - а)Г(с - b)T(a + п)Г(Ь + n)
Г(с + п)
1 riOQ
= -— / Г(а + s)T(b + э)Г(п - s)T(c -a-b- s)ds.
2ni J-ioo
Multiply both sides by (d)n/[n\(e)n] and sum with respect to n. The result is
Г(с-а)Г(с-Ь)Г(а)Г(Ь) (a,b,d
rw 3
1 Г°°
2tu J_@O
1 Г°° Г
2ni J_ioo T{e-d)
-s d
;
e
Г(е) r(a + s)r(b + s)r(ed + s)r(cabs)r(s)
B.4.11)
Take с = d, so that the 3F2 on the left becomes tF\. Thus we have
Г(а)Гф)Г(е)Г(е -а- Ь)Г(с - а)Г(с - b)
Г(с)Г(е-а)Г(е-Ь)
Г(е)
i J-ioo
-ds.
Г(е - с) 2ni
This is the required result after renaming the parameters. Note that some of the
operations carried out in the proof can be done only after appropriate restrictions on

92 2 The Hypergeometric Functions
the parameters. These restrictions can be removed later by analytic continuation.
The reader can check the details. ¦
A corollary of the proof of the previous theorem is the following interesting
formula:
Theorem 2.4.4
a,b,e-c
F (a'b'C-l) _ r(d)T(dab) / a,b,ec
3 2\ d,e ' ) r(d-a)T(d-bK 2\e,l+a + b-d'
r(d)T(e)r(d + e-a-b- с)Г(а +b-d)
+ T(a)T(b)T(d + e-a- b)T{e - c)
/d — a,d — b,d + e — a—b — с \
'3 2V d + e-a-b,d+\-a-b ' )'
Proof. As a consequence of Cauchy's theorem, B.4.11) is equal to
Г (a + n)T(b + n)T(e -d + n)T(c -a-b- n)(-l)n
п\Г(е
Г(е)
Т(е -d) Ln=Q .... ч- , .., n=Q
T(c-b + n)T(c-a+n)T(c + e-a-b-d + n)T(a + b-c- n)(-l)"
n\T(c + e-a -b + n)
Set this equal to the 3F2 on the left of B.4.11) and the theorem is obtained after
reduction. ¦
Corollary 2.4.5 Ifd + e = a+b+c+\, then
'a,b,c \ T(d)T(e)T(d-a-b)T(e-a-b)
d,e ) T{d-a)T{d-b)T{e-a)T(e-b)
1 T(d)T(e)
a + b-d T(a)T(b)T(d + e - a - b)
d-a,d-b,\ \
+ e —a —b,d+l—a—b' )
Proof. When d + e=a+b + c + l, the first 3 F-i on the right in Theorem 2.4.4
becomes iF\. Evaluate the 2^1 by Gauss's formula and get the result. ¦
Note that when a or b is a negative integer the second expression on the right
vanishes because of the factor l/[F(a)F(b)] and we recover the Pfaff-Saalschiitz
formula. Thus this result is the nonterminating form of the Pfaff-Saalschiitz
identity.

2.4 The Barnes Integral for the Hypergeometric Function 93
The second term on the right in Theorem 2.4.4 vanishes if we take с = e + n — l,
where n > 1 is an integer. The formula obtained is
/ a,b,l-n
/ a,b,ln \
T(d-a)T(d-bY \a + b-d+l,e' )'
B.4.12)
This leads to an interesting result about the partial sums of iF\(a, b; e; 1). Set
d = a + b + n+€ and let e -> 0 to get
a,b
; 1 [to n terms] =
)
п)Г(Ь + п)
; 1 .
)
xFo
\ e ) Г(а + Ь + п)Г(п) \ e,a + b + n
B.4.13)
A particular case, where a = b = | and e = 1, was given by Ramanujan in the
following striking form:
1
1.3
2Л
1
Bailey [1931, 1932] also proved the next, more general, theorem.
Theorem 2.4.6
Г(х + m)T(y + m) fx,y,u+m — l
3^2 ; 1
Г(т)Г(х + y + m) \ u,x + y + m
Г(п)Г(х
-3F2
x,y,u+n-l
to n terms
to m terms.
There is a simple proof from Theorem 3.3.3.
Remark The Mellin transform can be seen as the Fourier transform carried
over to the multiplicative group @, 00) by means of the exponential function.
In F(s) = f™xs-lf(x)dx, write s = a + it and x = е2ли to get F{a + it) =
2л- X!^(/(e2jr") ¦ elnau)elnitudu =: In /^ g(u)e2nitudu. A new feature in the
theory of Mellin transforms is that F(s) is analytic in a vertical strip.
Just as the gamma function, a special case of a Mellin transform, has a finite
field analog, so does the more general case. Let Fq denote a finite field with q
elements and F* its multiplicative part. Let / be a complex-valued function on
F*. Its Mellin transform is denned on the group of characters of F* as F(x) =
YlaeF* X(a)f(a)- The reader may verify that this has an inversion given by
f(s) ~ ~~[ S/ X(a)F(x)- There is also an analog ofBarnes's formula (Theorem
2.4.2) due to Helversen-Pasotto [1978]. A proof based on Mellin transforms is
given in Helversen-Pasotto and Sole [1993].

94 2 The Hypergeometric Functions
2.5 Contiguous Relations
Gauss defined two hypergeometric functions to be contiguous if they have the
same power-series variable, if two of the parameters are pairwise equal, and if the
third pair differ by 1. We use F(a±) to denote 2F\(a ± 1, b; c; x) respectively.
F(b±) and F(c±) are defined similarly. Gauss [1812] showed that a hyperge-
hypergeometric function and any two others contiguous to it are linearly related. Since
there are six functions contiguous to a given 2F1, we get (!j) = 15 relations. In
fact, there are only nine different relations, if the symmetry in a and b is taken into
account. These relations can be iterated, so any three hypergeometric functions
whose parameters differ by integers are linearly related. These relations are called
contiguous relations. In this section we show how Gauss's fifteen relations are de-
derived. Then we briefly point out connections with continued fractions and orthog-
orthogonal polynomials.
Contiguous relations can be iterated and we use the word contiguous in the more
general sense when the parameters differ by integers. It is easily verified that
d r(a,b \ ab /a + l,b+l \
-riFii ;x )=— 2FA ;x\. B.5.1)
ax \ с J с \ с + \ J
Since this 2F\ satisfies the equation
x(l - x)y" + [c - (a + b + \)x]y' - aby = 0,
we get the contiguous relation
l) fa + 2, b + 2
—2Fi[ ;x
c(c
By means of transformation formulas, this can be changed into other contiguous
relations.
Apply Pfaff's transformation
г, b \ _a fa,c — b x
с '*) ~ X 21V с 'x-1,
to each term in the above equation. After a little simplification where we set
и — x/(x - 1) and replace с — b by b, the result is
v(a,b \ c+(a-b+l)u fa + l,b
2ri[ ; и = 2^11 , i :'
\ с J с \ с +1
(a + l)(c — b + 1)m /a+2,b .
2fA . ' ;u . B.5.3)
c(c+l) \ c + 2

2.5 Contiguous Relations 95
This is a contiguous relation due to Euler, who derived it in a different way. If we
apply Euler's transformation
с — а,с — b
с
to B.5.2), then we get another contiguous relation:
'a,b \ c-{2c-a-b + Y)x ( a,b
' -x . B.5.4)
c(c + 1)
This is one of the relations Gauss obtained. Euler's method for obtaining B.5.3)
was to use his integral representation of 2^1- By direct integration he found a
formula of which the following is a particular case:
1
ta~\\ -t)c~a~\\ -tx)~bdt
0
= (c + (a + 1 - b)x) I ta(\- 0е"" A - txybdt
Jo
ta+\\ - t)c'a-\\ - txybdt. B.5.3')
0
This is identical with B.5.3). See Exercise 23 for a simple way of proving this
identity. As another example of how the integral can be used, observe that
A — xt)~~a = A — xt)~a~x{\ — xt) = A — xt)~"~l[l — x + A - t)x].
Substitute the right side in
; x =
)
; x ггмг
с ) Г(Ь)Г(с-
to obtain
/ Cl-
Clio
~ldt
\x )=(l-xJFi[ ¦x)+ 2^1 ,, \x).
с ) \ с ) с \ с + 1 )
The above examples show how contiguous relations arise. Now we give a
derivation of Gauss's basic contiguous relations. It is enough to obtain a set of
six relations from which Gauss's fifteen are obtained by equating the Сл pairs of

96 2 The Hypergeometric Functions
them. The first three in this set are
where
dF
x—=a(F(a+)-F), B.5.5)
dx
dF
x =b(F(b+)-F), B.5.6)
dx
dF
x— = (c-l)(F(c-)-F), B.5.7)
dx
a' ;xj and F(a+) := 2FX (° + ' ;x
and so on. The nth term of F(a+) - F is
\(a+\)n(b)n (a)n(b)n] „ (a + l)n-i(b)n
x = (a + n — a)x
I n\(c)n n\(c)n \ n\(c)n
а я!(с)„
and
„
X = > ———X".
dx ^ n\(c)n
This proves B.5.5). Formula B.5.6) follows by symmetry in a and b, and B.5.7)
is proved similarly. To obtain the other three equations, set 8 := x ? and verify
that the hypergeometric equation can be written as
[8(8 + c-l)-x(8+ a)(8 + b)]y = 0.
So,
[8(8 + с - 1) - x(8 + a - 1)(<5 + b)]F(a-) = 0.
Now
8(8 + с - 1) = (8 + a - 1H$ +c-a)-(a- l)(c - a)
so that
[(8+c-a)- x(8 + b)](8 + a - l)F(a-) = (c - a)(a - l)F(a-).
Apply B.5.5) in the form (8 + a - l)F(a-) — (a - l)F to the above equation to
get
[(8+c-a)- x(8 + b)]F = (c - a)F(a-),
or
dF
x(l - x)— = (c - a)F(a-) + (a - с + bx)F, B.5.8)
dx

2.5 Contiguous Relations 97
The remaining two relations are
dF
x(l-x) — = (c - b)F(b-) + (b - с + ax)F, B.5.9)
ax
dF
c{\ -x) — = (c - a)(c - b)F(c+) + c(a + b - c)F. B.5.10)
dx
Formula B.5.9) is obtained from B.5.8) by symmetry in a and b, and B.5.10)
is proved in a manner similar to B.5.8). Gauss's fifteen contiguous relations are
obtained by equating two values of x^ in B.5.5) to B.5.7) and in B.5.8) to
B.5.10). For example,
[c-2a-(b- a)x]F +a{\ - x)F(a+) - (c - a)F{a-) = 0
follows from B.5.5) and B.5.8), whereas B.5.5) and B.5.10) give
c[a - (c - b)x]F - ac(l - x)F(a+) + (c - a)(c - b)xF(c+) = 0.
We have seen a special case of the last contiguous relation. Set x = 1 to obtain
c(c-a-b)F[ ; 1 = (с - a)(c-
We used this relation in Section 2.2 to derive Gauss's formula for 2^1 (a, b; c; 1).
The reader should derive a few more contiguous relations as an exercise. It must
also be clear that once a contiguous relation is given, it is very easy to verify by
considering the coefficient of xn in each term. For example,
(a,b \ fa,b+l \ a{c-b) fa + l,b+l
2Fi{ ; x = 2^11 ; x \ Х2Г1 ; x
\ с J 2 \ c+\ ) c(c+\) \ с+ 2
B.5.11)
is true because
(a)n(b + !)„ a(c - b) {a + l)n_i(b + !)„-! = (a)n(b)n
n\(c + l)n c(c+l) (n — l)!(c + 2)n_! n\(c)n
Gauss used B.5.11) to obtain an interesting continued fraction for the ratio of two
associated hypergeometric series. Rewrite B.5.11) as
2Fi(a, b; c: x) a(c — b) 1
- j y .
— I —
_ Vxx
U2X

98 2 The Hypergeometric Functions
where
(a+n-l)(c-b + n-l) (b+n)(c-a+n)
u and vn =
(c + 2n~2)(c + 2n- 1) (c + 2n-l)(c + 2n)
Surprisingly, Euler had earlier found a different continued fraction for 2F\ (a, b;
c; x)/2Fi(a, b + 1; с + 1; x). This comes from B.5.3). Interchange a and b in
B.5.3) and rewrite it as
2F\{a, b; c\ x) (b + l)(c — a + \)x
We now have the continued fraction
{b + \)(c - a + \)x (b + 2)(c-a+2)x
с + A + fe я)х (с + j + B + fo _ fl)x)_ (с + 2 + C + b _ (
It is clear that numerous examples of continued fractions for appropriate ratios of
hypergeometric functions can be obtained in this way. Note that B.5.2), which is
just the differential equation for the hypergeometric equation, also gives a contin-
continued fraction for 2F\(a + I, b + 1; с + 1; x)/2F\(a,b; c; x). It involves quadratic
terms in x and is given in Exercise 26. We have not given conditions for conver-
convergence of these infinite continued fractions. The reader should see Lorentzen and
Waadeland [1992] for a discussion of convergence of continued fractions. Also
see Berndt [1985, pp. 136-137] for the reference to Euler's continued fraction and
for the work of Ramanujan on this topic.
One way of arriving at a connection between hypergeometric functions and
orthogonal polynomials (defined below) is through a formula of Jacobi, which we
now derive. Multiply the hypergeometric equation by xc~l(l — x)a+b~c and write
it as
— [x(l - x)xc-'(l - x)a+h-cy'] = abxc-\\ - x)a+b-cy,
dx
where у = 2F\(a, b\ c; x).
From B.5.1) it follows that the derivative of a hypergeometric function is again
hypergeometric, so that it satisfies the differential equation with a, b, с changed
toa + l,fe+l,c+l.By induction, this implies that
— \xk(l -х)кМуЩ =(a+k- l)(b + k- 1)**-Ч1 -x)k-lMy«-l\
dx J
where M = xc~x A — x)a+b~c. Consequently, we have the recurrence relation
— [х\1-х?МУ™}={а+к-\)ф + к-\I-ГГ1
= (a)k{b)kMy.

2.5 Contiguous Relations 99
Substitute
(k) _ (a)k(b)k /a + k,b + k
У (c)k l\ c+k
in the above equation to get
dk \ к,л ,кг, г fa + k,b + k \] fa,b
—- x A - x) M2F\ I ; x I = {c)kM2F\ I ;x
If b is a negative integer — n, then for к — n we have Jacobi's formula
2Fi(~n'a;x) =-—A~X) ^—[xc+n~\\ -х)а~с1 B.5.12)
\ с J (c)n dx"
For a more symmetrical expression in B.5.12) set x = A - y)/2, с = a + 1,
and a = n + a + fi + 1 to get
n,n+a+p + 1 1 -y
a + 1 ' 2
- v)-a(l + v)-^, _„ cf
Definition 2.5.1 The Jacobi polynomial of degree n is defined by
a + 1 ' 2
One of its fundamental properties is that
r+i
P(a,p)/ \p(a,p)/r\/i уЛаП -I- y\PHy
-1
+ р + 1)
1)Г(л + a + P + 1)л! ""*'
This is easy to prove. Use B.5.13) and integration by parts.
Remark 2.5.1 Formula E.13), which can be written as
A -xf(l +хI>Р<а-»(х) = t^-?.[(l -xf+a{\ +хГ^], B.5.13')
is often called the Rodngues formula for Jacobi polynomials. The particular case
for Legendre polynomials, where a = fi = 0, was published by O. Rodngues in
1816 in an Ecole Poly technique journal. Unfortunately, Rodrigues's paper did not
receive much attention. The formula was rediscovered independently by J. Ivory
in 1822 and Jacobi in 1827. It is amusing to note that Jacobi later suggested to
Ivory that they write a joint paper on this important formula and publish it in France
as it was not known there. Their paper appeared in Liouville's journal in 1837.

100 2 The Hypergeometric Functions
Interestingly, Rodrigues's teacher, Laplace, in the course of his work in probability
A810-11), found a similar formula [see F.1.3)] for Hermite polynomials. For
references, see Roy [1993].
A set of polynomials {pn(x)} is called orthogonal if there is a positive measure
d/x(x) with finite moments of all orders so that
oo
Pn(x)pm(x)dix(x) = 0, тфп.
—oo
Thus, the Jacobi polynomials are orthogonal with respect to the measure
d (x)= (A-х)аA+х)Ых, -1<х<1,
\ 0, otherwise.
We shall see in Chapter 5 that any set of orthogonal polynomials satisfies a
three-term recurrence relation:
хр„(х) = Anpn+i(x) + Bnpn{x) + Cnpn-i(x),
pO(x) = 1,/>_!(*) = 0,
An, Bn, Cn+i real, AnCn+i > 0, и = 0, 1,....
Conversely, any set of polynomials that satisfies this recurrence relation is orthog-
orthogonal with respect to a positive measure, which may not be unique.
The three-term recurrence relation for Jacobi polynomials comes from the con-
contiguous relation
2b(c-a)(b-a-
c
- [A - 2x)(b -a-lK + (b- a)(b + a- l)Bc -b-a- l)]2F{( °' *
V с
-2a{b-c)(b-a+\JFAa+ '*" ; x ) = 0, B.5.15)
V c )
after proper identification. In particular we require that a = —n, where и is a
positive integer. But B.5.15) continues to hold when the series does not terminate.
Another contiguous relation that gives a set of orthogonal polynomials is
a(l-xJFi(a+^b;x\ + [c - 2a - (b - a)x]2Fx( "' ; x
B.5.16)
We shall study orthogonal polynomials in detail in later chapters. That will
provide the natural setting for some of the contiguous relations in the sense implied
by the above remarks.

2.5 Contiguous Relations 101
Kummer [1836] considered the problem of extending the contiguous relations
to pFq, but he stopped with the remark that for 3F2 (a, b, c; d, e; x) the formulas
are more complicated. In particular, the linear contiguous relations require four
functions, a 3 F2 and three contiguous to it. Kummer also noted that only when
x = 1 did the formulas simplify. That is the key to three-term relations for some
higher p and q. These will be discussed in the next chapter.
Remark 2.5.2 It should be noted that the continued fraction from Gauss's re-
relation B.5.11) contains a continued fraction for arctanx as a special case. The
Taylor series for arctan x converges very slowly when x = 1, but the convergence
of the continued fraction is extremely rapid and was once useful in computing
approximations of n. See Exercise 25.
Remark 2.5.3 The orthogonality relation B.5.14) is a generalization of the well-
known fact from trigonometry:
cos тв cos n9d9 = 0 Ытфп. B.5.17)
о
This becomes clear by setting x = cos# and cosnd = Tn(x). Then Tn{x) is a
polynomial of degree n and B.5.17) becomes
/ Tm{x)Tn{x){\ -x)-1/2(l+x)-1/2dx=0 Ытфп.
It is not difficult to show that Г„(х) = CPn(-1/2'/2)(x),whereC = Bn)!/[22n(n!J].
Another set of polynomials is
n < m sin(n + 1N/
Un(cos9) =
Tn(x) and Un(x) are called Chebyshev polynomials of the first and second kind
respectively. The three-term recurrence relation for Tn (x) is
xTn(x) = -Tn+1(x) + -rn_i(x).
This is the trigonometric identity
2cos#cosn# = cos(n + \)в + cos(n - 1)#.
Several properties of the Chebyshev polynomials translate to elementary trigono-
trigonometric identities and they form the starting point for generalizations to Jacobi
and other sets of orthogonal polynomials. For this reason the reader should keep
them in mind when studying the "classical" orthogonal polynomials. A number
of exercises at the end of this chapter deal with Chebyshev polynomials. We also

102 2 The Hypergeometric Functions
refer to
sin{Bn + 1)9/2} cos{Bn + 1N/2}
Vn(cos0) = r—- and Wn(cos9) = —
sin #/2 cos #/2
as Chebyshev polynomials of the third and fourth kinds respectively. These poly-
polynomials are of lesser importance.
2.6 Dilogarithms
All the examples given in Section 2.1 of special functions expressible as hyper-
hypergeometric functions were either 2Fi or of lower level. The dilogarithm function
is an example of a 3F2. This function was first discussed by Euler and later by
many other mathematicians including Abel and Kummer. But it is only in the
past two decades that it has begun to appear in several different mathematical
contexts. Its growing importance is reflected in the two books devoted to it and
its generalization, the polylogarithm. See Lewin [1981, 1981a]. Here we give
a few elementary properties of dilogarithms. The reader may also wish to see
Kirillov [1994] and Zagier [1989]. The latter paper gives a number of interesting
applications in number theory and geometry.
The dilogarithm is defined by the series
:=2^—, for |x| < 1. B.6.1)
71 = 1 П
From the Taylor expansion of logA — t) it follows that
Li2(x) = - / — -dt. B.6.2)
Jo t
The integral is defined as a single-valued function in the cut plane С — [1, oo); so
we have an analytic continuation of Li2(x) to this region. The multivaluedness of
Li2(x) can also be studied easily. There are branch points at 1 and oo. If Li2(x)
is continued along a loop that winds around x — 1 once, then the value of Li2
changes to Li2(x) — 2ni logx. This is easily seen from the integral definition.
We now obtain the hypergeometric representation from the integral
Г (hi \ fl (hi \ /
Li:(x)= / 2Fi( 2 ¦,t\dt = xl 2FA ^ \xu\du = xiF1[
1,1,1
by formula B.2.2). Though it is possible to develop the properties of the diloga-
dilogarithm without any reference to the theory of hypergeometric series, we note one
example where Pfaff 's transformation is applicable.

2.6 Dilogarithms 103
Theorem 2.6.1 Li2(x) + Li2(x/(x - 1)) = -j[log(l - x)]2 (Landen's trans-
transformation).
Proof. By Pfaff's transformation (Theorem 2.2.5)
Set и = t/(t - 1) in the last integral to get
x/{x-\) /, ! \ du
2FA ;u
о V 2 У 1-й
The first integral is Li2(x/(x — 1)) and the second integral is
>dll = j x)
й 2
/о i-м 2
This proves the result. ¦
We give another proof since it involves a different expression for the dilogarithm
as a hypergeometric function. This expression is given by
Li2(x) = lim 1 Lfi ( *' ^; x J - l|. B.6.3)
Let x be in the region {x||x| < S < 1} П {x\\x/(x - 1)| < S < 1} = S5, where
5 > 0. Apply Pfaff's transformation to B.6.3) to get
Li2(x) — lim —
2
1 f / 6
= lim -{( 1 -t log(l -x) + -[log(l -x)]2 +
Now,
00 t / чи 00,/ \ n 00/-. -.
-1
X- 1

104
Thus,
Li2(x) = 1
2
(
The Hypergeometric Functions
: log(l — x) H [log(l — x)]
2
= ~[log(l-x)]z-Li2f X
The limit operation may be justified by the fact that for x e S$ and |e| < 1/2, the
relevant series represent analytic functions of x and e. This proves the theorem
again. There is another proof in Exercise 38.
Theorem 2.6.2
Ifw" = \, then -Li2(x") = V Li2(a/x). B.6.4)
n z—'
-Li2(x2) = Li2(x) + Li2(-x). B.6.5)
2
Li2(x) + Li2(l -x) = — -logxlog(l -x). B.6.6)
6
Proof. To prove B.6.4) start with the factorization A -1") = (I - t)(l - wt) ¦ ¦ ¦
A — con~lt). Take the logarithm and integrate to get
fx log(l -t") Г log(l -1)
- / dt = - dt
Jo * Jo t
fx log(l - wt) fx log(l - wn~lt)
- / dt / dt.
Jo t Jo t
A change of variables shows that the integral on the left is ^Li2 (x"). This proves
B.6.4), and B.6.5) follows by taking n = 2.
To derive B.6.6) integrate by parts:
?. , ч Г logd - 0 J , , /л Гх log?
Li2(x) = -/ -5 dt = -\ogxlog(\ -x)- / —^rff.
Jo t Jo 1 — t
The last integral, after a change of variables и = 1 — t, is
i-'-Mogd - и)
-<зм = Li2(l — x) —

2.6 Dilogarithms
105
But
2
by Theorem 1.2.4. The proof of the theorem is complete. ¦
Apparently, the only values x at which Li2(x) can be computed in terms of more
elementary functions are the eight values x = 0, ±1, |, ''^ , ^y^, ^^.
Theorem 2.6.3
Li2(O) = О,
Li2(D = ^-,
6
12
n
Г2'
2
Li2
Ll2
V5
15 4L""°V 2
'л/5-Р12
1
7Г2 1
л/5-l
V5+
B.6.7)
B.6.8)
B.6.9)
B.6.10)
B.6.11)
B.6.12)
B.6.13)
B.6.14)
Proof. Relation B.6.7) is obvious and B.6.8) was done in the proof of Theorem
2.6.2.
For B.6.9), observe that
7Г2
111
7T2
6 22 ' 6
\2
Set x = | in B.6.6) to get B.6.10).

106 2 The Hypergeometric Functions
The identities B.6.11) and B.6.12) can be derived as follows: Landen's trans-
transformation and B.6.5) combine to give
Li2 (j^-j-) + ^Li2(x2) - Li2(-x) = -^[log(l - x)]2. B.6.15)
Set the variables in the first two dilogarithmic functions equal to each other.
Then x/(x - 1) = x2 and x2 - x - 1 = 0. A solution of this is x = A - л/5)/2.
Substitute this x in B.6.15) to obtain
To find another equation involving Li2(C — л/5)/2) and Li2((l — л/5)/2) take
x = C - л/5)/2 in B.6.6) to arrive at
3-V5
Now solve these equations to obtain the necessary result. The proofs of the for-
formulas B.6.13) and B.6.14) are left as exercises. ¦
There are also two variable equations for the dilogarithm. The following is
usually attributed to Abel though it was published earlier by Spence. See Lewin
[1981a] for references. The formula is
- log(l-x)log(l-y). B.6.16)
This is easily verified by partial differentiation with respect to x or у and is left to
the reader.
More generally we can define the polylogarithm by the series
oo „
Limx:=V— for|x| < 1,/и =2, 3, .... B.6.17)
Z—/ „m
The relation
, n"
--Lim(x) = -Lim_
ax x
is easy to show and one can use this to define the analytic continuation of Lim(x).
The polylogarithm can be expressed as a hypergeometric function as well. The
formula is
Lim(x)=xm+1Fm( ' ""'
\ z,..., z

2.7 Binomial Sums 107
We do not go into the properties of this function any further but instead refer the
reader to Zagier's [1989] article and the books mentioned earlier.
2.7 Binomial Sums
One area where hypergeometric identities are very useful is in the evaluation of
single sums of products of binomial coefficients. The essential character of such
sums is revealed by writing them as hypergeometric series. Sums of binomial
coefficients that appear to be very different from one another turn out to be examples
of the same hypergeometric series. One reason for this is that binomial coefficients
can be taken apart and then rearranged to take many different forms. A few
examples given below will explain these points.
Consider the sum
n /k\fk-l-j\ n
+7
j=0 J ^ j=0
To write this as a hypergeometric series, look at the ratio cj+\/cj as we did when
we defined these series. A simple calculation shows that
cj (J-k+ 1H" + 2)
So
Л k-l
Now, as explained in Section 2.1, we could introduce j\ — A), in the numerator
and denominator to get
We have learned to sum two 3F2 series: the balanced and the well-poised series
(see Section 2.2). This is neither though it is actually "nearly" poised. However,
we have not yet considered this type of series. But there is another way out. Note
that the denominator has B)j, which can be written as (l)y+i. Then
S =
k-l\ (-k) *
k-l\ (-k)
y
(n + l)(k + I) j^ QM-k)t
(-к) f
!)(* + !) L V -к
f „l,*l _

108
2 The Hypergeometric Functions
The 2-Fi can be evaluated by the Chu-Vandermonde formula and after simplifica-
simplification we get
S =
1
Jfe+1 [\n
As another example take the sum
(-1)"
k>0
n+k \Bk\(-l)k
k>0
Here
after simplification. Thus
(k + n + \)(k+\)(k-m + n)
(jt + f + l)(t+=±i)(Jt + 2)
5 =
m
— 1 — n + m, n, —I
;
Г1
2 ' 2
The 3F2 is balanced and we can apply the Pfaff-Saalschiitz identity. We get
s =
m
(— -n) (-)
\ 2 /n+l-m V 2 /n+l-m _ i
(m=l) (- — n)
.V 2 /n+l-m V 2 "Jn + l-m
which simplifies to {nm_x)¦ The reader may verify that
E
k>0
n-k I \m +n
also reduces to an example of a Pfaff-Saalschiitz series.
As the final example take the series
In
= S,
where we are assuming that I = min(€, m,n). This reduces to the series
(/n - ?)!(/n + t)\(n - t)\(n +t)\
This is a well-poised series (see Section 2.2) that can be summed by Dixon's for-
formula. However, this result cannot be applied directly because we get a term

2.8 Dougall's Bilateral Sum 109
ГA — ?)/ ГA — 21) that is undefined. A way around this is to use the following
case of Dixon's formula:
-2г-2е,-т-г-е,-п-г-е
m-l-e + l,n-l-€ + l '
ГA - ? - е)Г{1 + m - ? - е)ГA + m +n + ? + e)
~ ГA -21- 2е)ГA + /п)ГA + и)ГA + т + п)
Apply Euler's reflection formula to the right side to get
+т -1-е)ГA +п-?-€)ГA+т+п
In the limit as e —> 0, this expression is
. B? - 1)! (m - l)\{n - ?)\{m
(?-\)\ m\n\(m+ri)\
Thus
(? + m + n)\B?)\Bm)\Bn)\
S =
(? + m)\(? + n)\{m + n)\?\m\n\'
Another example that gives a well-poised 3F2 is J2l=i ^H^I )(?+„)• Examples
can be multiplied but the discussion above is sufficient to explain how hyperge-
hypergeometric identities apply to the evaluation of binomial sums. See Exercise 29 in
Chapter 3.
2.8 Dougall's Bilateral Sum
The bilateral series
Г(а+п)Г(Ь + п)
is the subject of this section. In fact, the hypergeometric series 2^2 (a,b\ c; 1)
should be regarded as a special case of the bilateral series where d = 1, since
1/ Г(и) — 0 for nonpositive integers n. This explains why we introduced 1/n! in
the nth term of the hypergeometric series.
The above remark gives us a way of evaluating the above bilateral series. Ford =
1,2,..., the sum reduces to a series that can be evaluated by Gauss's summation
of iF\(a, b; c; 1). The following theorem of Carlson allows one to evaluate the
bilateral series from its values at d = 1,2,....

110 2 The Hypergeometric Functions
Theorem 2.8.1 Iff(z) is analytic and bounded for Re z > Oandif f(z) = Ofor
z = 0,1, 2,..., then f(z) is identically zero.
Remark The boundedness condition can be relaxed. We need only assume that
f(z) = O(e*|z|), where к < it. The simple proof given below of the particular
case is due to Selberg [1944].
Proof. As a consequence of Cauchy's residue theorem,
f(z)
= (a-\)(a-2)---(a-n) f°
(z-aKz-l)---(z-n)
for n > a > 0. Then, for a > 1,
Z
2л-
2nn\
Let n -»¦ oo to see that /(a) =0 for all real a > 1. This implies the theorem.
Theorem 2.8.2 (Dougall) For 1 + Re(a + b) < Re(c + d),
Г(а+п)Г(Ь+п) n2 F(c + d-a-b-l)
п=оо Г(с + n)V(d + n) sin л-a sin nb Г(с - а)Г(^ - а)Г(с - 6)Г(^ - &)
Proof. For Re<i > Re (a + b — c) + 1, the functions on both sides are bounded
analytic functions of d. Let m be an integer in this half plane. For d = m the
series on the left is
, Г(с + п)Г(т+п)
Г(а-т+ \)Г(Ь -т + \)^(а-т + \)еф -
T(c-m
This series can be summed by Gauss's iF\ formula. Thus Dougall's result can be
verified for d equal to an integer in the half plane. Carlson's theorem now implies
Theorem 2.8.2. ¦
Gauss's iF\ sum is itself a consequence of Theorem 2.8.1. We have to prove
that
iFA t"";l) =
с J Г(с-а)Г(с

2.9 Fractional Integration by Parts and Hypergeometric Integrals 111
This relation is true for b = 1, 2,... by the Chu-Vandermonde identity, and then
by the above argument the general case follows.
For a different example where Carlson's theorem applies, consider the formula
i:
xa~l{l-.
It is easy to prove this by induction when a is a positive integer. We saw this in
Chapter 1. The integral is bounded and analytic in Re a > S > 0 and so is the
right side of the formula. This proves the result.
2.9 Fractional Integration by Parts and Hypergeometric Integrals
Theorem 2.2.1 gives Euler's integral representation for a hypergeometric function.
One drawback of this representation is that the symmetry in the parameters a and
b of the function is not obvious in the integral. We observed that Erdelyi's double
integral B.2.5) gives the two different representations by changing the order of
integration. In this section, we show how fractional integration by parts can be
used to transform the integral for 2F\ (a, b; c; x) to another integral in which a and
b have been interchanged. In fact, fractional integration by parts is a powerful tool
and we also use it to prove a formula of Erdelyi. This contains some of the integral
formulas considered in this chapter as special cases. It also has implications in the
theory of orthogonal polynomials, which will be discussed in Chapter 6.
-iet
(//)(*) := / f(t)dt
Ja
and
(hf)(x) ¦= [ f f{h)dndt = f (x- t)f(t)dt.
J a J a J a
Inductively, it follows that, for a positive integer n,
(/„/)(*):= f [¦¦¦[" ' f(ta-i)dtn-i-..dtidt
J a J a J a
{x-tf-lf{t)dt.
(П ~ 1)! Ja
A fractional integral /„, for Re a > 0, is then defined by
'' Г (a) Ja
(/„/)(*) := ¦=$- f (x - t)a-lf{t)dt. B.9.1)

112 2 The Нуpergeometric Functions
The restriction Re a > 0 can be removed by using contour integrals. An in-
interpretation of Euler's integral for 2^1 (a, b; c; x) as a fractional integral is now
evident.
The fractional derivatives can also be defined by the formal relation
'^—w»-\ B.9.2)
dwv
when the right side is meaningful.
To state the formula for fractional integration by parts, suppose и and v are
functions defined by
00 00
Ar(x — ay , v = у
r=0 s=0
Then
fb dvv fb dvu
/ и dx = / v dx, B.9.3)
Ja d(b-x)v Ja d(x-aY
provided that the integrals exist. This can be verified directly by substituting the
series for и and v and their derivatives, which are obtained by applying B.9.2)
term by term. The two are seen to be identical after integration term by term. It is
noteworthy that if Re v < 0 in B.9.3), then B.9.3) is equivalent to the identity
fb \ fb 1 fb Г fx 1
/ u(x)\ I (y-x)"~1v(y)dy\dx= / v(x)\ / (x - y)"~1u(y)dy\dx,
Ja L J x J Ja L J a J
where v = —a. This formula holds because both sides equal the double integral
и u(x)v(y)(y - xf^dxdy.
We now show how to transform one integral representation of 2F1 (a, b; c; x) to
the other. It is clear from B.9.2) that
— f
-b)JQ
T(b)T{c
) f
-a)J0
»-\\-ty-b-\\-xt)-adt
Г{Ь)Г(с
By the integration by parts formula B.9.3), we get
Г(с)
r(b)r(c-a)J0
r1
/ a-ty-a-
Jo

2.9 Fractional Integration by Parts and Hypergeometric Integrals 113
The binomial theorem and B.9.2) give
db-atb-\{y_xtya ^ db-a ~ r(fl + r)f6+r-l xr
dtb-a ~ dtb-a f^ Г(а)г!
2. Г(в)г!
Substitute this in B.9.5) to obtain
r, *}? . ff-\\-ty—\\-xt)-bdt.
Г(а)Г(с-а) Уо
This is the expression in B.9.4) with a and ? interchanged; our claim is proved.
As another example of the use of fractional integration by parts, we re-prove
the following formula contained in Theorem 2.2.4:
:
d
when Rec > Red > 0, x ф 1, |arg(l - x)| < n. Use B.9.2) to see that B.9.4)
is equal to
= Г(с)
Г(Ь
Also,
dtb~d dtb~d f^ Г(а)г!
r=0
Substitute this in the last integral to complete the proof of B.9.6).
We now state and prove the formula of Erdelyi [1939] mentioned earlier.
Theorem 2.9.1 For Rec > Re д > 0, x ф 1, |arg(l — x)| < n, we have
) f t»-\\ - ty-»-\l - xtf—b
-ц) Jo
\xtJFi
Г(ц)Г(с
Х-а,Х-Ь

114 2 The Hypergeometric Functions
Proof. Apply Euler's transformation in Theorem 2.2.5 to the 2F\ inside the inte-
integral in B.9.6). The result is
'<*=*)-
By B.9.2) and the series representation of 2F\, we see that
k-a,k~b \ d^'k { t^~l /k-a,k-b
¦,xt)=-—r\-—2Fl( ;xt .
Substitute this in the last integral and use the fractional-integration-by-parts for-
formula B.9.3) to get
a,b \ Г(с) fx ^_, ^ (k-a,k-b
с "'
¦ -ТГ— Г 1 —^C1 - xt)k-"-b \ ,
d(\ -ty~k \ Г(с -к) J
Write the expression in curly braces as
T(c-k) V 1-
4 ' ^ r\T(c-k)
Take the (/л — A.)th derivative of this expression to obtain
'a + b-k,c-k (l-t)x
c-fi x-1
By Pfaff's transformation (Theorem 2.2.5), this is equal to
1 a + b-k,k-n (l-t)x
;
с -11 \-xt
Substitute this in the last integral. The result is proved. ¦
Exercises
1. Complete the proof of the second part of Theorem 2.1.2 concerning conditional
convergence.

Exercises 115
2. Suppose that YTk=\ak I anc^ S/Ui \ak I tendto infinity in the same way, that is,
< к
Е-
k=\
for all n and К is independent of n. Prove that
= lim **,
provided that the right-hand limit exists.
3. Use the result in Exercise 2 to prove Theorem 2.1.4.
4. (a) Show that I((l + x)" + A - x)") = 2F\(-n/2, -{n + l)/2; 1/2; x2).
Find a similar expression for |(A + x)" — A — x)").
(b) Show that A +x)" = 1 +nx2F\{\ -n, 1; 2; -x).
5. Derive the Chu-Vandermonde identity by equating the coefficient of x" on
each side of A -x)"a(l - x)~b = A -x)-(a+b).
6. Suppose that log(l + x) is defined by the series B.1.3). Use Pfaff's transfor-
transformation (Theorem 2.2.5) to show that log(l + x) = -log(l + x).
7. Show that Pfaff's transformation is equivalent to
-n,c-b \ {b)n
; 1 = -—-, n = 0, 1,....
с ) (с)„
This is one way of removing the restriction Re с > Re b > 0 used in the proof
of Theorem 2.2.5.
8. Prove the identities B.1.10) to B.1.13).
9. Show that xF\(a\ c; x) = ex,Fx(c - a; c\ -x).
10. Suppose x is a complex number not equal to zero or a negative integer. Show
that
;
x+1
r*) Jx
Note that the i F\ series exhibits the poles and residues of Г(х).
11. Show that
/•OO
/
JO
Г(а) (ах,...,ар,а х
b\,..., bq '
when p < q, Res > 0, Re a > 0, and term-by-term integration is permitted.

116 2 The Hypergeometric Functions
12. Show that
cos mx = 2F\[ 2', 2 ; sin2 x I,
V
/ \+m \-m \
sinmx = m sini2fi I 2 3 2 ;sin2xj.
V 2 /
13. Prove that the functions
and
/ 2a, 2fc, a + b
У2 = 3^2 r, , ~, , l , 1 ' X
\2a + 2b, a + b + \
both satisfy the differential equation
x2(x - \)y'" - 3
+ {[2(a2 + b2 + Aab) + 3(a + b) + l)]x - (a + b)Ba + 2b+ 1)}/
+ 4ab(a + b)y = 0.
Thus prove Clausen's identity (see Clausen [1828]),
a,b \J „( 2a,2b,a+b
Also prove that
2^1
' a-b+\,b-a
2' 2. v
14. Show that the Pfaff-Saalschiitz identity B.2.8) can be written in a completely
symmetric form as
(a,b,c \ 7r2r(rf)r(e)[cos7rJcos7re+cos7racos7rfecos7rc]
3 2V d,e ' J ~ r(J - a)Y{d - fe)r(J - с)Г(е - a)T(e - b)T(e - c)
when the series terminates naturally and d+e = a+b+c+l. This observation
was made by R. William Gosper.

Exercises 117
IS. Prove that
k=0
where Tn(x) is the Chebyshev polynomial of the first kind. Find a similar
expression for Un(x), the Chebyshev polynomial of the second kind.
16. Prove the following analog of Fermat's little theorem: If x is apositive integer
and p an odd prime then Tp(x) = T\ (x) (mod p).
17. Pell's equation is x1 — Dy2 = 1, where D is a square free positive integer.
Let S be the set of all positive solutions (x, y) of Pell's equation. Let (x\, yi)
be the solution with least x in S. Show that (Tn(x\), yiUn-\(x\)), n > 1, is
a solution and thus that Pell's equation has infinitely many solutions, if it has
one.
18. Let Sn(x) = Un-i(x) with U_i(x) = 0. Prove that gcd(Sn(x), Sm(x)) =
Sgcd(m,n)(x), where x, m, and n are positive integers.
19. (a) Show that
pn(a,-l/2)Bx2 _ 1)/J(x)A _ X2ydx = Oi
where p(x) is any polynomial of degree <2n — 1. Deduce that
(
(b) Show that
(e,e) _ ГBп + а + 2)п\
п+i U) - Г(л + a
(c) What do (a) and (b) mean for the Chebyshev polynomials of the first and
second kind respectively?
20. Prove that
(« + !)„/1+лсу /-n,-n-ptx-l
n! V 2 J - \ a+l x +
—n, —n — a
n\

118 2 The Hypergeometric Functions
21. Letx = cos 6». Prove that
dn~x sin2" в (-1)" Bn)!
sinn#. (Jacobi)
:=
dx"-x n 2nn\
22. Prove that the Jacobi polynomials Р„(а/3)О) satisfy the three-term recurrence
relation
2(n + l)(n + a + j9 + l)Bn + a f
= Bn + a + P+ l){Bn + «+/? + 2)Bn + a + p)x + a2 - /?2}
x Р^а-Р\х) - 2(n + a)(n + P)
xBn + a + p + 2)Pn("f'(x) =0, n = 1, 2, 3,....
(Compare this with B.5.15).)
23. Prove Euler's contiguous relation expressed as the integral formula B.5.3') by
observing that
0= / — (f"(l -t)c-"(l-tx)l~b)dt, Rec>Rea>0.
Jo at
24. Prove the following contiguous relations:
(a) (c-2a-(b- a)x)F + a(\ - x)F(a+) - (c - a)F(a-) = 0.
(b) (c - a - l)F + aF(a+) - (c - l)F(c-) = 0.
(c) (b - a)(l - x)F - (c - a)F(a-) + (c - b)F(b-) = 0.
(d) c(b - (c - a)x)F - bc{\ - x)F(b+) - (c - b)F(b-) = 0.
25. Prove
x l2x l2x 22x 22x
(b) log(l+x) = —
eV T 1+ 2+ 3+ 4+ 5+
x \2x2 22x2 32x2
(c) arctanx = —
1+3+ 5+ 7+
4 1+ 3+ 5+ 7+ "
The 9th approximant gives n/4 correctly up to seven decimal places.
x+1 2 - - -
(e) lOg - 2 3 4
X i x .x тЛ tX
arcsinx x l-2x2 1 • 2x2 3 ¦ 4x2 3-4x2

Exercises 119
26. Show that
fa,b
\ / fa + l,b+l
с V c+l
(a+2)(fc+2)
_ с - (a +b + l)x xu x; c(c+1) xu x-> (c+Q(C+2)
~~ r ' c+\-(a+b+3)x + c+2-(a+b+5)x +"''-
c+l c+2
27. Use Barnes's integral representation of 2F\ and Barnes's beta integral (Theo-
(Theorem 2.4.2) to prove formula B.3.13). Also consider the case where c — a — b
is an integer.
28. Multiply the equation in B.3.13)byxd(l-x)e"'/ and integrate over @, 1)
to obtain another proof of Theorem 2.4.4.
29. Prove the following formulas of Ramanujan [1927, paper 11]
Kx/O>+1)J l + (x/(b + 2)J j
Г
Jo
1 + (x/aJ 1 + (x/(a + I)J
Г0
X
(b) ' dx
(I + (jc/aJ)(l + (x/(a + I)J) • • • (I + (x/bJ)(l + (x/(b + I)J) • •
^Г(а)Г(а + {)Г(Ь)Г(Ь+{)Г(а + Ь)
r , a > U, b > U.
2 r(a+b+\)
30. Prove that for Re t > 0
oo ,
«=—oo v n——oo
ЯшГ: Denote the left side by /(a) and note that / has period one. Expand
/ as a Fourier series
oo
and observe that
^ A
.„ = Г e-'^e-^^dy.
J — oo
31. (a) Let x be a nontrivial even primitive character (mod ./V). Prove that
oo
n=— oo
where Re t > 0 and
a=l

120 2 The Hypergeometric Functions
Hint: First observe that
n=—00 a=l n=—00
Then apply the result of the previous exercise,
(b) Let x be a nontrivial odd primitive character (mod N). Prove that
32. (a) Show that л "¦Т(я)?Bя) is the Mellin transform of
oo
? e~" nt for Re s > 1.
n=l
(b) Use Exercise 30 to show that
s \ — s
(c) Observe that the expression on the right in (b) does not change under
s -+ I — s. Deduce the analytic continuation and functional equations of
the zeta function.
33. Obtain the functional equation of L(x, s), x primitive (mod N), using Exercise
31 and the idea of Exercise 32. (By Exercise 1.53 the functional equation is
ЩЛ-s)
Ш -_ 8(X) Bл У
VA' ' 2is \N J
Г(s)cosn(s-S)/2'
where S = 0 or 1 depending on whether x is even or odd.)
34. Assuming the functional equation of the zeta function, apply Mellin inversion
to prove that
35. Evaluate the Mellin transform of 2F\ (a, b\ c\ —x) as follows:
f
Jo
Jo
a,c-b_ x
Set и = x/(x + 1) and integrate the 2F\ series term by term.

Exercises 121
36. Let Ft (s) be the Mellin transform of /, (x). Show that
-J-r / Fl(s)F2(s)F3(s)ds
2ЛI Jk-ioo
uv J uv
Deduce Barnes's continuous extension (Theorem 2.4.3) of the Pfaff-Saalschiitz
identity.
37. (a) Prove that
J2 , Ima > O,Res > 1,
n=l n=—oo
by using Carlson's theorem. (The formula is due to Lipschitz.)
(b) Expand 2^°oo(a + n)"f, Im a > 0, as a Fourier series Y,Ame27Tima. Ex-
Express Am as an integral over (—oo, oo). Use the result in (a) to deduce
Hankel's formula for 1/ F(s). See Exercise 1.22.
38. (a) Verify Theorem 2.6.1 by differentiating both sides of the equation.
(b) Verify the Abel-Spence identity B.6.16).
(c) Prove the identities B.6.13) and B.6.14).
39. (a) Prove that
/y\ fl-xl ГуA-х
Li2(x) - Li2()O + Li2 [y- )+ Li2 - Li2
л2 [1-х
= -г -logxlog
6
г logxlog
6 \_\-y
(b) Prove that
Prove that Li2
, x(\-y)
— jc)J
= Li2 \-x---^ +Li2 —
1-
i
+ Li2 J- + -log 2y. (Kummer)
1-х 2
See Lewin [1981a].
(c) Prove that
Li2(x) + Li2(y) - Li2(xy)
T. (x(\-y)\ t T. (y{\-x)
= Ll2 + Ll;
V 1 - xy J
(Rogers)
(d) Show that if 0 i;x < 1 and f(x) e C2(@, 1)) and satisfies B.6.6) and the
functional relation in part (c), then /(x) = Li2(x). See Rogers [1907].

122 2 The Hypergeometric Functions
40. Suppose X is a primitive Dirichlet character mod N. Show that
N
n N
n=l n=\
where g(x) is the Gauss sum defined in Exercise 31.
41. Suppose и is a positive integer. Define
[logo, n = pk, a power of a prime,
Л(и) = <
[0, otherwise.
(a) Show ?(я) = ПрО - P~"TX for Res > 1.
(b) Show that
for Re s > 1.
X
(c) Let i[r(x) = Yln<x Л(п). Show that rjr{x) = 0(x) and
x~s~xilr{x)dx = , Res
s
(d) Prove the inversion formula
yjr{x) = / ds, c>l and x not an integer.
2лi Jc_ioo * ?(•*)
(e) Let
f{t)dt.
x
Show that
rco 1 C'(s) n
, Res
0 5E -'
(f) Prove the inversion of (e),
1 rc+ioo xs
= — -—г /
2ЛI Jc-ioo S(S+
ds, c>\ and x not an integer.
(g) Show that the Mellin transform of J2T=iA(n)e~"x is -T(s)$'(s)/Z(s).
(h) Prove the Mellin inversion formula of (g), that is,
i pc+ioo t'(i) ^L
/ rT(s)—ds=VA(n)rH, о 1, x>0.
2ni Jc_loo y't;{s) V

Exercises 123
42. Let a, b, c, d be complex numbers.
(a) Suppose с ф an integer and Re {a + b) > — 1. Prove that
K=—oc
Л
(a + c+ \)T{b-c+
(b) Suppose Re {a + b + с + d) > -1. Prove that
oo -
/-" T{a-k+ \)T{b -k+ 1)Г(с + к+ l)T(d + k+l)
T(a + b + c + d+\)
~ Г (a + с + 1)Г(Ь + с+ 1)Г(а + d+ 1)Г(Ь + d + 1)'
(с) Which beta integral is extended to the sum in part (b)?
43. This problem gives a quick way of obtaining the differential equation for
у = 2Fi(a, b\ c; x). From the series for у and Euler's transformation, we
have
(a) y' = — 2Fi(a + l,b+l;c+l;x)
с
= —A - x)c-a-b-\Fx{c -a,c-b;c+l; x),
and
(b) ^-[xc-x2Fx{a,b;c;x)} = (с - l)xc-22Fx(a, b; с - 1; x).
dx J
Use (a) and (b) to see that
— [x'(l - x)a+b+l-cy'] = abxc-l2Fx(c -a,c-b;c; x)
dx
= abxc-\l-x)a+b-cy.
The differential equation is obtained after computing the derivative on the
left side.
44. Evaluate
/¦7Г/2
J-Tit
1-71/2
by use of the binomial theorem and Theorem 2.2.1, and so show that
лТ(а + ? + 1)
/ (cos 0)a+fl cos(a -
Jo
1)'

Hypergeometric Transformations
and Identities
Gauss's work on the hypergeometric equation contains a discussion of the mon-
odromy question for the solutions of this equation. Gauss found and analyzed a
quadratic transformation of hypergeometric functions; this apparently led him to
the problem of monodromy. Unlike the linear (fractional) transformations of these
functions, of which Pfaff's formula in Theorem 2.2.5 is an example, quadratic
transformations exist only under certain conditions on the parameters. Neverthe-
Nevertheless, they are important and useful. We have given some applications of these
transformations after deriving a few basic formulas. An interesting application
deals with the problem of proving Gauss's arithmetic-geometric mean to be ex-
expressible as an elliptic integral.
This chapter also contains a discussion of some methods for the summation
of certain types of hypergeometric series. We use a quadratic transformation to
obtain Dixon's identity for a well-poised 3F2 at x = 1. We then apply a method of
Bailey to derive identities for special types of p+\ Fp with 2 < p < 6, including
Dougall's identity, which was mentioned in Remark 2.2.2 in the previous chapter.
An important transformation formula due to Whipple is obtained by the same
method. Just as Barnes's integral on the product of gamma functions was an analog
of Gauss's 2 F\ sum, these identities also have integral analogs and we discuss them.
The hypergeometric identities provide a systematic approach to the evaluation of
single sums of binomial coefficients.
Contiguous relations for hypergeometric series contain an enormous amount of
hidden information. There are three-term relations for balanced 4F3 functions as
noted by Wilson [ 1977] and independently by Raynal [ 1979] and for 3 F2 functions
at x — 1, a fact pointed out much earlier by Kummer. We describe Wilson's simple
technique for deriving the contiguous relations that contain the three-term recur-
recurrence for Wilson polynomials. These 4F3 polynomials contain a whole range of
classical orthogonal polynomials as special or limiting cases. We devote a section
of this chapter to the definition and orthogonality of Wilson polynomials. They are
124

3.1 Quadratic Transformations 125
orthogonal with respect to a weight function that occurs as the integrand in an
integral analog of a 5 F4 identity.
Gosper, Zeilberger, and Wilf have done significant work toward devising com-
computer algorithms for finding and proving hypergeometric identities. We discuss the
Wilf-Zeilberger method and compare it with that of Pfaff; both are applications
of contiguous relations.
3.1 Quadratic Transformations
Exercise 2.19 asks the reader to prove the following relation satisfied by Jacobi
polynomials:
lJn p(g,-l/2),2 2 _ n ГЗ 1 П
This important formula can be stated in hypergeometric form as
/-2и,2и + 2а +1 \ (-n,n + a + \ Д
2FA ^ ;(l-x)/2)=2F,( + . 2\l-x2 . C.1.2)
Note that the 2F\ on the left is linear in x and the one on the right is quadratic in
x. This is an example of a quadratic transformation. In this section we give the
fundamental results on quadratic transformations with two free parameters.
It is natural to suspect that C.1.2) continues to hold when the two series do not
terminate. This can be shown directly by rewriting C.1.2) as
2a,2b \ ( a,b \
, ;x =2^1 , ;4x(l -x) C.1.3)
:+fo+2 / \a + b+2 J
and expanding the right-hand side as a power series in x. The coefficient of x" is
a balanced 3F2, which can be summed by the Pfaff-Saalschiitz identity (Theorem
2.2.6). However, two cases have to be considered: n even and n odd. An equivalent
identity C.1.4) does not have this problem.
Apply Pfaff's transformation (Theorem 2.2.5) to the left-hand side and re-label
the parameters and variable. The result we want to prove is
Theorem 3.1.1 For all x where the two series converge,
a,b \ „ ч_„ „/a/2, (l+a)/2-b -4x
C.1.4)
u-oti (i — X)" j
Proof. Write the series on the right as
y^ Kul^)kK~" T V" T L)l*-)k , ,fc.j ч-я+2*
^ Ща-Ь+\)к
^л (a/2)k(—b + (a + l)/2)t j.^(a + 2fe); •
= > (-Ax) > -x1.
L~l k\(a-b+\)k z—' /'!
к=0 y=0

126 3 Hypergeometric Transformations and Identities
It is easy to see that the coefficient of x" in the last expression is
(-b + (a + l)/2)t(-4)*(fl + Ы)п-и
f^ (a - b + l)kk\(n - k)\
Now observe that
(a)n+k {a)n(a+n)k
( ' '
ia + 2k)-k= (aht -2Ща/2Ы(а
So C.1.5) is the same as
{a)n v^ {—b + (a + \)/2)k(a + n)k(-n)k
n\ f^0 (a-b+l)k«a+l)/2)kk\
Application of the Pfaff-Saalschiitz identity shows that this balanced 3 F2 is equal to
n\(a-b+ l)n'
Clearly, this is also the coefficient of x" on the left side of C.1.4). This proves the
theorem. ¦
Corollary 3.1.2 (Kummer [1836])
Proof. Let x —> —1 in C.1.4) and conclude by Abel's continuity theorem that
a-b+ V ) \ a-b+
Now sum the 2 Fi on the right by Gauss's summation formula (Theorem 2.2.2). The
corollary follows after an application of Legendre's duplication formula (Theorem
1.5.1). ¦
The quadratic transformation in Theorem 3.1.1 holds when the two series
converge. This is no longer true for C.1.3). Both sides of C.1.3) converge for
I < x < 1. Letting x —> 1 on both sides gives
2a,2b \
, 1 ; 1 = 1.
+ fc+ )
= 1. C.1.6)
This implies, by Gauss's summation, that
T(a +b + \)Г(\ - a -b) _ cosn(a-b)
Г(Ь — a + 5Jr(a - b + \) cosn(a+b)
This identity is not true in general, although it is true when a or b is an integer.
When a or b is a negative integer, C.1.3) holds for all x. Otherwise it holds in the
connected component of x = 0 when both \x\ < 1 and |4x(l — x)\ < 1. Gauss
was the first to remark on this, although his paper remained unpublished until after
analytic continuation was discovered. Gauss understood that a hypergeometric

3.1 Quadratic Transformations 127
function is many valued, citing sin x as a similar case; thus, he saw that C.1.3)
does not hold for all x where the two sides converge. See Exercise 6 for the correct
identity when i < x < 1.
The set j < x < 1 in C.1.3) maps to x < — 1 in C.1.4). The function on the right-
hand side is evaluated at a point inside the unit circle, but on the left an analytic
continuation is needed to make sense of the function.
Our first evaluation of 2F1(a, b; c; x) at x = 1 depended on Euler's integral
representation
^ f t-\l ~ 0—41 - xt^dt.
с-а) Jo
It is possible to derive Kummer's identity in Corollary 3.1.2 by taking x = — 1 in
the integral. In this case, с = a — b + 1, which makes the powers of 1 — t and
1 — xt equal. The other fundamental quadratic transformation comes from taking
the powers of t and 1 — t to be the same, that is, с = 2a.
Theorem 3.1.3 For all x where the series converge
Proof. The left side equals
ГBо)
Г(а)Г(а)
Substitute s = 1 — 2t or t = A — s)/2 and simplify to get
ГBо)A - {x/2))~b П
J-A x-
ГBо)A -
When n is odd, the last integral is zero. Otherwise it is
Jo
where n = 2m. So
(a,b \ ГBа)(\-(х/2)Гь J^ (ЬJтГ(т+ к)Г(а) ( х N 2m
242e:xj = -
_ rBa)r(I)(l-(x/2))-fe /fo/2,(fo+l)/2. / x \2\
Г(а)Г(а + |J2«-1 21\^ a + i ' \2 - x) )'
C.1.8)
An application of Legendre's duplication formula proves the theorem. ¦

128 3 Hypergeometric Transformations and Identities
Remark It is worth observing that Legendre's duplication formula follows from
C.1.8) by taking *=0.
Theorems 3.1.1 and 3.1.3 contain the basic quadratic transformations. Others can
be derived from these two by using the fractional linear transformations or the three-
term relations connecting different solutions of the hypergeometric differential
equation.
Apply Pfaff 's transformation to the right side of C.1.4) to obtain
) 4 a-b+l ;(
Replace 4jc/A + xJ with x to derive the equivalent formula
\ a,b
C.1.10)
A combination of C.1.10) and C.1.7) is another useful transformation, namely,
a b Ax \ , (a,a+\ -b
: ; (i+^Ja2^i Д ;
2F1
To prove this, replace x with 4x/( 1 + xJ in C.1.7) and interchange a and b to get
By C.1.10) it follows that the right side of the last formula is equal to the right
side of C.1.11). This proves C.1.11).
To see how three-term relations can be used to derive more quadratic transfor-
transformations, recall formula B.3.13),
Г(с)Г(с-а-Ь) ( a,b
Г(с)Г(а + fo -?) A _ x)C-a-b2F.f c-a,c-b
Apply this to C.1.3). The result is
2a,2b x+l\ Г(а + Ь+\)Г(\) ^(a,b 2
( )() (a+\,b+\ 2
2*\ X
Г(а)Гф) 2*\ 3/2 >
C.1.12)

3.1 Quadratic Transformations 129
Some general remarks about quadratic transformations are now in order. There
are two sides to the standard quadratic transformations: the linear side and the
quadratic side. On the linear side, the variable is linear or the image of linear under
the linear fractional transformation x{x — I). The parameters on the linear side
are restricted by one condition, which comes from writing the 2F\ as an integral
and equating two of the exponents of the functions in the integrand. This gives
с = 2а, с = a — b + I, and a + b = 1. To obtain the complete list, symmetry in
the parameters a and b and the Pfaff transformation are used. The complete list of
conditions is
с = 2a,c = 2b, c = a-b+l,c = b-a + l,a + b = \,c = .
C.1.13)
On the quadratic side, the variable appears in a quadratic form; the parameters are
restricted by requiring that those in the numerator differ by j, or the denominator
parameters differ by \ (that is, с = \ or с = |), or the Pfaff transformation is
used. The conditions are
1
2'
1
a + -,c -a
1
+ +-,c-a
1
+ ~ 2'C
1
~ 2'C
3
~ 2"
C.1.14)
Note that we use a, b, с to denote generic parameters and that they may differ in
a specific formula.
The two sides of the quadratic transformation are split into two groups; one has
с = la or с = 2b on the linear side and any of the other conditions C.1.13) in
the other group. On the quadratic side, one set has с = | or с = |, and the other
set has the remaining conditions in C.1.14). Notice that C.1.7) connects с = 2a
with b = a + \ and that C.1.4) connects с = a -b+ 1 with с = (a + b + l)/2.
The transformations that go to the group с = \ and с = | come from three-term
relations. An example of this is C.1.12).
Notice a simple way to distinguish those quadratic transformations that come
from C.1.4) from those that come from C.1.7) via linear transformations. In
C.1.4) the denominator parameters are equal; in C.1.7) they differ except for
one value. Observe that C.1.11) is not one of these quadratic transformations.
Both sides have conditions in C.1.13), and they were shown to be equal by show-
showing that each is equal to a function with one of the conditions in C.1.14) satis-
satisfied.
Let us return to the proof of Theorem 3.1.1 and note that the balanced 3F2 that
occurs in its proof has one specialization not needed to sum it. There is likely to be a
more general quadratic transformation with one more degree of freedom. It would
be nice if this were a transformation of a general 2^1, but our discussion of the

130 3 Hypergeometric Transformations and Identities
connection between the quadratic transformation and Euler's integral makes this
unlikely. The extra freedom comes at the 3F2 level. Whipple found the following
quadratic transformation:
F ( a,b,c
31 \a-b+l,a-c+l'X
l_Li'a °, 1 ; 71 ^)- С3-1-15)
a—b+l,a—c + \ A— xJ )
The proof of this formula is very similar to that of Theorem 3.1.1. The coefficient
of x" in the expression on the right of C.1.15) is again a balanced 3 F2. The details
are left to the reader. There exist examples of cubic transformations, though these
are not as well understood as quadratic transformations. Two examples are given
in Exercise 38.
Kummer [1836] gave some quadratic transformations of 2F\ s with one free pa-
parameter. The parameter was chosen so that more than one transformation could be
applied to the 2 F\ function. Consider the following two fundamental transforma-
transformations:
and
The 2^is on the left become identical when we make the denominator parameters
equal. This means 2b = a — b + 1 orb = (a + l)/3. In this case
= f аЛа+D/3
2Г\Bа+2)/3'Х
The last equation was obtained by an application of the Pfaff transformation. Let

3.1 Quadratic Transformations 131
x —> —1 in the first and last expressions of C.1.16) to get
F"/  \" ' //. I I I г "/ "' \" ¦ // . I
Ba + 5)/6 '9j \3J 2 \ 2(a
2)/3)
4)/6)Г((а + 1)/2)"
Now use Gauss's quadratic transformation C.1.3) and then Pfaff's transformation
B.2.6) to obtain
Recall that Gauss's formula holds in the connected component of the region
— x)\<\ that contains the origin. Thus Vl — 4x + Ax2 = \ —2x. (A sim-
similar argument applies in the derivation of the last equation in C.1.16).) Combine
C.1.16) and C.1.17) to get
Let x ->¦ | in the first equation to arrive at
(a /fl/2,(a + l)/6
2(fl + l)/3 '9y~V3/ V 2(fl + l)/3 '
The result in B.3.13) connects the following three
fl/2,(fl + l)/2 \ /fl/2,(fl + l)/2.
2(a + l)/3 У' 2 Л Bа + 5)/6 '
and
G-2fl)/6

132 3 Hypergeometric Transformations and Identities
Thus the third 2F1 can also be computed. In Exercise 2 we give a few more results
of Kummer on quadratic transformations with one free parameter.
3.2 The Arithmetic-Geometric Mean and Elliptic Integrals
Definition 3.2.1 LetO < к < 1. Following Legendre, the integral
dt
for xe [-1,1] C.2.1)
о УA - t2){\ - k2t2)
is called an elliptic integral of the first kind. Ifx = 1, this definite integral is called
a complete elliptic integral and denoted by K. Thus
К := K(k) := Г f * = Г f M . C.2.2)
Jo Ja-t2)(i-k2t2) Jo J\ -k2sm2e
To fully understand the integral C.2.1), one has to study its inverse, which is a
Jacobi elliptic function. This is similar to looking at the integral /q* A — t2)~l/2dt as
the inverse of the sine function. We shall take a very brief look at the Jacobi elliptic
functions in Chapter 10. Here we consider how some theory of hypergeometric
functions can be used to obtain interesting results about integral C.2.2). First note
that the binomial expansion of the integrand A — k2 sin2 в)~1/2, integrated term
by term, gives
K1/21/24 (з-2-з)
Now replace x with A — Vl — x2)/(l + Vl — x2) in the quadratic transformation
C.1.11) to obtain
2b'" J V 2 У *"y b+l/2
Apply this to the hypergeometric function in C.2.3). The result is
C-2'4)
where k'2 = 1 — k2. To iterate this result, we introduce the following notation:
ko:=k,k'm:=Jl-k2m, and *m+1 := iz^-, M = 0, 1,2,.... C.2.5)
1 + Km
It follows from C.2.4) and C.2.5) that
K(k) = fl Y^TK(km+i)-
m=0 m

3.2 The Arithmetic-Geometric Mean and Elliptic Integrals
133
Observe that
So
Moreover,
^n < i - (i -4) <4-
km < к and km —> 0 as m -> oo.
l-K.
m—0 m=0
converges, since J^mLo^ ~~ л/^ ~~ ^m) < SmU)^2™ converges. Thus
on _ oo
7Г
ш=0 ш ш=0
The fc^s can be obtained successively from
C.2.6)
The right side is the ratio of the geometric mean and the arithmetic mean of 1
and k'm.
Now suppose there are two sequences {an} and {bn} with a0 = 1 and bo = к'
such that
b^ = k'n and *±L =
C.2.7)
Then
and
an an_i
n-l
-П
oo , ,,
Moreover, fe^ -> 1 implies that
m=0
lim fon = lim an.
n->oo n->oo
From C.2.6) and C.2.7) it follows that
an +bn
and bn+i = \janbn.
C.2.8)

134 3 Hypergeometric Transformations and Identities
Conversely, it is easy to see that if two sequences {an} and {bn} satisfy C.2.8)
with uq = 1 and bo = k', then C.2.7) also holds. The common limit of the two
sequences is called the arithmethic-geometric mean of the sequences and it is equal
to
2 Г1
л Jo
dt
This result is due to Lagrange and Gauss independently. See Cox [1984]. We state
it as a theorem after formally defining the arithmetic-geometric mean.
Definition 3.2.2 Suppose {an} and {bn} are two sequences such that a =: ao and
b =: bo are real with a > b > 0 and an+\ = (an + bn)/2, bn+\ = y/anbn. Then
the two sequences converge to a common limit M(a, b), called the arithmetic-
geometric mean.
Theorem 3.2.3
dd
1 _ 2 Г1
a, b) ж Jo
M(a
Proof. It is clear from the definition that M(Xa, Xb) — kM(a, b). So M{a, b) =
aM(l, bid). The theorem follows from the result in the previous paragraph by
taking k' = b/a. ¦
The fact that the two sequences in Definition 3.2.2 have a common limit can be
obtained directly. It is easy to see that
a = a0 > fli > a2 > • • • > an > ¦ ¦ ¦ > bn > bn_\ > • • • > b0 = b.
Also,
an + bn an - bn
an+\ — t>n+i S an+\ — on = — bn = —-—.
Thus
a-b
an-bn<
2"
and the two sequences converge to the same limit. From the equation
(an - bnJ
4(an
+i
it follows that the convergence is quadratic. Gauss saw this rapid convergence by
computing the numerical example with a — -s/2 and b = 1. The first few values

3.2 The Arithmetic-Geometric Mean and Elliptic Integrals 135
of an and bn are
n
0
1
2
3
4
an
1.41421356237
1.20710678118
1.19815694809
1.19814023479
1.19814023473
К
1.00000000000
1.18920711500
1.19812352149
1.19814023467
1.19814023473
Gauss calculated up to twenty-one decimal places but the above table is sufficent
to illustrate our point. Somewhat later he calculated the ratio л/со where
со
= 2 f\l-x4r1/2dx.
Jo
On May 30, 1799, he noted in his diary that Af (a/2, 1) and ж/сЪ agreed to eleven
decimal places and he conjectured that they were equal. Later he proved the more
general result contained in Theorem 3.2.3. Once one has made this conjecture,
however, it is not too difficult to prove the result. If I (a, b) denotes the integral in
the theorem, then it is enough to prove
I(a,b) = I(a\,b\), C.2.9)
for then
I(a,b) = I{a\,b\) = •¦¦ — I{am,bm) = ¦•¦ — lim I(am,bm).
m->oo
To prove C.2.9), Gauss defined a new variable в\ by
2a sin 0,
a + b + (a - b) sinz 0]
and asserted that
(a2cos20 + fc2sin20)~1/2d0 = (a2cos20! +fc2sin201)~1/2rf01.
This requires some computation, and C.2.9) follows.
We started with an elliptic integral and then introduced the arithmetic-geometric
mean. Clearly Gauss was approaching the problem from the opposite direction,
as presented above. Let us now consider another way of arriving at the elliptic
integral from the arithmetic-geometric mean. Gauss also studied the function
/w- '
M(l+x, 1 -x)
Now
2/
MA+,M-,2) = ('+'V(>'). <3.2.,0)

136 3 Hypergeometric Transformations and Identities
Assume that / is analytic about t = 0. We want the analytic function that satisfies
/@) = 1 and the functional relation in C.2.10). Since / is clearly even, f(x) =
g(x2) for some function g and
Replace t2 with x to get
Ax
- )=(l+x)g(xz).
Write g(x) = J2T=o anx"> and use this functional equation to get
I2 ¦ 32 I2 • 32 ¦ 52
d\ = #o/4, по = ~T^ 7^"Я(Ь 6E3 = — — —-dc\
2 * 4 2 • 4 • о
This suggests that
+ X) ) = A + X) lF\
Formula C.1.11),
Ffa'b-
lF\ 2b '
suggests the same identity. It is possible to show directly that f{x) = g(x2) is
analytic. See the first few pages of Borwein and Borwein [1987]. However, it is
easier to do the argument in the other direction, as done above.
Definition 3.2.4 The complete elliptic integral of the second kind is defined as
rn/2
E:=E(k):= A - k2 sm2e)l/2d6. C.2.11)
Jo
A theorem of Legendre connects the complete elliptic integral of the first kind
with that of the second kind. Before proving it we state a lemma about the
Wronskian of an hypergeometric equation. (For the reference to Legendre's book,
where the result appears, see Whittaker and Watson [1940, p. 520].)
Definition 3.2.5 If y\ and y2 are two solutions of a second-order differential
equation, then their Wronskian is W(y\, y2) := у\у'2 — Угу\-
Lemma 3.2.6 Ify\ and y2 are two independent solutions of the hypergeometric
equation y" + (c — (a + b + l)x)y' — aby = 0, then
A
xc(l-x)a+b-c+1'
where A is a constant.

3.2 The Arithmetic-Geometric Mean and Elliptic Integrals 137
Proof. Multiply the equation
jcA - x)y'2' + (c-(a + b+ \)x)y'2 - aby2 = 0
by y\ and subtract from it the equation obtained by interchanging yt and уг- The
result is
x(l - x)(yiy'i - y2y'l) +(c-(a + b+ \)x)(yiy'2 - y2y[) = 0
or
jcA - x)W'(yi, y2) + (c-(a + b+ l)x)W = 0.
Now solve this equation to verify the result in the lemma. ¦
We shall need particular cases of the following two independent solutions of the
general hypergeometric equation:
fa, b \
У1 =2Fii c ;xj, C.2.12)
C.2.13)
Observe that from C.2.11)
IT /I -I Л
C.2.14)
Theorem 3.2.7 EK' + E'K - KK' = f, w/геге /iT' := ?(?'), ?' := ?(*'), and
ка=1- к2.
Proof. Set x — k2 so that 1 — x = /c'2. The contiguous relation B.5.9) gives us
Similarly,
х(\-х)^=1-Е-\-(\-х)К. C.2.15)
ax 2 2
-X{l-X)d?=l-E'-X-K>. C.2.16)
Multiply C.2.15) by K' and C.2.16) by К and add to get
EK' + E'K - KK' = 2jcA - x)W(K', K).
With a = b = | and с = 1, Lemma 3.2.6 gives ЩАГ', K) = A/x{\ - x). So
EK' + E'K — KK' is a constant. An examination of the asymptotic behavior of
К as л; ->¦ 1 shows that the constant must be (?r/2). This is left to the reader. ¦

138 3 Hypergeometric Transformations and Identities
It is possible to prove the following more general result of Elliot [1904] in
exactly the same way. The proof is left as an exercise. The formula in Theorem
3.2.7 is called Legendre's relation.
Theorem 3.2.8
„/\+a,-\-c \ „/'\-a,c+\
iFA l , , \x\2FA 2, ,l; 1 — x
\ a+b+l )l \ b + c+l
a+b+l ) \ b+c+l
a + \ c \ г
e+V+i >XJF
Salamin [1976] and Brent [1976] independently combined Legendre's relation
with the arithmetic-geometric mean to find an algorithm for approximating ж. We
conclude this section with a brief sketch of this application. Some of the details
are left for the reader to work out.
Lemma 3.2.9 If{an] and {bn} are sequences in Definition 3.2.2, then
2Jn+i — Jn— anbnln,
where
Jn := f (a2 cos2 в + b2n sin2 в) 1/2d6,
Jo
and In has — | as the power in the integrand.
Lemma 3.2.10
E(k)= M-
V и=0
where c2n = a2n - b\.
Proof. From C.2.9) we know that /„ = I (a, b) =: I. By Lemma 3.2.9,
2(jn+1 - a2n+1l) - (Jn - a2nl)
= (а„Ьп-2а2п+1+а2пI

3.2 The Arithmetic-Geometric Mean and Elliptic Integrals 139
Rewrite this equation as
Z [Jn+l an+\1) Z V" anl)—L Cn1
and sum it from n = 0 to n = m to get
r-iAl, C.2.17)
и=0 /
where J := Jq. Now,
*¦ \Jm+\ -am+1lm+i) — I cm+1
Since c^,+1 tends to zero quadratically, the last term tends to zero. Let m -*¦ oo in
C.2.17) and then take a = 1 and b — k'. The lemma is proved. ¦
Theorem 3.2.11
7Г =
where с2 = a2 — b2 with ao = 1 and bo = 4=.
n n n " " v 2
Prao/ Take к = J=. Then /c' = J= and Legendre's relation becomes
- = BE-K)K, where ? = ?D=) and K = K[~
2 VV2/ VV2
This implies
и=0
Since /i: = 7T/BMA, 4^)), the result follows.
An algorithm based on this theorem has been used to compute millions of digits
of it. Define
лт '¦=
_ Vm 2"r2
2^n=0Z Cn
Then 7rm increases monotonically to n. Note that cn = y/a2 — b2 = c2_x/Aan.
The an and bn are computed by the arithmetic-geometric mean algorithm. For
more information on the computation of n, see Berggren, Borwein, and Borwein
[1997].

140 3 Hypergeometric Transformations and Identities
3.3 Transformations of Balanced Series
In the previous chapter we saw how a general 2F1 transforms under a fractional
linear transformation and how to evaluate the sum of the series when x = 1. In the
case of quadratic transformations, there were restrictions on the parameters. For
higher p+iFp, transformations and summation formulas do not exist in general.
There are, however, two classes of hypergeometric series for which some results
can be obtained.
Definition 3.3.1 A hypergeometric series
is called k-balanced where к is a positive integer, if x = 1, if one of the a,s is a
negative integer, and if
p p
/=0 /=1
The condition that the series terminates may seem artificial, but without it many
results do not hold. The case к = 1 is very important, and then the series is called
balanced or Saalschiitzian.
Definition 3.3.2 If the parameters in the hypergeometric series satisfy the rela-
relations
a0 + 1 = a\ + bi = ¦ ¦ ¦ = ap +bp
the series is called well poised. It is nearly poised if all but one of the pairs of
parameters have the same sum.
In Section 3.4 we shall give a connection between the two kinds of series consid-
considered in Definitions 3.3.1 and 3.3.2. We begin with a study of balanced series. The
main theorem of this section is the following result of Whipple, which transforms
a balanced 4F3 to another balanced 4/%
Theorem 3.3.3
(e - a)n(f - a)n
d,e,f ' ) («)„(/)„
—n,a, d — b,d — с
**F\d,a+\-n-e,a
where

3.3 Transformations of Balanced Series 141
Proof. Start with Euler's transformation:
Rewrite this with different parameters:
Suppose c — a — b = f — d — e and multiply the two identities to get
f-d.f-e \ (c-a,c-b
The coefficient of x" on the left side is
(a)k(bMf ~ d)n-k(f - e)n-t
This expression can be rewritten as
(f-d)n(f-e)n ( a,b,l- f-n,-n
Equating this to the coefficient of x" on the right side, we obtain
/ -n,a,b, -f-n+ 1
4 \ d — f — +1 — f — +1
-n, с — а, с — b,\ — f — n
U-d)n{f-e)n \ c,l-d-n,l-e-n ' )'
This is equivalent to the statement of the theorem. This result is due to Whipple
[1926]. For a different proof see Remark 3.4.2. ¦
The next result was given by Sheppard [1912].
Corollary 3.3.4
-n,a,b \ (d-a)n(e-a)n ( -n,a,a + b - n - d - e + 1
3^2 j . , .. . . ,; 11-
d,e 'lJ- (d)n(e)n 3i\a-n-d+l,a-n-e+l
Proof. Let / -*¦ oo but keep / — с fixed in Theorem 3.3.3 so that
f-n,a,b,c \ f-n,a,b \
4^3 , , ; 1 ->¦ 3^2 , ; 1 •
V d,e,f ) \ d,e J

142 3 Hypergeometric Transformations and Identities
A similar change takes place on the right side and the end result is
—n,a,b \ (e — a)n
d,e J (e)n
Sheppard's transformation is obtained by applying this transformation to itself.
The corollary is proved. ¦
The formula in Corollary 3.3.4 has some interesting special cases. For example,
suppose the left side is ^-balanced, that is,
d + e = k — n + a + b.
Then the right side is a sum of к terms. In particular, к = 1 gives back the Pfaff-
Saalschiitz identity.
Corollary 3.3.5
a,b,c \ _ r(e)T(d + e— a—b — c) ( a,d-b,d — c
d,e ' J ~ T(e -a)T(d + e - b -cK \d, d + e - b - с '
when the two series converge.
Proof. Let n -> oo and keep f + n fixed. Since the number of terms of the series
tends to infinity, Tannery's theorem may be used to justify the calculation. The left
side in Theorem 3.3.3 becomes
, ;
\ d,e
To find the right side, write
(e-a)n(f-a)n T(e - a + n)T(f - a + n) Г(в)Г(/)
(«)„(/)„ Г(е-а)Г(/-а) Г(в + и)Г(/+и)
Г(е) Г(д-/+1)Г(-и-/ + 1) Г(е-д+и)
Г(е-а)Г(а-/-и + 1)Г(-/+1) Г(е + и) '
where Euler's reflection formula was used to derive the second equality. Recall that
l-f-n — d + e-a-b-c. So
(e - a)n(f - d)n j:{e)Y(d + e - a - b - с) / Г(д - / + 1)Г(е -а + п)
As и -*¦ oo andn + /is fixed, we have— / -» oo and the expression in parentheses
equals 1 in the limit. The corollary follows. ¦
Corollary 3.3.5 was given by Kummer [1836]. If we apply Kummer's transfor-
transformation to itself we get a theorem of Thomae [1879]:

3.4 Whipple's Transformation 143
Corollary 3.3.6
\ d,e J Г (a)
where s = d + e — a — b — с
3.4 Whipple's Transformation
The main result of this section is an important formula of Whipple [1926] that
connects a terminating well-poised jF6 with a balanced 4/% We prove it by a
method of Bailey, which requires that we know the value of a general well-poised
3F2 at x = 1. In Chapter 2, we showed that the latter result, known as Dixon's
theorem, is a consequence of Dougall's theorem. See Remark 2.2.2 in Chapter 2.
Because Dougall's theorem is itself a corollary of Whipple's transformation, it
would be nice if we had a directproof of Dixon's formula, that is, one that does not
use Dougall's formula. Several such proofs are known. We give one that follows
from a quadratic transformation given in Section 3.1.
Theorem 3.4.1
f a,-b,-c
3 2l и
= Г((а/2) + 1)Г(а + 6+1)Г(а-
~ Г (a + 1)Г((а/2) + * + 1)Г((а/2) + с + 1)Г(а + b + c+l)'
Prvof. If a — —n, a negative integer, in the quadratic transformation C.1.15),
then both sides of the equation are polynomials in x. Take x — 1. If a is an even
negative integer, then we get
-2n,b,c Л Bn)\(b + с + n)n
\-2n-b,l-2n-c' ) (b + n)n(c + n)nn\
If a is an odd negative integer, then
/ -2n-l,b,c \
>F\-2n-b,-2n-c'l)=0-
Thus Theorem 3.4.1 is verified when a is a negative integer. Now suppose that с
is a positive integer and a is arbitrary. In this case both sides are rational functions
of a and the identity is true for an infinite number of values of a. Thus, we have
shown that the identity holds if с is an integer and a and b are arbitrary. In the
general case, for Re с > Re(—a/2 — b— 1), both sides of the identity are bounded
analytic functions of с and equal for с = 1, 2, 3, By Carlson's theorem the
result is proved. ¦

144 3 Hypergeometric Transformations and Identities
Kummer's identity, which gives the value of a well-poised 2^1 at x — —1, is a
corollary of Dixon's theorem. To see this, let с -> oo.
Remark 3.4.1 We noted earlier that the balanced identities in their simplest form
come from the factorization
A - x)-a(l - x)-b = A - x)-"-b.
In a similar sense, the well-poised series comes from
Equate the coefficient of x2n from both sides to get
k\Bn -k)\
or
-2n-b' ) n\(bJn
This is Kummer's identity for
a,b;-\
>F\a-b.
when a is a negative even integer. As in the proof of Theorem 3.4.1, we can now
obtain the general result of Kummer. The 2^1 result was so special that Kummer
failed to realize that series of a similar nature could be studied at the 3 F2 and higher
levels. This is not surprising for well-poised series are not on the surface.
We also need the following lemma to prove Whipple's theorem. It is proved by
Bailey's [1935] method mentioned at the beginning of this section.
Lemma 3.4.2
Ff a,b,c,d, —m
\a — b+ l,a — с + l,a — d + l,a + m + I'
(a/2), a -b-c+l,d,-m
((a/2)+ \)m{a-d + \)m \a - b + I, a - с + \,d - m - a/2'
Proof. By the Pfaff-Saalschiitz identity for a balanced 3 F2 we have
(-n)r(a-b-c + l)r(a + n)r (b)n(c)n
r\(a -b + \\(a -c+\)r ~~ (a-b + 1)„(а - с + 1)„ '

3.4 Whipple's Transformation 145
So
/ a,b,c,d,-m
5 \a-b+l,a-c + l,a-d + l,a + m + l'
^-^ (а)„(а)„(—от)и ^-^ (—n)r(a — b — с + l)r(a + n)r
n\(a — d + \)n(a + m + 1)и ^ r!(a — & + l)r(a — с + l)r
(— l)r (a)n+r(d)n(—m)n{a — b — с + l)r
г — r)\r\{a — b + l)r(a — с + \)r(a — d + \)n(a + m + l)n
(sett =n -r),
(a)t+2r(d)t+r(-m)t+r(a -b-c+ l)r(-l)r
r=0 (=0 l !r!(a - *+ l)r(a - с + l)r(a - J + l),+r(a + m
\(a-b+ \)r(a -c+ \)r(a -d+ \)r(a + m + l)r
(a+2r),(fif + r),(-m+r)(
The inner sum can be computed by Dixon's identity (Theorem 3.4.1). An easy
calculation then gives the required relation. ¦
Lemma 3.4.2 transforms a terminating well-poised 5F4 to a balanced 4F3.
Corollary 3.4.3
a,(a/2) + l,c,d,-m \ (a + l)m(a - с - d + l)m
; 1
а/2,а-с+1,а-а" + 1,а+от-|-Г ) {a-c+\)m(a-d+\)m
Proof. Take fe = (a/2) + 1 in Lemma 3.4.2. The 4F3 reduces to a balanced
3F2. ¦
Theorem 3.4.4
a,(a/2) + l,b,c,<i,e,-от
7 \a/2,a -
(a + \)m(a -d -e+ \)m r ( a-b - с+\,d,e,-m
(a~d+ \)m{a -e + l)m4'

146 3 Hypergeometric Transformations and Identities
Proof. The proof of this theorem is exactly the same as that of Lemma 3.4.2
except that one uses Corollary 3.4.3 instead of Dixon's theorem. Thus
a, (a/2) + 1, b, c, d, e, — m
a/2, a -b+l,a-c+\,a-d+l,a-e+l,a+m + l'
^ (a)«((a/2) + l)n(d)n(e)n(-m)n
f^Q и!((а/2))„(а - d + 1)„(а - e + 1)„(а + m + l)n
E(-n)r(a - b - с + 1)Да + n)r
r\(a - b + l)r(a - с + l)r
After a calculation similar to the one in Lemma 3.4.2, this sum equals
^U (аJг((а/2) + l)r(d)r(e)r(-m)r
yC (a + 2r),((a/2) + r + !),(«/ + r),(e + r),(-m + r),
¦^ t\((a/2) + r),(a — d + r + l),(a — e + r + I),(a + m +r + 1),'
The inner sum can be evaluated by Corollary 3.4.3 and the result follows after a
straightforward calculation. ¦
Since the two elementary identities in Remark 3.4.1 are not related, we see the
very surprising nature of Whipple's identity. In a later chapter we give a more
natural proof of Whipple's theorem as a consequence of some properties of Jacobi
polynomials. We refer to the above 7 Ff, as a very well poised 7 F(,. The word "very"
refers to the factor
((a/2) + l)t = а + 2k
(a/2)t ~ а
The 5F4 in Corollary 3.4.3 is also very well poised.
Remark 3.4.2 Theorem 3.3.2 is a particular case of Theorem 3.4.4 because of
the symmetry in the parameters b, c, d, e in the 7^.
Whipple also stated a more general form of Theorem 3.4.4.
Theorem 3.4.5
( a,(a/2) + l,b,c,d,e,f
1 \a/2,a-b+l,a-c + l,a-d + l,a-e+l,a- /+Г
Г (a - d + 1)Г(а -е+ 1)Г(а - / + 1)Г(а -d-e-f + l)
Г (a + 1)Г(д -e-f+ 1)Г(а - а" - e + 1)Г(а - а" - / + 1)

1 \л ts ^i-,»*,^,^ .11

3.5 Dougall's Formula and Hypergeometric Identities 147
provided the series on the right side terminates and the one on the left con-
converges.
Proof. This is a consequence of Carlson's theorem and Theorem 3.4.4. The reader
should work out the details or see Bailey [1935, p. 40]. ¦
Theorem 3.4.6
F( a,(a/2) + l,b,c,d,e
6 \a/2,a-b+l,a-c + l,a-d+l,a-e + l'
_ Г(а-дЧ-1)Г(а-е+1) / a - b-с+ l,d,e
d-e + lf \a - b+ I,a - с + Г
Proof Let m -> oo in Theorem 3.4.4 to prove the result. ¦
Observe that Theorem 3.4.6 connects a general 3F2 at x = 1 with a very well
poised 6^5 at x = —1.
3.5 Dougall's Formula and Hypergeometric Identities
Set 2a + 1 = b + c + d + e — min Theorem 3.4.4. The 4F3 reduces to a balanced
3F2, which can be summed. The result is Dougall's formula.
Theorem 3.5.1
F( a,{a/2) + \,b,c,d,e,-m
1 \a/2,a-b+\,a-c+ l,a-d+l,a-e+l,a + m + l'
= (a + l)m(a -b-c+ l)w(a -b~d+ l)w(a - с - d + l)w
(a - ft + l)m(a - с + l)m(a - J + l)m(a - ft - с - d + l)m '
when la + 1 =b + c + d + e — m. This formula sums a 2-balanced very well
poised 7 Tv
In the following identities, convergence conditions need to be imposed. They
are not explicitly stated, since they are easy to work out in each case.
Corollary 3.5.2
( a,(a/2)+l,c,d,e
5 \a/2,a-c+l,a-d + l,a-e + l'
_Г(а-с+ 1)Г(а - d + 1)Г(а -е + 1)Г(а -c-d-e + 1)
~ Г (а + 1)Г(а -d-e+ 1)Г(а - с - e + 1)Г(а - с - d + 1)'
Prao/ Substitute b = 2a — c — d — e + m + lin Theorem 3.5.1 and let m-»oo.
This procedure may be justified by Tannery's theorem. The corollary follows. ¦

148 3 Hypergeometric Transformations and Identities
One may also derive this corollary from Corollary 3.4.3 of the previous section
by an application of Carlson's theorem. Dixon's formula follows from Corol-
Corollary 3.5.2 by taking e = a/2. The next corollary gives the value of a very well
poised 4F3 at x = — 1.
Corollary 3.5.3
( a, (a/2) , .,о,„ . _
a/2,a-c + l,a-d+ Г / Г (a + 1)Г(а - с - d + 1)
Proof. Let e -> —00 in Corollary 3.5.2 and the result follows; or else take
Z? + c = a + lin Theorem 3.4.6. ¦
Here are a few more summation formulas.
Theorem 3.5.4
<¦¦* M/2U
Г(с/2)Г((с+1)/2)
+ в)/2)Г((с-в-
Proof Let x -> — 1 in Pfaff's transformation (Theorem 2.2.5),
to get
There are two ways in which the series on the left becomes well poised so that
it can be summed by Kummer's identity (Corollary 3.1.2). These cases are (i)
2c — b — a + 1 and (ii) a + с = с — b + \ or a + b = 1. The two parts of the
theorem follow immediately. ¦
Theorem 3.5.5
ГA/2)Г(с + A/2))Г((а + b + 1)/2)Г(с -(a+b-
Г((а + 1)/2)Г((* + 1)/2)Г(с - (a - 1)/2)Г(с - (b -

3.5 Dougall's Formula and Hypergeometric Identities 149
,'a,b,c
(и) з^2 , ; 1
лТ(е)Г(/)
22сГ((а + е)/2)Г((а + /)/2)Г((Ь + е)/2)Г((Ь -
when а + Ъ - 1 and е + / = 2с + 1.
Proof. These results follow from Thomae's formula (Corollary 3.3.6) by choos-
choosing parameters appropriately. To obtain (i), choose the parameters so that the right
side becomes well poised. Thus d — (a + b + l)/2 and e = 2c and Thomae's
formula gives
a,b,c Л Г((а+Ь+1)/2)ГBс)Г(с-(а+Ь-1)/2)
(а+Ь+1)/2,2сУ ) Г(а)Г(с-(а-Ь-1)/2)ГBс-(а+Ь-1)/2)
/2c-a,(b-a + l)/2,c-(a+b-l)/2 \
Now apply Dixon's identity to get (i). Note that we require с > (a + b - l)/2.
The 3F2 also exists without this condition when с = — n, a negative integer, if the
series is taken to terminate with n + 1 terms. The value of this series can be found,
but it is not what one gets by letting с tend to —и.
(ii) To prove this identity choose the parameters so that the right side is the 3F2
given by (i). Thus take a + b = 1 and e + f = 2c + 1 to get
(a,b,c \ _ Г(е)Г(/)Г(с) /e-a,f-a,c
3 V,/' ) Г(а)Г(Ь + с)ГBс)Ъ \ b + c,2c '
An application of (i) at this point gives (ii). ¦
Note that Theorem 3.5.4(i) and (ii) are limiting cases of Theorem 3.5.5(i) and
(ii) respectively. Let с -> 00 to see this. Part (i) of the last theorem is due to Watson
[1925], who proved it for the case where a = —n, a negative integer. Watson's
theorem can also be obtained by equating the coefficient of xn on each side of
the quadratic transformation C.1.11). The general case then follows by Carlson's
theorem. Another way is to multiply Equation C.1.3) by (x — x2)c~l and integrate
over @, 1). This only works in the terminating case.
Remark We end this section with the following comments on well-poised series.
Let
l ~n,aua2,. aq
\ -n -a\,..., 1 -n —aq )

150 3 Hypergeometric Transformations and Identities
Then the polynomial f(x) satisfies the relation
fix) = (-iynx"fil/x).
A polynomial g (x) that satisfies g(x) = xng(l/x)is called a reciprocal polynomial
of degree n. These polynomials have the form
gix) = ao + a\x + a2x2 + ¦ ¦ ¦ + a2x"~2 + a\xn~x + uqx" .
Note that fix) is reciprocal if either q or n is even. It is easy to check that if g is
the reciprocal polynomial given above then
gix) = oo(l + x)" + aixil + x)"-2 + ¦¦¦ + avxvil + *T~lv C.5.1)
for some an, й\,..., av which can be defined in terms of an, a\, Here v =
[и/2]. It can be shown that
aJ= 1^
0<2r<n
and
0^, 7"' Jn-j-i
Note that C.5.1) can also be written as
) Uk 2k- C-5-4)
Observe the connection of C.5.4) with the quadratic transformations C.1.4) and
C.1.15).
3.6 Integral Analogs of Hypergeometric Sums
In Chapter 2 we saw two integrals of Barnes that were continuous analogs of
Gauss's 2 F\ identity and the Pfaff-Saalschiitz 3 F2 identity. There are other Barnes-
type integrals that are analogs of the higher pFq sums we have considered in this
chapter. The path of integration in the integrals will be parallel to the imaginary
axis but suitably deformed so that the increasing sequence of poles of the integrand
is separated by the contour from the decreasing sequence of poles.
The following theorem of Bailey is an analog of Corollary 3.5.2, which sums a
very well poised 5F4 at x = 1. See Bailey [1935, p. 47].

3.6 Integral Analogs of Hypergeometric Sums 151
Theorem 3.6.1
2) + 1 + s)T(b + s)T(c + sW(d + s)r(b -a- s)r(-s)ds
J_ Г Г(а
Ъп J
Г((а/2) + «)Г(а - с + 1 + «)Г(а - d + 1 + s)
Гф)Г(с)ГУ)Гф + с~ а)Гф + d-a)
Proof. The proof is similar to that of Theorem 2.4.2. The reader should fill in
the details. The residues at the poles of Г(Ь — a — s)V(—s) to the right of contour
give the integral as the sum of two very well poised 5 F4. These can be summed by
Corollary 3.5.2. The result follows. ¦
A different and useful form of Theorem 3.6.1 was given by Wilson [1978]. We
note it here.
Theorem 3.6.2
! + s)T(a - х)Гф + х)Гф - s)T(c + s)T(c - s)V(d + s)F(d - s)
1 f Г (а
bd J
T{2s)T{-2s)
2Г(а + b)T(a + с)Г(а + d)Yф + с)Гф + d)V(c + d)
¦ds
Here the contour is along the imaginary axis but suitably deformed. As always,
we are assuming that a, b, c, d are such that this can be done.
Proof. In Theorem 3.6.1 replace a with 2a; b, c, d with b + a,c +a, and d + a
respectively; and s with s — a. We get
Г (а + s)T{s + l)F(b + s)T(c + s)V(d + s)V(b - s)V(a - s)
— f
\ni J
- d + s)
Г (а + Ь)Г(а + с) Г (а + d)V(b + с)Г(Ь + d)
-ds
2ГA-c-d)r(a + b +
Use Euler's reflection formula to rewrite this as
Г (а + s)V(b + s)V(c + s)V(d + s)V(a - s)V(b - s)V(c - s)V(d - s)
2ni J rBs)r(-2s)
sin(c — s)n sin(d — s)?t
¦ ds
—smsn cos sn sin(c + d)n
_ 2Г(а + Ь)Г(а + с)Г(а + d)T{b + c)T{b + d)T{c + d)
~ Г (a + b + с + d)
The trigonometric expression in the integrand is
sin en sin dn cos2 sn + cos en cos dn sin2 sn
1 -
sin(c + d)n sin sn cos sn

152 3 Hypergeometric Transformations and Identities
Now observe that the second term involving the trigonometric functions changes
sign when s is changed to —s. Hence that part of the integral vanishes and the
theorem is proved. ¦
If a, b, c, d are positive or a = b and/or с = d and the real parts are positive,
then we can write Wilson's formula as
-I
Г (a + ix)F(b + ix)F(c + ix)F(d + ix) z
dx
FBix)
_ Г(д + b)F(a + с)Г(д + d)F{b + c)F(b + d)F(c + d)
F(a+b + c + d)
The result in C.6.1) was given also by de Branges [1972, 1972a]. Formula C.6.1)
continues to hold when one of the parameters is zero. The following corollary is
the result of letting d —> oo:
2n Jo
Г (a + ix)F(b + ix)F(c + ix)
dx = F(a+b)r(a+c)r(b
FBix)
C.6.1)
There is also an analog of the Dougall 7 F6 formula. This is the next theorem, also
given by Bailey.
Theorem 3.6.3
Г (a + s)F((a/2) + 1 + s)F(b + s)F(c + s)V(d + s)
In i J
Г ((a/2) + s)V(a - с + 1 + s)F(a - d + 1 + s)
F(e + s)F(f + s)F(b -а- ^г(~^)^
Г (a - e + 1 + s)F(a - f + 1 + s)
F(b)F(c)F(d)F(e)F(f)F(b + c-a)
2F(a -d-e+ 1)Г(а -с-е+ 1)Г(а - с - d+ 1)
F(b+d- a)F(b + e - a)F(b + / - a)
" Г(в - с - / + 1)Г(а -d-f+ 1)Г(а - e - f + 1)'
when 2a+ 1 = b + c + d + e + f.
C.6.2)
Proof. It is not possible to evaluate this integral as in Theorem 3.6.1. The two 7 F6
series cannot be summed by Dougall's formula unless they terminate. However,
if they terminate, a contour separating the increasing and decreasing sequences
of poles cannot be constructed. It is possible, nonetheless, to give a proof that
is the integral analog of the proof of Theorem 3.4.4. Start with the formula in

3.6 Integral Analogs of Hypergeometric Sums 153
Theorem 2.4.3 in the following form:
Г (а - d + 1+ s)V(a -e+l+ s)V(a - f+l+s)
_ 1
~ Г (a - d - e + 1)Г(а - J - / + 1)Г(а - e - f + 1)
Г(</ + г)Г(е + г)Г(/ + ?)Г(а -rf-e-/ + l- г)Г(* - t)
2ni J
dt.
C.6.3)
The left side of C.6.4) is a part of the integrand in C.6.3). Substitute this in C.6.3)
to get
1 f V(d + t)T(e + t)V(f + t)V(a -d-e-f+l-t)
2ni J V(a-d-e+ 1)Г(а - d - f + 1)Г(а - e - f
1 f Г (а + *)Г((а/2) + 1 + s)T(b + s)V(c + s)
ч!
r((a/2)+s)F(a-c+l+s)
r(b-a-s)r(s~t)r(-s)J
¦ dsdt.
Evaluate the inner integral by Theorem 3.6.1. The resulting integral then reduces
to 1, which can be computed by C.6.4) when 2a + l=b + c + d + e + f. The
theorem follows. ¦
An analog of Dixon's well-poised 3F2 can also be derived. Take d = a/2 in
Theorem 3.6.1 to get the required result:
1 ГГ(а+ s)V(b + s)V(c + s)V(b -a- s)V(-s) J
ds
2ni J
Г(Ь)Г(с)Г(а/2)Г(Ь + с - а)Г(Ь - u, -„
= C.6.4)
2Г((а/2)~с+1)Г(Ь + с-(а/2))
To obtain a more symmetrical form, replace a with 2a and b, с with b + a,c + a
respectively and s with s — a to get
Г (а + s)V(b + s)V(c + s)T(b - s)V(a - s)
ds
b)r(a + __z C65)
2ГA~с)Г(а+Ь + с)
Now apply the reflection formula and do the simplification (as employed in the

154 3 Hypergeometric Transformations and Identities
reduction from Theorem 3.6.1 to Wilson's integral) to get
1 f
/ Г (a + s)T(b + s)T(c + s)F (a - s)T(b - s)T(c - s) cos snds
2л i J
Г(а)Г(Ь)Г(с)Г(а + b)T{a + c)T(b + c)
= 2Г(а+Ь + с) •
If a, b, and с are positive, we can write the formula as
- / \Г(а + ix)T(b + ix)T(c + ix)\2 coshлxdx
л- Jo
Г(а)Г(Ь)Г(с)Г(а + Ь)Г(а + с)Г(Ь + с)
Г(а
3.7 Contiguous Relations
C.6.7)
In the previous chapter we gave the three-term contiguous relations of Gauss for
the 2F{ functions. More generally, there are (q +2)-term relations for pFqs with
p < q + 2. Under certain conditions, these relations become three-term relations.
Kummer observed that it was possible to obtain such relations for the 3F2s when
x = 1. Bailey [1954] gave a procedure using the differential equation satisfied by
the 3f2 to produce these relations. A simpler method was given by Wilson [1978].
This applies to more general pFqs. In this section we use Wilson's method to
derive his results on the three-term contiguous relations for balanced 4F3S. These
contain the three-term recurrence relations for a set of orthogonal polynomials due
to Wilson, which contain the "classical" orthogonal polynomials as special cases.
Before describing Wilson's method, we note that it is possible to obtain three-
term relations for the 3 F2S from the contiguous relations for the 2 F\ s by integration.
For example, multiply the equation
(b - d)F + aF(a+) - bF{b+) = 0
by xd~l(l - x)e~d~l and integrate over @, 1) to get
,, . „(a,b,d \ (a + \,b,d Л , „(a,b+\,d Л „
(b-aKF2{ ;1 )+a3F2[ ¦,\)-b3F2[ ; 1 = 0.
V c,e ) \ c,e J \ c,e J
As another example, apply this procedure to
а,Ь \ (c-b)ax (a+\,b+\
\x\ 2
е/ c(c + l)
to arrive at
a,b + l,d \ rfa,b,d Д a(c - b)d fa + \,
;1Ч ;1Ч
c+l,e ;1ЛЧ с,е ;1Г^ПO3Ч с + 2,е

3.7 Contiguous Relations 155
Let us now turn to Wilson's procedure for systematically deriving all the con-
contiguous relations of a balanced 4F3. Note that if only one of the parameters in a
balanced 4F3 is altered, the new 4F3 is not balanced.
Definition 3.7.1 Given a balanced 4F3, a contiguous 4F3 is obtained by altering
two parameters by±\ in such a way that the new series is also balanced. As before,
a relation among contiguous functions is called a contiguous relation.
Denote the balanced 4F3(a, b, c, d; e, f, g; 1) by F. By definition, one of the
numerator parameters is — n and the sum of the denominator parameters is one
more than the sum of the numerator parameters. There are2 x G2) = 42 4F3s
contiguous to F. Consider the difference F(a—, b+) — F. Since
(a - \)k(b + l)k(c)k(d)k
k\(e)k(f)k(g)k k\(e)k(f)k(g)k
(a)k-i(b + l)t
k\(eMf)k(g)k
(a-b-l),
(k - \)\(e)k(f)k(g)k
we have
Т7, и,л it (a-b- \)cd
F(a-,b+)- F = F+(a-), C.7.1)
efg
where F+ is obtained from F by increasing every parameter by 1. Similarly,
F(a-,e-)-F=f^-F+{fl-), C.7.2)
(e - \)efg
„, . ., j, (e-a)bcd
F(a+, e+) - F = — —— F+(e+), C.7.3)
e(e+l)fg
(e - f + \)abcd
F(e+, f-)-F= ^—^-777 F+(e+)- C-7-4)
e(e + \)fg
By symmetry in the parameters, we now have expressions for all the differences
between F and a function contiguous to it. Some of the contiguous relations are
immediate corollaries of these relations. By C.7.1),
Therefore, equating the two expressions for F+(a~), we get
b(a-c- l)(F(a-, b+) - F) = c(a - b - l)(F(a-, c+) - F).
The other contiguous relations would follow once we find an equation connecting
F with F+(a—) and F+(e+). Of course, this equation would imply the other

156 3 Hypergeometric Transformations and Identities
necessary relations by symmetry. To derive the required relation, take a = —n and
apply the transformation in Theorem 3.3.1 to obtain
(f)n(g)n „ c( a,b,e-c,e-d
(f-b)n(g-b)n \e,e + f-c-d,e + g-c-d'
a, b + 1, e — с, е — d
-) =4^3
(/-*)„(*-*)„
and
(/+l)»-l(g +!)„-!
(f - b)n-i(g - b)n-i
a + l,b+l,e-c + l,e-d+l
Now the connection between F, F(b+, e+), and F+(e+) is given by C.7.3).
This implies the relation among F, F+(a—), and F+(e+), which is given by
fgF - (/ - a)(g - a)F+{a-) + п(в ~ **"?*" ^ F+(e+) = 0. C.7.5)
In this derivation we also used the fact that a+b + c + d+\ — e + f + g. We
note that for C.7.5), it does not matter which numerator parameter is a negative
integer. This is true because F is a rational function of the five free parameters.
Thus we also have
t г lf ил, им? ,и л , He - aXe-cXfi - d)
fgF - (/ - bXg - b)F+{b-) -\ F+(e+) = 0.
e(e + 1)
Eliminate F+(e+) from the last two equations to get
b(e - aXf ~ a){g - a)F+(a-) - a(e - bXf ~ b)(g - b)F+(b-)
+ {a- b)efgF = 0. C.7.6)
The final relation we need is similarly obtained from C.7.5). It is
(e - ci)(e - bXe - cXe - d)
(/ - aXf ~ bXf ~ cXf ~ d) p /fi^ ,
f,f + V) F+(f+) + g(e- f)F = 0. C.7.7)
All the contiguous relations can now be obtained from C.7.1) to C.7.7). As one
more example, substitute the values of F+(a—) and F+(b—) from C.7.1) into

3.8 The Wilson Polynomials 157
C.7.6) to get
Ke-a)(f-a)(g-a)i
a-b-l
a(e - b)(J - b)(g - b)
(F(a+,b-)~ F)+cd(a-b)F = 0. C.7.8)
b-a- 1
The contiguous relations for the 3 F2 s are found by letting n -» 00 in the relations
for the 4F3S. One may also write down the fundamental relations corresponding
to C.7.1) to C.7.7) and derive the others from these. We give a few examples. In
C.7.1), let a — —n and n -» 00 to get, after renaming the parameters,
F(a+) -F = ^F+, C.7.9)
de
where F stands for a general 3F2 at x = 1. In C.7.2), let d = —n and e =
—n + a + b + с — f — g — 1 and n -> 00. After renaming parameters,
F(«-) - F = -~F+{a-). C.7.10)
de
It is possible to get C.7.10) from C.7.1) too. Similarly, we have
-) -F= abC F+ C.7.11)
(d — \)de
and
F(d+)-F = - f\ F+(d+). C.7.12)
a(a + l)e
By taking limits in different ways in C.7.5) we get two more relations:
deF -a(d + e-a-b-c- \)F+ - (d - a)(e - a)F+(a-) = 0 C.7.13)
and
eF~(e- a)F+(a~) - ^ ~^~ C) F+{d+) = 0. C.7.14)
The rest of the relations can be found in a similar manner. In fact, all the 3 F2
contiguous relations follow from C.7.9) to C.7.14) if symmetry in the parameters
is also used.
3.8 The Wilson Polynomials
Consider the polynomial pn of degree n > 0 defined by the relation
;;l) C.8.1)
a + b,a + c,a + d I

158 3 Hypergeometric Transformations and Identities
where a, b, c, d are real and positive. From the contiguous relation C.7.8) we find
that pn satisfies the three-term recurrence relation
An(pn+i(x) - pn(x)) + С„(р„_,(х) - pn(x)) + (a2+x)pn(x) = 0, C.8.2)
where
(n + a+b + c + d- \){n + a + b)(n + a + c)(n + a + d)
" Bn + a+b + c + d-l)Bn + a
and
n(n + c + d- \){n + b + d- \){n
cn =
c+d- 2)Bn + a+b + c+d-\)'
As we have remarked before, since AnCn+x > 0 for n > 0, the polynomials pn are
orthogonal with respect to some positive weight function. In fact, it will be shown
that
2л-
о
Г (a + ix)T{b + ix)r(c + ix)T{d + ix) 2 2 2
pn(xz)pm(xz)dx
rBi*)
C.8.3)
= ^m,nn\{n + a +b + с + d — 1)„
Г(а + b + п)Г(а + с + п) ¦ ¦ ¦ Г (с + d + п)
х ,
Г(а +b + c + d + 2n)
where
р„(х2) - (а + Ь)п(а + с)„(а + d)npn(x2).
The relations given above continue to hold when a — b and/or с = d and the real
parts of these parameters are positive.
Definition 3.8.1 The Wilson polynomials pn (x) are defined by
Pn(x2) = (a + b)n{a + c)n{a + d)n
( —n, n + a + b + c + d — 1, a — ix,a + ix
\ a + b,a + c,a + d
where a,b,c,d are complex parameters.
It is evident from the definition that pn(x) is symmetric in b, c, and d. An
application of Theorem 3.3.3 shows that symmetry in all four parameters a, b, c,
and d exists. The Wilson polynomials are orthogonal with respect to the integrand
of Theorem 3.6.2 as a weight function. We denote the integrand by f(s).

3.8 The Wilson Polynomials 159
Theorem 3.8.2 With the contour and the parameters a, b, c, and d as in Theo-
Theorem 3.6.2,
t— / f(s)pn(-s2)pm(-s2)ds = 8m,n2n](n
2m J
Г (a + b + п)Г(а +c + n)T(a + d + п)Г(Ь + с + n)T(b + d + п)Г(с + d + n)
Г(а + b + c + d + 2n)
Proof. First observe that we can write
m
pm(-s2) = ^Ak(b- s)k(b + s)k,
*=0
where Ak are suitable constants. We compute
^*-r f f(s)pn(-s2)(b - s)k(b + s)kds
2ni J
= (a + b)n(a + c)n(b + с)пУ
(a+b)j(a+c)j(b-
¦ ^— I f(s)(a - s)j(a + s)j(b - s)k(b + s)kds. C.8.4)
The integral in the sum can be rewritten as
1 f Г(о + j + s)T(a + j - s)T(b + к + s)T(b + к - э)Г(с + s)T(c - s)r(d + s)T(d - s) ,
J с
This integral can be evaluated by Theorem 3.6.2. After a little simplification we
see that C.8.4) is equal to
2Г(й + b+ к)Г(а +с + п)Г(а + d + п)Г(Ь + с + к)Г(Ь + d + к)Г(с + d)(a + b)n
„ ( -n, n + a + b + c + d - l,a + b + k
¦3F2I ; 1
\ a+b, a + b + c + d +k
This 3F2 is balanced and so can be summed by the Pfaff-Saalschiitz identity. We
then get
Ini J
f(s)pn(-sz){b - s)k(b + s)kds = 2(-k)n
T(a + b + k)T(a + с + n)T(a + d + n)T(b + с + k)T(b + d + к)Г(с +d + n)
T(a + b + c + d+n+k)
The factor (—к)„ is zero for к < п. By symmetry in a and b we know that
Pn(s2) = {-=^-{n +a+b + c + d- \)n(b - s)n{b + s)n
n\
k=0
This completes the proof of the theorem.

160
3 Hypergeometric Transformations and Identities
The result in C.8.3) follows from Theorem 3.8.2 when a, b, c, and d are positive
or when a — b, с = d, and the parameters have positive real parts. The Wilson
polynomials contain many sets of orthogonal polynomials as limiting or special
cases. Here we show how the Jacobi polynomials (introduced in the previous
chapter) are derived.
Set a = b = (a + l)/2, с = d = ф + l)/2 + ico, and x = «V(l - 0/2 in
the 4F3 in Definition 3.8.1. Let со ->• oo. We get, except for a constant factor, the
Jacobi polynomial
a + 1
3.9 Quadratic Transformations - Riemann's View
Riemann exploited to the fullest degree the idea that a function is determined to a
large extent by its singularities. An example of this was given in the previous chap-
chapter in a discussion of the hypergeometric differential equation. Here we show how
the ideas developed there give Riemann's basic result on quadratic transformations.
The hypergeometric equation has regular singularities at 0,1, and 00 but no other
singularities. Suppose the exponents at 0 are 0 and 1/2. By a change of variables
x = t2, the solutions at 0 become analytic. But now singularities are introduced at
t = ±1 and we get another hypergeometric equation. To see the details, note that
B.3.4) of Chapter 2 shows that
C.9.1)
C-9-2)
is the set of solutions of the
d2y /1 \-bi-
dx2 \2x x-\
The change of variables x =
dy 1 dy
dx 2t dt
( 0 oo
p\ 0 a
I 1/2 c2
equation
b2\dy | ( b
)dx + {x
= t2 implies
and ^У
1
h
b2
-i
l
2t
A
j
\ ее) У
-Idy 1 d2y
2t2 dt 2t dt2
Substitute this in C.9.2) to obtain an equation that can be written as
d*y
dt2
t -1
2bxb2
l-bi-b2\dy
~dt
C.9.3)

3.9 Quadratic Transformations - Riemann's View 161
Apply Theorem 2.3.1 to conclude that the set of solutions of C.9.3) is
It follows that
Theorem 3.9.1
= P{ bi 2ci bi t) C.9.4)
-1
b\
b2
0

b2
oo
2c2
oo
2ci
2c2
1
1
?l f >
*2 J
1
h ^
*>2
This is Riemann's theorem on quadratic transformations. Let us derive the two
basic quadratic transformations for the 2F\ contained in Theorems 3.1.1 and 3.1.3.
Write C.9.4) as
= P { с а 0 1-х2
x- 1
Set с = 0, and replace 2b with 1 — 2b + a, 2a with a, and J with b —ato get
.6-0^-6 1/2
or

162 3 Hypergeometric Transformations and Identities
Since there is only one solution analytic at 0 with value 1 at that point, we get
b 4x
il-xJF\a-b+l'X)=2F\ a-b+l
This is the result of Theorem 3.1.1. Similarly, we have
0 oo 1 j Л ( 0 oo 1
P I a О с — \ = P I 2a с с
or
b 1/2 d ~ ) {2b d d 1+X
Oool
P I la с с х > = P I a 0
2b d d ) {b 1/2 d y2 X
With appropriate changes in the parameters we can write this as
or
2-х
This proves Theorem 3.1.3 once again.
The differential equation with regular singularities at — 1, 1, and oo but no other
regular singularities when written as
A - x2)y" - 2xy' + [v(v + 1) - -^-J у = 0
is called Legendre's differential equation. It is clear that the set of solutions is
( -1 oo 1 |
P I pull v + 1 fi/2 x \ . C.9.5)
[-H/2 -у -д/2 J
A comparison of C.9.5) with C.9.4) shows that quadratic transformations apply
to the solutions of the Legendre equation.

3.10 Indefinite Hypergeometric Summation 163
3.10 Indefinite Hypergeometric Summation
We have derived a number of hypergeometric identities in this chapter. In this
section we consider the problem of evaluating partial sums of hypergeometric
series. Gosper [1978] has given an algorithm that yields the value of such a sum
provided it is a hypergeometric term. To make this more precise, suppose that
YH=i ck is a partial sum of a hypergeometric series. The problem is to find a
function Sn such that
ck=Sk-Sk-U C.10.1)
when it is assumed that SkjSk-\ is a rational function of k. We refer to such an Sk
as a hypergeometric term. From C.10.1) it is clear that
*=1
Write
Ck Pk 4k
C.10.2)
Ck-l Рк-l Гк
where pk,qk, and rk are polynomials in к satisfying
= 1, C.10.3)
for all integers j > 0. The necessity of this condition will become evident later,
but qk and rk can be chosen so that C.10.3) is true. Condition C.10.3) implies
that if к + a and к + fi are factors of qk and rk respectively, then a - ft cannot
be a nonnegative integer. Suppose that, initially in the decomposition C.10.2),
gcd(qk, rk+j) = gk- Then replace qk, rk, and pk with
, qk , rk
qk = —. rk = —. and Pk =
qk . k
gk gk-j
It is easy to check that
It is now clear that after a finite number of repetitions of this process, condition
C.10.3) will hold for all j > 0. The next step is to write
Sk = —fkck. C.10.4)
Pk
Substitute in C.10.1) and use C.10.2) to find a relation for fk, which is the only
unknown in C.10.4). Then
pk = qk+ifk ~ rkfk-i- C.10.5)

164 3 Hypergeometric Transformations and Identities
By the condition on Sk, we see that fk is a rational function of k. We show that it
is a polynomial. Let
fk = tk/mk,
where lk and mk are polynomials in к with no common factors. Suppose mk is
not independent of k. Let j be the largest nonnegative integer such that к + X and
к + X + j are both factors ofmk. Substitute the expression for fk in C.10.5) to get
pkmkmk-i = qk+xlkmk-X - rklk-\mk. C.10.6)
Since к + X — 1 | mk-i, the last equation implies that к + X — 1 | rktk-\mk. But
^_i, tk-i) = 1 and к + X — 1 does not divide mk by the maximality of j. So
* + A.-l|rt. C.10.7)
Similarly, the fact that к + X + j | qk+\lkmk-\ implies that
k + X + j\qk+loi k + X + j -l\qk. C.10.8)
By C.10.7) and C.10.8), gcd(^b rk+j) ф 1. This contradicts C.10.3) and we can
conclude that / is a polynomial of degree d, say, given by
fk=aokd + aikd-l+--- + ad. C.10.9)
Substitute this expression for fk in C.10.5) to obtain a system of linear equations
satisfied by ao, a\,..., a^. If this is a consistent system the values ao, ai,..., a<j
are obtained by solving the equations. From fk we get Sk by C.10.4).
To obtain the possible degrees of the polynomial fk, write C.10.5) as
z Afk + fk-l) , . , .(fk — fk-\) /Т1ПШЧ
Pk = (Чк+i ~rk) + (qk+i+rk) . C.10.10)
There are two cases. First suppose that
deg(qk+i + rk) < deg(?i+i - rk) = d'.
Since deg(fk - fk-\)/2 < d, it follows that
d = deg pk - d!.
Now suppose that
(qk+l + rk)/2 = bkd' + ¦¦¦, ЪфО
and
(qk+l - rk)/2 = ckd'~x +¦¦¦.

3.10 Indefinite Hypergeometric Summation 165
Use these expressions in C.10.10) to obtain
If aoc + a0bd/2 ф 0, then
d = deg pk-d' +I.
Otherwise,
d = —2c/b and d > deg pk — d! + 1.
The last value of d is used only if it is an integer greater than deg pk—d' + l. This
completes Gosper's algorithm. It decides whether a partial sum of a hypergeometric
series can be expressed as a hypergeometric term and gives its value if it does.
Zeilberger [1982] extended the scope of this algorithm by taking ck as a func-
function of two variables n and к rather than just k. We discuss the Wilf-Zeilberger
method here and in the next section. This method is very powerful in proving
hypergeometric identities.
Suppose the identity to be proved can be written as
к
where A{n) ф 0 and n > 0. Divide both sides by A(n) and write the identity as
^F(n,fc)=l. C.10.11)
к
This implies that
Earlier we were trying to express F(n,k), or rather T(n,k), as the difference
«Sjt+i — <Si, but this is often not possible. As an example, consider the sum
Л (-\f(-n)k
?-> k\
k=0
when j < n. By going through the steps in Gosper's algorithm, it can be seen that
this sum is not expressible as a hypergeometric term.
In Zeilberger's method one tries to write the difference F(n + 1 ,k) — F(n,k) as
<Sjt+i — <Sjfc. This improves the situation. Suppose there is a function G(n, k) such
that
F(n + 1 ,k) ~ F(n, k) = G{n, к + 1) - G(n, k). C.10.12)

166 3 Hypergeometric Transformations and Identities
This function G can be determined by Gosper's algorithm. Then
к
^ (F(n + 1 ,k) ~ F(n, k)) = G(n, K + \)- G(n, -L). C.10.13)
k=-L
If we assume that G satisfies the property
lim G(n,k) = 0, C.10.14)
k—r±oo
then it follows that
У~] F(n,k) — constant.
k
It is then sufficient to verify the identity for one value of n, say n = 0.
Thus to prove C.10.11), find a G that satisfies C.10.12) and C.10.14). If this G
exists, then C.10.11) is known to be true after it is verified for n = 0. This method
works for a very large class of identities. The reader may consult Petkovsek, Wilf,
and Zeilberger [1996] and Nemes, Petkovsek, Wilf, and Zeilberger [1997] for
examples and further results.
As an example, consider the identity
2—i ьпп
k\2n
к
Here
F(n'k)=
к\2п
and one can show that
G{n,k)=^—
satisfies C.10.12) and C.10.14). The identity is therefore verified since it is true
for n = 0.
The next section contains further illustrations of the Wilf-Zeilberger method
and a comparison with a related method.
3.11 The W-Z Method
In a series of papers, Zeilberger (sometimes jointly with Wilf) developed a tech-
technique he called creative telescoping. The method is often referred to as the W-Z
method in that an important component of this work was presented in the Wilf-
Zeilberger paper. The method is surprisingly easy to describe in full, but for sim-
simplicity we shall apply it to some elementary identities and then compare this method
with that of Pfaff.

3.11 The W-Z Method 167
Suppose that there is a linear, homogeneous recurrence relation that we wish to
prove for a particular sum. In other words, suppose
/t=-oo
where for each n, F(n, k) is zero for all but finitely many k. Suppose we expect
that
a(n)S(n) = P(n)S(n - 1)
(so that in fact S(n) = 5@) П"=1 (?(./)/«(./))• Then the w~z method constructs
a function G{n,k) (which again for each n is 0 for all but finitely many k) so that
a(n)F(n, k) - P(n)F(n - 1 ,k) = G(n, k) - G(n, к - 1).
Then the desired recurrence follows immediately. Therefore,
a(n)S(n) - p(n)S(n - 1) = J](«(n)F(n, k) - p(n)F(n - 1, k))
к
к
= 0
because the final sum telescopes. This example illustrates the appropriateness of
the label "creative telescoping."
The best way to appreciate this is through some examples. Consider the Chu-
Vandermonde summation. We wish to prove
Sv(n) = ~ , where Sv(n) = J] Fv(n, k)
(b)n k=o
and
Fv(n,k)= —.
k\(b)k
In other words, we wish to find G(n,k) so that
(b + n- l)Fv(n, k)-{b-a + n- l)Fv(n - 1 ,k) = G(n, k) - G(n, к - 1).
C.11.1)
Zeilberger has fully implemented on the computer an algorithm for finding G(n,k).
For problems involving hypergeometric series such as Sv(n), G(n, k) has been
shown by Zeilberger to be of the following form:
G(n,k) = R(n,k)F(n- l,k).

168 3 Hypergeometric Transformations and Identities
However, we can easily work out the value of G(n, k) by inspection. In C.11.1)
set к = 0. Then, assuming G(n, — 1) = 0, we see
G(n, 0) = (b + n- l)Fv(n, 0)-(b-a + n- \)Fv(n - l, 0)
= a.
Set* = 1 in C.11.1). Then
G(n, 1) = G(n, 0) + (b + n- l)Fv(n, l)-(b-a + n- l)Fv(n - 1,
In this manner we can work out (either by hand or with the use of a computer
algebra system) the conjecture that in fact
G(n,k) = (a+k)Fv(n-l,k). C.11.2)
Once conjectured, the proof of C.11.2) is pure algebra. We have
(b + n- l)Fv(n, k)-(b-a + n- l)Fv(n - 1, *)
(b + n- l)(-w)t(q)fc (b-a + n- 1)(-и + 1)к(а)к
k\(b)k k\(b)k
(~" * uik~l(a)k № + n- l)(-n) -(b-a + n- \)(-n + k)]
k\(b)k
(n\,
(-an-k(b-a+n-\)). C.11.3)
к\(Ъ)к
However,
G(n,k)-G(n,k- 1)
(a + k)(l - n)k(a)k (a + k-
k\(b)k (*-l) !
l)ii(q)t
{-an -k(b-a + n-
k\(b)k
= (b + n- l)Fv(n, k)-(b-a + n- l)Fv(n - 1, k),
(by C.11.3)). Hence creative telescoping shows us that
(b + n- l)Sv(n) = (b-a+n- l)Sv(n - 1),
and so by iteration Sv(n) = (b — a)n/(b)n, as desired.

3.11 The W-Z Method 169
Now turn to the Pfaff-Saalschutz summation. This summation was initially
treated in Chapter 2. It may be stated as follows:
(c-a)n(c-b)n
Ьр(П)- (c)n(c-a-b)n'
П
where Sp(n) = ^ Fp(n, k) with C.11.4)
t=o
{-n)k{a)k(b)k
FJn.k) = .
л k\(c)k(\-n + a + b-c)k
Note that C.11.4) is clearly equivalent to
(c + n- l)(c -a-b + n- l)Sp(n)
= {c-a+n- l)(c - 2> + n - l)Sp(n - 1).
We proceed as before. We let
Gp(n,k) = Rp(n,k)Fp(n-l,k)
and we wish to construct the rational function Rp(n, k) so that
(c + n - l)(c - a - b + n - l)Fp(n, k)
-(c-a + n- l)(c -b + n- \)Fp{n -\,k) = Gp(n, k) - Gp(n, к - 1).
C.11.5)
Consequently, with к = 0 in C.11.5) we get
Gp(n, 0) = (c + n - l)(c -a-& + n-l)-(c-a+n- l)(c -b + n-I)
= (c + n- l)(-a) + a(c - fc + n - 1)
= -afc.
Now by C.11.5) with A; = 1,
G,(n, 1) = Gp(n, 0) + (c + n - l)(c -a-b + n- 1)
- (c - a + n - l)(c - b + n - 1)
cB — n + a+fo — c)
B -
cC - n + a + b - c)
Thus as in C.11.2) we may conjecture
Gp(n, k) = -(a + k)(b + k)Fp(n - 1 ,k). C.11.6)
The proof of this conjecture is again merely an algebraic exercise. In each case it

170 3 Hypergeometric Transformations and Identities
turns out that
(c + n- l)(c-a-b + n- l)-(c-e+n- l)(c-b+n- \)Fp(n
= Gp(ri,k)-Gp(n,k-l)
k)(l-n)k(a)k(b)k
k\(c)kB-n + a + b-c)k
(a + k- \)(b + к - 1)A - w)t-i(a)t-i(b)t-i
which simplifies to our desired result. So C.11.4) has been proved by the W-Z
method.
We now turn to Bailey's 4F3 summation, which is somewhat more difficult. Our
object here is to prove
(b - a)n
ft/2, (Ы-l)/2, e + 1 •1J=
Note that this is a balanced 4 F3. In notation suitable for the W-Z method, we wish
to prove
(b - a)n
where SB(n) = ELo fb("' *) with
+ n)k(-n)k
In this instance, our first expectation is that we can prove
(b + n- l)SB(n)-(b-a+n-
However, the method utilized in the previous two cases fails initially. It is at this
moment that we realize how useful computer algebra is in such matters. It turns
out that we can use the W-Z method to obtain a second-order recurrence, namely
(n + 1)(—n — b + a)FB(n, k)
+ (-a2 + ba - a + 2nb + 3b + 2 + An + 2n2)FB(n + l,k)
-(b + n + l)(a +n + 2)FB(n + 2, k)
= GB(n,k)-GB(n,k-l),
where
GB(n,k) = FB(n,k).
(b + n)(n -k + \)

3.11 The W-Z Method 171
This implies that
(n + l)(-n -b + a)SB(n) + (-a2 + ba - a + 2nb + 3b + 2
2n2)SB(n + l)-(b + n+ \)(a + n + 2)SB(n + 2) = 0. C.11.8)
Also, since SB@) — 1, SB(l) = (b — a)/b, and since (b - a)n/(b)n satisfies the
above recurrence, we see that C.11.7) is proved.
A couple of observations should be made here. First, an identity like C.11.7)
often arises in practice as a conjecture. In other words, if C.11.7) is true, then
something useful follows (c.f. Andrews and Burge [1993]). Consequently, one
usually knows the form of the desired summation identity before looking for a
proof. Second, suppose that we are dealing with a more complicated identity
where we perhaps had not determined exactly what the summation should be like.
Here Marco Petkovsek has produced an auxiliary algorithm for the W-Z method. It
finds the minimal recurrence satisfied by the sum in question. Thus in this instance
Petkovsek's algorithm applied to C.11.7) would reveal C.11.8).
There is a somewhat different summation method due to Pfaff. This method
is less algorithmic than the W-Z method. However, it spreads out the algebraic
complications to systems of recurrences. Consequently, it may provide new sum-
summations in addition to the one we wish to prove and it may allow the required
algebra to be considerably simpler than that required by the W-Z method. Pfaff's
method rather resembles the W-Z method; however, it allows the various additional
parameters in the summation to play an important role. Pfaff's method begins very
simply. We merely subtract term by term the sum at n — 1 from the sum at n.
Consider the Chu-Vandermonde summation once again. We phrase the problem
slightly differently. Let
Now we note
ySv(n-l,a + l,b+l). C.11.9)
b

172 3 Hypergeometnc Transformations and Identities
Note that C.11.9) together with Sv@, a,b) = 1 uniquely defines Sv(n, a, b). But
if
av(n,a,b) =
then
(b - a)n (b - a)n-i
av{n, a, b) — crv(n — 1, a, b) =
(b)n
(b-a)n-i
(-a)
a (b — я)л-1 я
= — Т. ~~п = — т~ ^и (и — 1, я
Hence
Sv(n, a, b) = av(n, a, b) — (b - a)n/(b)n
as desired.
Pfaff-Saalschiitz summation by Pfaff's method is as follows: Let
Now we note
5p(n, a, fe, c) — Sp(n — 1, a, b, c)
j^ j\(c)j(\ -n + a + b-c)j jl(c)jB -n + a + b-c)j
= "^2 . J J~l —((-и)A -n + a + b-c + j)
j=0 ^ 'J^ 'J~X
n(a + b+l-cJ_^
c(l-
У=о
n +
(
me
n(a
Я +
+ 1
-(a
+
b
b
C
+ 1
c)B
-и
-с)яй
1)у(Ь+1)у
-с)я^
Sp(n - 1, a + \,b + 1, с + 1).
— и + a + b — c)B — n + a + b — c)
C.11.10)

3.11 The W-Z Method 173
The rest follows by showing that (c — a)n(c — b)n/[(c)n(c -a — b)n] satisfies the
same recurrence and is equal to 1 when n equals 0.
This is precisely the proof given by Pfaff for this formula in 1797. We finally
look at Bailey's 4F3 summation by Pfaff's method.
Just as the W-Z method did not work as expected, so too does Pfaff's method
require a new twist. Here we wish to prove that
SB(n,a,b)= (*~g)", C.11.11)
where
Subtracting term by term we find
SB(n,a,b)-SB(n-l,a,b)
a(l-b- In)
b(b + 1)
where
TB(n-l,a + 2,b + 2), C.11.12)
We have thus introduced a new sum. Calculation of this sum for и = 1,2, and 3
suggests that
TB(n, a, b) = " . C.11.14)
(b + 2n- l)(fc)
We now try term-by-term comparison of TB(n,a,b) with SB(n,a,b) and
SB(n — 1, a, b). The second comparison yields
(b + n- l)(a + n)
TB(n,a, b) - SB(n - 1,a, *) = + -TB(n -
C.11.15)
The two recurrences C.11.12) and C.11.15) together with the initial values
SB@, a,b) = TB@,a,b) = l completely define SB(n,a,b) and TB(n, a, b). It is
now again an easy algebraic exercise to see that (b — a)n/{b)n and (b — a)n/((b +
2n — \)(b)n-\) satisfy the same recurrences and initial conditions. Consequently,
we have not only proved C.11.11) but we have also proved C.11.14).

174 3 Hypergeometric Transformations and Identities
3.12 Contiguous Relations and Summation Methods
From B.5.6) and B.5.8) we get one of Gauss's contiguous relations, namely
{c - a - b)F = {c - a)F(a-) - b{\ - x)F(b+).
Set a = -n + 1 to find
(c + n- lJFi(-n, b;c;x)-(c-b + n- lJFi(-n + 1, b; c; x)
Now put x = 1 to see that precisely the creative telescoping of Zeilberger re-
reduces the right-hand side to 0, and we have produced the W-Z proof of the Chu-
Vandermonde sum. Similarly,
(c + n — l)(c — a — b + n — \)T,F2(—n, a, b; с, 1 — n + a + b — c; x)
- (c - a +n - l)(c-b + n - lKF2(-n + l,a,b;c, 2-n + a+b-c;x)
j>0 v U - l)!(c);-iB -n+a + b- c);_i
j\(c)jB- n + a+b -c)j
= -ab{\ - xKF2(l -n,a + l,b+l;c,2-n + a + b-c;x),
and if we set x = 1 we get the W-Z proof of the Pfaff-Saalschiitz summation.
Thus we see how the W-Z method is an effective algorithm for discovering useful
instances of contiguous relations. In the case of Bailey's 4F3, the W-Z method fails
to find a first-order recurrence because there is no three-term contiguous relation
relating
b/2Ab+l)/2,a
to a third 4 Ft, multiplied by a factor including A-х). However, when one moves
to four terms, such a relation holds and the W-Z proof of Bailey's summation
follows by setting x = 1.

3.12 Contiguous Relations and Summation Methods 175
Pfaff's is even more obviously a method of contiguous relations. In his method
x is set equal to 1 before we begin. The full Pfaff proof of Chu-Vandermonde is
given by directly establishing the contiguous relation
-и.о. Л r/-n + U Л a р/-« + М
The Pfaff proof of Pfaff-Saalschiitz is just
-n,a,b \ / \-n,a,b
iFl[ - ' ¦ - ¦ -- -"> *"\с,2-п+а+Ь-с'
n(a+b+l-c)ab ^, > ,., „ ,.,„,. .
c(l -n + a+b-c)B-n+a + b-c) \c +1,3 - n + a + b - с
The proof of Bailey's formula relies entirely on two contiguous relations:
/a/2, (a + l)/2, b + n, -n \ _ fa/2, (a + l)/2, b + n - 1, 1 - n .
4 3V b/2, (b+l)/2,a+l ' ' 4 3|
a{\-b-2n) Па/2)+\,{а + Ъ)/2,Ъ + п,\п
4 3V Ъ/2+\ф + 3)/2а + 2
b(b+\) 4 3V Ъ/2+\,ф + 3)/2,а
and
fa/2,(a + l)/2,b + nl,n
4 \ */2 (*+l)/2,a
fa/2,(a + l)/2,b + nl,ln
4 3V b/2,(b+\)/2,a + l
(b + n-\){a+n) Па/2) + I, (a + 3)/2, b + n, I -n
+ 2 ;
The discoveries of Wilf and Zeilberger truly revolutionized the study of sum-
summations of terminating hypergeometric series. An important offshoot of this work
is the MAPLE implementations of these algorithms also prepared by Zeilberger.
Peripheral to this accomplishment has been a philosophical debate by one of us
(Andrews [ 1994]) with Zeilberger [ 1994] about the implication of these discoveries
for artificial intelligence.
Currently the internal constraints in MAPLE have prevented a W-Z proof of
j-, ( — 2n, x + 2n + 1, x — z + \, x + n — 1, z + n + 1
\_ (x/2) + 1, (x + l)/2, 2z + In + 1, 2x — 2z
(\/2)n{2z-x)nBz-x + n + 2)n _ 2]
(x + 1)„_зО + 2n- 2KO - z)n(z + n

176
3 Hypergeometric Transformations and Identities
The Pfaff-method proof involves a gigantic simultaneous treatment of twenty iden-
identities established in the manner discussed earlier. Undoubtedly, the improvements
of software and hardware will eventually yield a W-Z proof of C.12.1).
It is to be hoped that this contest between methods will serve to make clear that
progress occurs when human thought aided by machines applies itself to any given
problem. What should not get lost in the shuffle, however, is the observation that
the Pfaff and W-Z methods are valuable applications of the classical theory of
contiguous relations to summation problems. It would be hard to believe that these
are the only such methods buried in contiguous relations, and further investigations
are clearly merited.
Exercises
1. Verify the following quadratic transformation formulas:
(ь,
(с)
a,b
2b
a,b
a-b+Y
\х \ = (\ -
a,2b-a
ь + \ ;
a,a-b + ±
2a - 2b + 1 ' A +
(e)
= 2*1
+ 2F1
/2a, -2b+1 1
\ a-b+l ' 2V l+x 2)
2. Verify the following one-parameter transformations of Kummer [1836]:
4х
B(e

Exercises 177
a,Da-
x\-"a/2,(a-
2 2 Ч Bа+5)/6 '\2-x
a/2, (a + l)/2 Ax
a + C/4) ; (l+x]
3. Show that
^/2,B-fl)/6. Л ^а/2 „ fa/2, (a
Г(Bа+2)/3)ГA/2)
Г((а
(b) ^BТ+\1^г
Г(Bа + 3)/4)Г(Bа-
4. Deduce Kummer's identity in Corollary 3.1.2 from Euler's integral for 2F1.
5. Note that C.1.2) is equivalent to
Here и is a positive integer.
Multiply by xc~l A - x)d~c~l and integrate over @,1) to get
-2n,2b, с \ ( -n,b,c,d-c
Deduce that
2a,2b,-k \ ( a,b,-k,d
l)F
where к is a positive integer,
(d) Let d = — - + e and & ->• 00 to see that (a) holds without restriction on n.

178 3 Hypergeometric Transformations and Identities
6. Show that for 0 < x < \,
7. Prove formulas C.2.3) and C.2.14) for the elliptic integrals E and K.
8. Prove Elliot's result contained in Theorem 3.2.8.
9. Prove Lemma 3.2.9.
10. ^О \
l?l 1.
(a) Prove that (92(q) + 042(<?))/2 = 032(?2) and /Цч)^) 4(^)
(b) Deduce that the arithmetic-geometric mean ofe%(q) and ^(g) is 1 (i.e.,
M@22(<?) A2(<?)) = 1).
(c) Prove that 9%(q) - 9%(q2) = 9%(q2).
(d) Deduce from (c) and (a) that 9%(q2) - 9$(q2) = 0%(q) and 9%(q) =
(e) For 0 < q < 1, let к := k(q) := 9$(q)/9i(q). Prove that 0 < it < 1 and
Af A, it') = 03(?), where jfc'2 = 1 - к2. Also prove that
K(k) = ~92(q).
11. (a) Use Exercise 2.30 to show that VJc 9ъ(е~пх) = 9ъ(е~ж1х)апйфс 9г(е~пх)
= 94(е-л/х).
(b) With k(q) as in Exercise 10, show that к(е~лх) = к'(е-ж'х).
(d) Show that the unique solution of 9$(q)/92(q) = к for 0 < к < 1 is
q = e-nK',K
12. Prove formula C.2.9), that /(a, b) = I ((a + b)/2, Jab).
13. Let a < b < с Prove that
'* dx ж
/
J a
f
Jb
— b)(x — c) M(Vc - a, ~Jc — b)'
dx in
-a)(x —b)(x -c) M{Jc -a, -Jb -a)

Exercises 179
14. Multiply Euler's transformation
vfa'b \ ,л ^с-а-ь „fc-a,c-b
lF\ с <Х) = A-ХУ 2^^ c ;
by xd~l{\ - x)e~d~l and integrate over @, 1) to obtain Corollary 3.3.5.
15. Prove that
-n,-a,-b . Л = Ф + с - \)n(a + c)n
3 2| cl-n-a-Ь-с1 ) (a + b + с - 1)„(с)и
аи
1 +
(b+c- Ща+с + п- 1)_
16. Prove Watson's identity in Theorem 3.5.5(i) by multiplying Equation C.1.2),
by xc 1(l — x)c 1 and then integrating over @, 1). (Note that и is a
nonnegative integer in the above formula. Otherwise the formula does not
hold over @, 1).)
17. (a) Show that
pf a,b,c,d,-n j
5 \l+a-b, 1+a- c, 1+a-d, 1 + a + я '
when За + 2 = 2(b + с + d - n).
(b) Prove that
a,b,c,-n
+a-fe, l+a-c, l+a + n'
Г (a - 2fc + 2n + 1)Г(и + ^)Г(д - fr + 1)Г(д +n + 1)Г(^ - b)
T{a -2b+ 1)Г(§)Г(а - b + n + 1)Г(в + In + 1)Г(| - b + n)
when 1 + 2a = 2b + 2c — 2n.
(c) Deduce that
-k,a,b,c \ Ba)kBb)k(a +b)k
; i =
4 3| 1 - а - it, 1 - b - k, 1 - с - к' V (a)t(*)tBe + 2b)k'
when 1 - 2c = 2a + 2b + 2k.

180 3 Hypergeometric Transformations and Identities
(d) Deduce Clausen's identity:
a,b
2a,2b,a+b
(Hint: Equate the coefficient of x" on both sides.) Note that a different
proof of this identity was given in Exercise 2.13.
18. Prove that
(a-d-e+\)n
(a)
i-d + \)n(a -e + \)n
( a — b — с + \,d,e,—n
\tз| ; l
\a — b + \,a — с + \,d + e — a — n
(a-b-c+\)n
(a-b+\)n(a-c+\)n
( a — d — e+\,b,c,—n \
4 \a-d+\,a-e+l,b + c-a-n' )'
a, (a/2) + I, b,с Л Г (a - b + 1)Г(а - с + 1)
;1 =
^a/2,a-b+l,a-c+l' ) Г(а + 1)Г(а - b - с + I)'
19. (a) Use the method of Lemma 3.4.2 to prove that
a,b,c,-n \ (d-a)n
(b)
(d)n
a-d + l,a/2,(a + l)/2,a-b-c+l,-n .
+ l,a—c+l,a—d — n + l)/2, (a — d — n)/2 + 1'
This transforms a nearly poised 4F3 into a balanced 5 Fa,. Deduce that
d — a — n — \)(d — a)n-i
a,fe,n \ (a-2*)B
3F2 _ ,,,„, , ', 1 =
(а-Ь+\)п((а/2)-Ь)п(-2Ь)п
~ (a - b + l)n(-2b)n'
(a - 2* - l)B(
(a-b+ l)B

Exercises 181
-n,b,c,e \ (d-e)n
\-n-b,\-n-c,d ) (d)n
( e,l-n-b-c,-n/2,(l-n)/2,l-n-d \
'5 \l-n-b, l-n-c, (l+e-d- n)/2, (e-d- n)/2 + 1' )'
(Whipple)
See Bailey [1935, §§ 4.5 and 4.7] for the reference to Whipple.
20. Prove that
{Hint: Apply Exercise 19(f) to the 4F3 that appears after squaring the 2F1.
Then apply Theorem 3.3.3.)
21. Obtain the transformation in Exercise 19(a) by multiplying Whipple's qua-
quadratic transformation for 3F2 in C.1.15) by xd+n~l and equating the coeffi-
coefficients of x".
22. Derive the formula
/ a,a/2+l,b,c,-n \ = (d - a -n - l)(d - a)n-i
5 \a/2ab + lac+ld' )~
a/2,a-b + l,a-c+l,d' ) (d)n
( a/2+l,(a + l)/2,abc + l,ad + l,n
'5 \(a-d-n + 3)/2, (a-d-n+ 2)/2, a-b + l,a-c + V
by using the formula in Exercise 19(b) instead of Dixon's theorem in the
proof of Lemma 3.4.2. The above formula transforms a nearly poised 5 Fa, into
a balanced 5F4. See Bailey [1935, §4.5].
23. (a) By letting a ->• 0 in Corollary 3.5.2, evaluate the sum
(b)n(c)n(d)n
\-b)n(\-c)n{\-d)n
(b) More generally, show that
(a + b)n(a + c)n(a + d)n
(a-b + \)n(a -c+l)n(a-d + 1)„
Г(а - b + 1)Г(а - с + 1)Г(а -d+ 1)ГA - b - с - d)r(b)r(c)F(d)
Г (а + Ь)Г(а + с)Г(а + d)F(l -Ъ- с)ГA - Ъ - d)F(l -c-d)

182 3 Hypergeometric Transformations and Identities
24. (a) Observe that Dougall's formula can be written as
/ k,k/2+l,k + b-a,k+c-a,k + d-a,a+n,-n
7 6[k/2,a-b + l,a-c+l,a-d+l,k-a-n + l,k+n+l'
(k + \Ub)n(c)n(d)n
(a - k)n(a -b + \)n(a - с + \)n(a - d + 1)„ '
when k = 2a—b — c — d+l.
(b) In the proof of Lemma 3.4.2 use (a) instead of the Pfaff-Saalschutz's
identity to get
/ a,a/2+l,b,c,d,e,f,g,-n \
9 ^8 I " 1 I
\a/2, a-b+ \,a-c+\,a-d+\,a-e + \,a-f+\,a-g+\,a + n+l )
(а +\)„(к-е+ 1)„(к - f + 1)„(к - g + 1)„
{k+\)n{a-e + 1)„(а - / + 1)„(а - g +
/ k,k/2 + l,k + b-a,k + c-a,k + d-a,e,f,g,-n
Fs\k/2,a-b + l,a-c+\,a-d+ l,k-e+ \,k - f+ l,k - g+ l,k + n + l'
when к = 2a-b-c -d + 1 and b + c + d + e + f+g-n =
(c) Deduce Theorem 3.4.5 from (b).
See Bailey [1935, §4.3].
25. (a) Show that
/ a,a/2+l,b,c
4 \a/2,a-b+l,a-c+l'
Г (а - b + 1)Г(а - с + 1)Г((а + 1)/2)Г((а + 1)/2 - ?> - с)
~ Г (а + 1)Г(а - & - с + 1)Г((а + 1)/2 - &)Г((а + 1)/2 - с)'
(b) Add this identity to the one in 18(b) to obtain a formula for
/ a a + \ a b b+l с d+l
2' 2 '4+ '2' 2 '2' 2
а а — i> a — b+l 1 а — с , а — с + I '
26. Show that
\ 4' ^ + ' 2 ' Г
1\3 /1-3\3 /1 - 3 • 5N 3
23/2
(c) 1-5 - +
1\5 „/1•3\5 _/1 - 3 • 5Ч 5
2) Ч2-4У 'Л2-4-6У {ГC/4)}4-

Exercises 183
27. Show that
(a) s + (s +:
\
Г((* + 1)/2)Г(A-3«)/2)
/s\ fs(s + l)l sinm
(b) s-(s + 2)(-\ +(s+4)l \ 2 > =-^~¦ (Dougall)
For Exercises 26 and 27, see Hardy [1940, pp. 105-6] or Bailey [1935, p.
96].
28. Prove that
2 ~ 22"+1(n!J
— = 1 + V ——. (Takebe Kenko)
4 ^ Bn+2)!
See Roy [1990] for reference.
29. Evaluate the sums
2n + l \ fp +
к V
2n
Л

184 3 Hypergeometric Transformations and Identities
'n\2 fm + 2n-ks
(i) ?
^ 2n (n + k\Bk\{-
30. Prove that
/ a, I + a/2, d/2, (d + l)/2, a-d, I + 2a-d + m,-m
7 6W2, 1 + a - ^/2, a + A - ^)/2, l+d, d-a-m, l+a + m'
(l+a)m(l+2a-2d)m
l+a-d)m(l+2a-d)
See Bailey [1935, p. 98].
31. Prove that
(a) 2f i
2a,2b,a+b
= sF21
(b) 2Fi(
\a-r v — j / \ы^о— 2 /
/ 9/7 9J, — 1 л д. A, _ 1 \
(Orr)
— 2, a
See Bailey [1935, p. 86].
32. Prove Theorem 3.6.1.
33. Prove that
Г(х+т)Г(у + т) (x,y,v + m-\ \
3F2 ; 1 to n terms
Г(т)Г(х + у + т) у v,x + y + m J
Г(х+п)Г(у + п) (x,y,v + n~\ \
= ---—- -3^2 ;1 torn terms.
Т{п)Г{х+у + п) у v,x + y + n J
34. Prove formula C.8.2).
35. Prove that Wilson's polynomial in Definition 8.1 is symmetric in a, b, c, and
d.

Exercises
185
36. Show that
a, (a + 2)/2, b, c, d, e, -n
(a + \)n(a - b - c)n(a - b - d)n(a - с - d)n
(a + 1 - b)n{a + 1 - c)n(a + 1 - d)n{a - b - с - d)n
n(n + 2a — b — с — d)(a — b — с — d)]
{a-b- c)(a -b- d){a - с - d) J'
when e — 2a +n - b — с — d. The 7F6 is 4-balanced and very well poised.
37. Prove formulas C.5.2), C.5.3), and C.5.4) connected with reciprocal polyno-
polynomials.
38. Prove Bailey's cubic transformations:
(a)
a,2b-a-l,a+2-2b x\
/ a q+± a+2
,-a ^ 3' з ' з -27*
(b)
(a,b-\,a + \-b
3 2\ 2b,2a + 2-2b '
For comments on these cubic transformations and for the reference to Bailey,
see Askey [1994].
39. Show that
la
a,a+l/2.
2a + ( '
and
40. Define
a, a +1/2
2a
2a-1
n-\
ф(х;п) =
j=o
ф(х;0) =

186 3 Hypergeometric Transformations and Identities
If
show that
See Gould and Hsu [1973].
41. By appropriate choices of g(&) and </>(&; и), show that
-n,n+a,l+a-b-c .A = Ф)п(с)„
\+a-b,\+a-c ' )- (\+а-Ъ)п{\+а-с)„
gives the sum of the terminating very well poised 5F4.

Bessel Functions and Confluent
Hypergeometric Functions
In this chapter, we discuss the confluent hypergeometric equation and the related
Bessel and Whittaker equations. The Bessel equation is important in mathemati-
mathematical physics because it arises from the Laplace equation when there is cylindrical
symmetry. The confluent hypergeometric equation is obtained when we start with
a second-order differential equation whose only singularities are regular singular-
singularities at 0, b, and oo; we let b —> oo. The resulting equation has oo as an irregular
singular point obtained from a confluence of two regular singularities. Thus, the
confluent equation can be derived from the hypergeometric equation by chang-
changing the independent variable x to x/b and letting b ->• oo. The solutions are \Fy
functions, and some properties of these functions are limits of properties of 2^1
functions. However, it is often easier to derive the results directly than to justify
the limiting procedures.
Whittaker transformed the confluent equation to one in which the coefficient of
the first derivative is zero. Solutions of this equation are called Whittaker functions.
We find their series and integral representations and their asymptotic behavior and
then give some important examples such as the error function and the parabolic
cylinder function.
The Bessel equation can be derived from a particular Whittaker equation and
can be solved to obtain the Bessel functions to which we devote a good por-
portion of this chapter. These functions are also important for their role in Fourier
transforms in several variables. We present some integral representations of
Bessel functions due to Poisson, Gegenbauer, and others. Later, we discuss
some interesting finite and infinite integrals involving Bessel functions as inte-
integrands. Some of these are really limits of generating functions for Jacobi poly-
polynomials.
The sine and cosine functions are particular cases of Bessel functions. Thus, it is
useful to look for generalizations of formulas for these trigonometric functions to
Bessel functions. Nicholson found a remarkable extension of sin2 x + cos2 x = 1
to Bessel functions; he expressed it as an integral formula. We present Nicholson's
187

188 4 Bessel Functions and Confluent Hypergeometric Functions
formula and later show how Lorch and Szego used it to derive results about zeros
of Bessel functions.
We end this chapter with a discussion of some work of Saff and Varga on zero-
free regions for the sequence of polynomials that are partial sums of the exponential
function and, more generally, of the i F\ functions.
4.1 The Confluent Hypergeometric Equation
It is easily seen that the hypergeometric series
) DЛЛ)
is a formal solution of the differential equation
[8(8 + bl-l)---(8 + bq-l)-x(8 + al)---(8 + ap)}y = 0, D.1.2)
where
When p > 2 or q > 1, this equation is of order max(p, q + 1) > 2, and the
resulting equation is not as useful as the hypergeometric equation. When q = 1
and p = 0 or 1, the equation is still of second order with a regular singular point at
x = 0, but the other singular point is at x = oo and is an irregular singular point.
Although irregular singular points cause serious problems, it is still possible to say
something about the solutions near them.
We consider the case where p = q = 1. Then the equation is
or
xy" + (c - x)y' -ay = 0. D.1.3)
This equation can be obtained from the hypergeometric equation
x(l - x)y" + {c-(a + b+ \)x}y' -aby = 0
by the following process. Replace x with x/b, so that the new equation has singular
points at 0, b, and oo. Now let b —>¦ oo so that infinity is a confluence of two
singularities. The resulting Equation D.1.3) is called the confluent hypergeometric
equation.

4.1 The Confluent Hypergeometric Equation 189
When с is not an integer, two independent solutions of the hypergeometric
equation around x — 0 are
/ a + l-c,b+l-c
and x C2F\ I ; x
V 2-е
Replace x with x/b in these expressions and let b —»¦ oo to get
a;x] and j'-^J
c У V 2-е
These are two independent solutions of D.1.3) around x = 0. They are valid over
the whole complex plane, since i F\ is an entire function. One has to be somewhat
more careful to find the solutions around infinity. A solution of the hypergeometric
equation about infinity is given by
a+l-b ' x
When x is changed to x/b and b ->¦ oo, this expression tends termwise to
This series diverges, so it does not directly give a solution of Equation D.1.3).
However, it is possible to find an integral representation of a solution of D.1.3)
that has this series as an asymptotic expansion. To find this integral representation,
start with Euler's hypergeometric integral
a + 1-е,a.
a + l-b >
-/ (l + ~') ta-l(l + -) dl. D.1.5)
Jo \ xj V b/
Г(а)ГA-Ь)
It is possible to let b ->¦ -oo in the integral, though D.1.5) no longer makes sense
in the limit. The right side of D.1.5) tends to
r-a />oo / f\c-a-l , /-oo
Г (a) Jo \ xj Г (a) Jo
D.1.6)
This integral converges for Re a > 0 and Re x > 0. It is easy to verify that D.1.6)
is a solution of the confluent equation D.1.3).
Remark 4.1.1 Here is another way of arriving at D.1.6). Let
/•OO
y(x) = / e~xtf(t)dt.
Jo

190 4 Bessel Functions and Confluent Hypergeometric Functions
Then
xy" + (c — x)y' — ay
= / (xt2 - (c - x)t - a)e~x'f(t)dt
Jo
= / e-xl{[t2f(t)]' + [tf(t)]' ~(a + ct)f(t)}dt = 0.
Jo
The last equation holds when
f'(t) _ a - 1 + (c - 2)/ _ a - 1 с - а - 1
ДО
or
and we get integral D.1.6) once again.
Suppose x > 1 in D.1.6). By Taylor's theorem
Vм
where 0 < 6
1 f00
where
П V
The integral
x)
1 < l.So
) —
converges
_
+
and
i \
k=l
(-1)"
1-е)
)xn
(a + 1
и!
¦it — г
Jo
[a +
k\
- с
^
1-е
)« t"
JC"
Я+ 1
a+n-1
¦)*
(¦
-
к I
(]
tk
xk
+ X )
c)k(a)k
JC J
Thus we see that, except for a constant factor,
, ._a „ /a, a+ 1 -c
V —
gives an asymptotic expansion of a solution of the confluent hypergeometric equa-
equation when jc > 1 is large. In fact, we need not restrict ourselves to positive x if

4.1 The Confluent Hypergeometric Equation 191
instead of D.1.7) we use
D.1.8)
This holds as long as 1 + Ms not a negative real number. To remove the restriction
Re* > 0, which is necessary for convergence in D.1.6), consider the integral
Г@+) / {\ c-a-l
or
x~a / e-'f"! 1 + - ) dt. D.1.10)
ioo V */
These integrals are also solutions of the confluent equation, but without the restric-
restrictions on x and a needed in D.1.6). From D.1.10) and D.1.8), we can once again
obtain the 2^0 asymptotic expansion for large |x|, when |arg x\ < л — 8 < л.
Relations among solutions of the hypergeometric equation suggest correspond-
corresponding relations among solutions of the confluent equation. These can then be proved
rigorously. Similarly, transformations of hypergeometric functions imply transfor-
transformations of the 1 F{ function. The following are a few examples.
In Pfaff's transformation,
change x to x/b and let b ->• 00 to get Kummer's first transformation,
D-i.li)
A similar procedure applied to the quadratic transformation,
ab \x \ , (a,a + \ -b A
U:)(i+j[^4 } CU1>
leads to Kummer's second transformation,
a . \ ъ
Finally, the three-term relation
'a,a + I - с 1
(-xTa2Fu
a+1-b x
-с)Г(я + 16) (<*>Ь
1\ с '
Г (с - 1)Г(а + \-Ь) ,_с (а + \-с,Ъ+\-с
Г(а)Г(с-Ь)

192 4 Bessel Functions and Confluent Hypergeometric Functions
suggests that
2-е '
D.1.13)
Formulas D.1.11) and D.1.12) can be proved directly. Thus, the coefficient of jc"
on the right side of D.1.11) is
()(l)* = 1 (-n,c-a
n\2 \ с '
(c)kk\{n-k)\ и! V c
(a),
n\{c)n'
which is the coefficient of jc" on the left side. There is a similar proof of D.1.12).
We give a proof of D.1.13) in the next section where we approach this topic from
a different point of view.
4.2 Barnes's Integral for jfi
We can find the contour integral representation for \F\{a; c; jc) by computing its
Mellin transform. This is similar to finding such a representation for the hyperge-
ometric function. Let
/= / x-1lFdu;-x)dx.
=
By Kummer's first transformation D.1.11),
/= Гxs-le-xiFi(C~a;x )dx
Jo Vе
o e^ (От
= f(sJFi ; 1 =
с ) Г (а) Г(с-5)
By Mellin inversion, we should have
or
— (-x)sds. D.2.2)

4.2 Barnes's Integral for , F, 193
Of course, once we have seen Barnes's integral for a 2^1, this can be written by
analogy. In D.2.2) we have — x > 0, but this can be extended. The next theorem
gives the extension and is due to Barnes.
Theorem 4.2.1 For |arg(—je)| < л/2 and a not a negative integer or zero,
where the path of integration is curved, if necessary, to separate the negative poles
from the positive ones.
The proof follows the same lines as that of Theorem 2.4.1. The reader should
work out the details.
Again as in Chapter 2, this representation of 1F1 can be used to obtain an
asymptotic expansion by moving the line of integration to the left. The residues
come from the poles of Г (a + s) at л = —a — n. The result is contained in the
next theorem.
Theorem 4.2.2 For Rex < 0,
Corollary 4.2.3 For Re x > 0,
Г(с)е* (c-a,l-a
Proof This follows from Theorem 4.2.2 after an application of D.1.11).
Now note that the 2^0 in Theorem 4.2.2 suggests the integral
J = / Г(-я)ГA -c-s)T(a+s)xsds. D.2.3)
2л-/ J_ioo
Again the line of integration is suitably curved. By moving the line of integration
to the left and picking up the residues at s = —k - a, where к > 0 is an integer,
we get
1 r— a— n+ioo
—: / r(-s)T(l-c-s)T(a + s)xsds. D.2.4)
?711 J —a—n—ioo
To ensure the validity of this formula, we need an estimate of the integrand on
5 = a + iT, where T is large and — a — n < a < 0. By Stirling's formula (see

194 4 Bessel Functions and Confluent Hypergeometric Functions
Corollary 1.4.4),
Y
—
M
The expression on the right-hand side dies out exponentially when |argx| <
37Г/2 — S < 37Г/2. We assume this condition and D.2.4) is then true. The last
integral in D.2.4) is equal to
Г (a + n - s)T(\ +a-c+n- s)F(s - n)xsds = 0(x~"'n)
2тп J_,
when \x | is large. Thus
Г(а)ГA +a-
the asymptotic expansion being valid for |argx| < Зтт/2.
However, when the line of integration is moved to the right, it can be seen that
/ = Г(а)ГA - с) iFi ; x )+ Г (я + 1 - с)Г(с - l)xl~ciFi
a + 1 — с
2-е ;'
D.2.6)
when |argx| < 37Г/2. This proves the next theorem.
Theorem 4.2.4 For |arg x \ < Зл"/2, D.2.5) and D.2.6) hold and
Г(а)ГA - c)ifi Г°; x\ + Г (a + 1 - с)Г(с - Djc'^iJ
Observe that this is the same as relation D.1.13).
This theorem gives the linear combination of the two independent i F\ that
produce the recessive solution of the confluent equation. This is of special interest
in numerical work. To clarify this point, consider the simpler equation y" - у = О,
which has independent solutions sinhx and coshx as well as ex and e~x. This
equation has an essential singularity at oo and, in the neighborhood of this point
for Re x > 0, e~x is the recessive solution. Any other solution independent of
e~x is a dominant solution. Thus the combination Aex + Be~x can be computed
very accurately from values of 0е and e~x. However, A cosh x — В sinh x creates
problems, especially when А яа В and x has a large positive real part.

4.3 Whittaker Functions 195
4.3 Whittaker Functions
Whittaker [1904] gave another important form of the confluent equation. This is
obtained from Kummer's equation D.1.3) by a transformation that eliminates the
first derivative from the equation. Set у = ex/1x~c/2co(x) in D.1.3). The equation
satisfied by со is
Two independent solutions of this equation can have a more symmetric form if we
set
с
с = 1 + 2m, - — a = к
or
c- 1 1
m = —j—, a = -+m-k.
The result is Whittaker's equation:
' = 0. D.3.1)
From the solutions D.1.4) of D.1.3), it is clear that when 2m is not an integer, two
independent solutions of D.3.1) are
Mk,m(x) = e^2x^lFl ( 5 ++w2 * ; ^ D.3.2)
and
Mk,.m(x) = e-'^x^F! ( 5 -^ k;xy D.3.3)
The solutions Mk,±m{x) are called Whittaker functions. Because of the factors
x 5±m, the functions are not single valued in the complex plane. Usually one restricts
x to |argx| < n.
Formulas for \F\ obviously carry over to the Whittaker functions. Kummer's
first formula, for example, takes the form
х~1"тМКт{х) = {-хГ^тМ_Кт{-х). D.3.4)
A drawback of the functions M^±m (x) is that one of them is not defined when
2m is an integer. Moreover, the asymptotic behavior of the solution of Whittaker's
equation is not easily obtained from these functions. So we use the integral in

196 4 Bessel Functions and Confluent Hypergeometric Functions
D.1.10) to derive another Whittaker function, Wktin(x). This is defined by
Wk,m(x) := - —-I
r(o+)
7
J CO
(-0-*~i+m(l + ->) 2 е"'Л, D.3.5)
where argx takes its principal value and the contour does not contain the point
/ = —x. Moreover, |arg(—1)\ < n, and when t approaches 0 along the contour,
arg(l + t/x) ->¦ 0. This makes the integrand single valued. It is easily verified
that Wk<m(x) is also a solution of D.3.1). Note that Whittaker's equation D.3.1) is
unchanged when x and к change sign. Thus W-k,m(—x) is also a solution and is
independent of Wk,m № ¦ This is clear when one considers the asymptotic expansion
for Wk,m(x). The reader should verify that the remarks after D.1.10) imply that
) -oo, D-3.6)
when |argx| < ж — S < тт. Consequently,
This shows that Wk,m(x) and W-kiin(—x) are linearly independent.
4.4 Examples of iFi and Whittaker Functions
This section contains some important examples of i F\ and Whittaker functions
that occur frequently enough in mathematics, statistics, and physics to be given
names.
(a) The simplest example is given by
ex = 1Fi(a;a;x). D.4.1)
(b) The error function is defined by
erfx = —= / e~'2dt = 1 - erfcx (x real), D.4.2)
л/ТГ Jo
where
2 /"
erfcx = —— I e~' dt.
V* Jx
It is easy to see that erf x = ^1^A/2; 3/2; x).

4.4 Examples of i F] and Whittaker Functions 197
To express the error function in terms of Wk<m (x) we need to write D.3.5)
as an integral over @, oo). Assume that Re (k — | — m) < 0; then D.3.5) can
be written as
n , ( 1 \ sin7T (k + A — m)
Wk,m{x) = e'x'2xkT (k+--m\ ^p >-
/¦OO
¦ / e-'rk-Li+m(l+t/x)k-L2+mdt
Jo
= —г —— e-'rk-^m(\+t/x)k^+mdt. D.4.3)
Г(г - к + m) Jo
The integral converges for Re (k — \ - m) < 0 and Re x > 0. Note the relation
of D.4.3) with the integral in D.1.6). Now set t = u2 — s2, where и is the new
variable. Then
wk,m
Set к
Thus
(X)
=
e-x/2xk2es<
T(\-k + m)
-1/4, m = 1/4,
erf x =
'/У
andx
1--
= s2 to get
mfx + u2-s2
V x
/•00
1 e~u2du.
Js
/2W_1/4,l/4(*2).
D.4.4)
An asymptotic expansion for erf x can be derived from this formula and D.3.6).
(c) The incomplete gamma function is defined by
r-x /-oo
(a,x)= e-'t"-ldt = Г (a) - / e^t^dt = Г (а) - Г(а,х).
Jo Jx
D.4.5)
After expanding e~' as a series in / and term-by-term integration, it is clear
that
y(a,x) = —iFi(a;a+l;x). D.4.6)
a
The reader may also verify that
T(a, x) = e-xl2x°-^Wa^a_(x). (АЛЛ)
(d) The logarithmic integral li(x) is defined by
dt
о bg/

198 4 Bessel Functions and Confluent Hypergeometric Functions
Check that
If x is complex, take |arg(—logx)| < ж.
Additional examples of Whittaker functions such as the integral sine and
cosine and Fresnel integrals are given in Exercise 4.
(e) The parabolic cylinder functions are also particular cases of the Whittaker
functions. To see how these functions arise, consider the Laplace equation
д2и д2и д2и
^+^+^=°- DA8)
The coordinates of the parabolic cylinder |, r], z are defined by
x = ^2-r12), у = $4, z = z. D.4.9)
Apply the change of variables D.4.9) to the Laplace equation. The result after
some calculation is
д2и д2и\ д2и
This equation has particular solutions of the form U(^)V(r])W{z), which
can be obtained by separation of variables. The equation satisfied by U, for
example, has the form
where a and X are constants. After a slight change in variables, this equation
can be written as
= 0. D.4.10)
Equation D.4.10) is called Weber's equation. It can be checked that
?>n(x) = 2t+?x-2W§+i_i(x2/2) (|argx| < Зтг/4) D.4.11)
is a solution of D.4.10). The constant factor is chosen to make the coefficient
of the first term in the asymptotic expansion of Dn (x) equal to one. Dn (x) is
called the parabolic cylinder function. When л is a positive integer, Dn(x) is
e~'tx2 times a polynomial, which, except for a constant factor, is Hn(x/^/l),
where Hn(x) is the Hermite polynomial of degree n. These polynomials will
be studied in Chapter 6.

4.5 Bessel's Equation and Bessel Functions 199
(f) In the study of scattering of charged particles by spherically symmetric poten-
potentials, we can take (see Schiff [1947, Chapter V]) the solution of the Schrodinger
equation
h2 2
V и + Vu = Eu
2(i
to be of the form
ОС - ч
Ей(г)
P|(COS#),
t=0
where Pi is the Legendre polynomial of degree t and yi satisfies the equation
By a change of variables we can take к — I. The Coulomb potential is given
by U(r) = 2r]/r, so the equation for у is
d2y г 2г, Щ +
+
1
r
Comparison of this equation with Whittaker's equation D.3.1) shows that
уi = rt+le"iriFi(i + 1 - irj; 21 + 2; 2ir).
The function
<J>t(r),r) := e~"\F\(l + 1 - irj; 21 + 2; 2ir)
is called the Coulomb wave function.
4.5 Bessel's Equation and Bessel Functions
Bessel functions are important in mathematical physics because they are solutions
of the Bessel equation, which is obtained from Laplace's equation when there is
cylindrical symmetry. The rest of this chapter gives an account of some elementary
properties of Bessel functions.
When к = 0 and m = a in Whittaker's equation D.3.1), we get
d2W Г 1 1/4-a2
dp [ 4
If we set y(x) = ^/xWBix), then у satisfies the equation
dx2 xdx

200 4 Bessel Functions and Confluent Hypergeometric Functions
This equation is called Bessel's equation of order a. It is easily verified that
Г(а +1) ya+ 1
is a solution of D.5.1). Ja(x) is the Bessel function of the first kind of order a.
From D.1.12), we have another representation of Ja{x). This is
Equation D.5.1) is unchanged when a is replaced by —a. This means that /_a (x)
is also a solution of D.5.1). One can check directly that when a is not an integer,
Ja(x) and J-a(x) are linearly independent solutions. When a is an integer, say
a = n, then
/_„(*) = (-1)"/„(*). D.5.4)
Therefore /_„ (x) is linearly dependent on /„ (x). A second linearly independent so-
solution can be found as follows. Since (—1)" = cos пж, we see that Ja (x) cos na —
J-a(x) is a solution of D.5.1), which vanishes when a is an integer. Define
/a(x)cos7ra - J-a(x)
Ya(x) := : . D.5.5)
Sin7TO!
When a = n is an integer, Ya (x) is defined as a limit. By L'Hopital's rule,
D.5.6)
а-и ж [ да да
Note that
оо
This implies that Ja (x) is an entire function in a. Thus the functions j? and ^jj2-
in D.5.6) are meaningful. Moreover, as functions of x, Ja (x) are analytic functions
of x in a cut plane. Thus we can verify that Yn (x) is a solution of Bessel's equation
D.5.1) when a = n is an integer. We can conclude that D.5.5) is a solution of
D.5.1) in all cases. Ya(x) is called a Bessel function of the second kind.
Substitution of the series for Ja(x) in D.5.6) gives, after simplification,
2k -n
1 °° Г-П*
- - X) i— №& + к + 1) + ф(к+ l)](x/2Jk+n. D.5.7)
Here л is a nonnegative integer, |argx| < ж, and ifr(x) = Г'(х)/Г(х).

4.5 Bessel's Equation and Bessel Functions 201
Note that Bessel's equation can be written
Suppose a is not an integer. It is easy to deduce from D.5.8) that
/-„(*)— x— - Ja(x)— x— = 0
dx \ dx ) dx \ dx )
or
x[j-a{x)J'a(x) - Ja(x)J'_a(x)] = С = constant.
To find C, let x -> 0 and use the series D.5.2) and Euler's reflection formula. The
result is
С = 2 sin ал /тт.
Thus the Wronskian W(Ja(x), J-a(x)) = Ja{x)J'_a{x) - J_a(x)J^(x) is given
by
W(Ja(x), /_ц(х)) — —2sina7r/7rx,
for а ф integer, and
W(Ja(x), Ya(x)) = 2/nx
not only when а ф integer, but also for a = n by continuity.
Many differential equations can be reduced to the Bessel equation D.5.1). For
example,
u=x"Ja(bxc)
satisfies
и = 0. D.5.9)
X [ Xz J
When x = 1/2, b = 2/3, с = 3/2, and a2 = I/a, this equation reduces to
u" + xu=0. D.5.10)
This is the Airy equation and it has a turning point at x =0, so solutions oscillate
for x > 0 and are eventually monotonic when x < 0. As such, solutions of the
Airy equation can be used to approximate solutions to many other more compli-
complicated differential equations that have a turning point. For example, the differential
equation after F.1.12) has x = \/2n + 1 as a turning point. Airy functions can be
used to uniformly approximate Hermite polynomials in a two-sided neighborhood
of the turning point. See Erdelyi [1960].

202 4 Bessel Functions and Confluent Hypergeometric Functions
4.6 Recurrence Relations
There are two important differentiation formulas for Bessel functions:
d r«Kr\ у-(-1)"B*
dx a(> ^ Г(и + а
n=l)
nx
and
' 2
and, similarly,
—x~aJa(x) = -x~aJa+i(x). D.6.2)
From the series for cosine and sine,
D.6.3)
cosx. D.6.4)
Rewrite D.6.1) and D.6.2) as
aJa(x) +xJ'a(x) = xJa-i(x)
and
-a/a(x)+x/^(x) = -xJa+l(x).
Elimination of the derivative J'a gives
2a
Ja-i(x) + Ja+i(x) = —Ja(x). D.6.5)
Elimination of Ja(x) gives
Ja-i(x) - Ja+l(x) = 2J^(x). D.6.6)
It follows from D.6.1) and D.6.2) that
and
/1 И \n
I I (x~a Ja(x)) = (— l)nx~a~nJa+n(x). D.6.8)
\x dx J
When these are applied to D.6.3) and D.6.4), we obtain
9 /1 И \ п /
\ X J
D.6.9)

4.7 Integral Representations of Bessel Functions 203
and
/ 2 ,, /1 d \" /cosx\
J-n-ipix) = J — xn+l (-—) ). D.6.10)
V 7ГХ \x dx J V x J
The following two formulas can now be proved by induction (the details are left
to the reader):
Г'"" i-l)kin + 2k)\
ГТI
= J — < sin(x - ИЛ-/2)
itx ' ^i2k)Mn-2k)M2x)
D.6.11)
J-n-\n(x) = \l^{ COSQt + Л7Г/2) V^ Ц—
1 VttxI ' f^o i2k)\in - 2k)li2xJk
¦ , „ч "v^ (— 1)*(и + 2k + 1)! I
— SinQt + Л7Г/2) >^ 2А+Г f "
D.6.12)
4.7 Integral Representations of Bessel Functions
Set у — xau in Bessel's equation
„ 1 , 9 9
/ + -/ + A - cr/x2)y = 0.
x
Then м satisfies the equation
хм" + Bа+ 1)м' + хм=0. D.7.1)
Since equations with linear coefficients have Laplace integrals as solutions, let
u = A / ext fit)dt,
Jc
where A is a constant. Substitute this in D.7.1). Then
0 = / fit)ixt2 + Ba + 1)/ + x)ex'dt
Jc
f \ 2 9 ] xt
~ Jc l V dt " T
= [extit2 + l)fit)]c + / ext\ --[it2 + l)/@] + Ba + l)tf(t) \dt
Jc V & J

204 4 Bessel Functions and Confluent Hypergeometric Functions
after integration by parts. This equation is satisfied when
[ex'(t2 + \)f(t)]c = 0 D.7.2)
and
^1 D.7.3)
at
Equation D.7.3) holds when f(t) = (t2 + 1)"~1/2. Replace t with *J^At so that
D.7.2) holds when С is the line joining —1 and 1 and Re a > —1/2. The above
calculations imply that
/•i
y = Axa I elx'(l - t2)a-llldt (A.IA)
is a solution of Bessel's equation. We assume that arg(l — t2) = 0. To see that
D.7.4) gives an integral representation of Ja(x) for Rea > —1/2, expand the
exponential function in the integrand as a series and integrate. The result, after an
easy calculation involving beta integrals, is
An application of Legendre's duplication formula (Theorem 1.5.1),
22кГ(к + 1)Г(к + 1/2) = т/пBкI,
gives
y(x) = А^Г(а + \/2JaJa(x).
Therefore,
)a f eixt(\ - t2)"-
D.7.5)
when Rea > —1/2. Put t = cos# to get the Poisson integral representation
Ja(x) = — ]2{х/2)а I eiXCOS9s[n2aede
= —= (x/2)a [ cos(jccos6»)sin2a6»rf6l, D.7.6)
у/лГ(а + 1/2) Jo
for Re a > — 1 /2. An important consequence of D.7.5) is Gegenbauer's formula,
which gives a Bessel function as an integral of an ultraspherical polynomial. The
formula is
D.7.7)
for Re v > —1/2. When v ->• 0, we get Bessel's integral D.9.11) for Jn(x).

4.7 Integral Representations of Bessel Functions 205
To prove this, take a = v + n in D.7.5) and integrate by parts n times to get
т , ч апх/2у "'
7-i dt"
By Rodrigues's formula B.5.13),
d"(\-t2y+"-l/2 _ (-2)"п!Г(у + п + 1/2)ГBу) _ 2
IT" ~ r(v + l/2)rBv + n)
Use of this in the previous integral gives Gegenbauer's formula D.7.7).
Condition D.7.2) is also valid when С is a closed contour on which e'x'(t2 —
l)<*+i/2 returns to its initial value after t moves around the curve once. We take С
as shown in Figure 4.1, and write an integral on С as
/
Ja
d+,-1-)
f(t)dt.
Here 1+ means that 1 is circled in the positive direction, and —1— means —1 is
circled in the negative direction. We are interested in the integral
У(х)
Ja
(l+.-i-)
eixt{t2-l)"-1/2dt,
which is denned for all a, since С does not pass through any singularity of the
integrand. When Re a > — 1 /2, we can deform С into a pair of lines from — 1 to 1
and back. We choose arg(r2 - 1) = 0 at A; then
y(x) = xa U eixt[(l - fte-'T-^dt + J eix'[(l - ,
= хаИйп(]--а\п f eixt(l-t2y-1/2dt
V2 / 7-i
Mx).
Figure 4.1

206 4 Bessel Functions and Confluent Hypergeometric Functions
This gives Hankel's formula:
1.7.8)
when а ф ^p, n = 0, 1, 2,..., and arg(r2 - 1) = 0 at A.
We now prove another formula of Hankel,
r / n Г(\/2-а)е™{х/2)а
2я1 Ji00
eix'(t2 - If
, D.7.9)
when Rejc > 0, —Зтг < arg(r2 — 1) < я, and a + \ ф 1, 2, The contour in
D.7.9) is shown in Figure 4.2. It is assumed that the contour lies outside the unit
circle.
To prove D.7.9) we need the following formula of Hankel (see Exercise 1.22):
1
1
/
Г (г) 2ТГI J.
@+)
l /•(<>+)
= — \ e'rzdt,
2ТГI J-ooe.fi
D.7.10)
for|0|<;r/2.
The expansion of (r2 — I)"/2 in powers of 1/r converges uniformly on the
contour. We have
(_„
k[
-^ < argr < |. So
*¦ • Jioo
D.7.11)
Figure 4.2

4.7 Integral Representations of Bessel Functions
207
Figure 4.3
Figure 4.4
Since Re* > 0, we have |argjt| = \в\ < \n. Set и = ei7l/2xt. Then
K-1+.1+) /-@+)
/
J—o
= (-l)ke-°"Tix2k-2a-
e и
271 i
ГA -2а+2к)'
Substitute this in D.7.11) and apply Legendre's duplication formula. This proves
D.7.9).
Now modify the paths in D.7.7) and D.7.8) to those shown in Figures 4.3 and
4.4 respectively. Take Rejc > 0 in D.7.7) and D.7.8). When the horizontal parts
of these curves are made to go to infinity we get
Ja(x) =
Г(\-а)(х/2)а
2л i
/•()+) /Ч-1-)
/ + / eixt(t2 - \)a~l'2dt
Jl+ioo J— 1+ioo
D.7.12)

208 4 Bessel Functions and Confluent Hypergeometric Functions
and
f + + [ eixt(t2 - lf-1/2dt
Jl+ioo J—l+ioo
D.7.13)
In D.7.12) arg(?2 -1) is 0 at A and it at B, whereas in D.7.13) arg(r2 -1) is 0 at A
and -it at B. To make aig(t2 - 1) = it at В in D.7.13), we multiply (t2 - I)"/2
in the second integral by the factor e-2(«-!/2)^ pormuia D.7.13) may now be
written (after reversing the direction of the contour in the second integral) as
+ e-"ai f eixt(t2-lf-Ut]. D.7.14)
J— l+ioo J
The form of D.7.12) and D.7.14) suggests the following two functions, which are
called Bessel functions of the third kind or Hankel functions:
^-al« Ja(x) - J-a{x)] D.7.15)
sinajr
and
sinajr
\eaniJa(x) - J-a(x)l D.7.16)
These are more simply written in terms of Ja {x) and of Ya (jc), the Bessel function
of the second kind. Thus,
H$\x) = Ja{x) + iYa(x), D.7.17)
HB\x) = Ja{x) - iYa(x), D.7.18)
H$\x) = V2_ /(x/2H1 / eix'(t2 - l)a-l/2dt, D.7.19)
•Jititi Jl+ico
and
eix'(t2 - l)a-1/2dt. D.7.20)
J-
— l+ioo
The integral formulas for H^\x) and H^2)(x) hold when Rejc > 0, a + \ ф
1,2, Moreover,arg(r2 - 1) = — it at l + j'ooandarg(?2 — 1) = it at— 1+гоо.
Note also that
H%(x) = J—(cosx + i sinjc) = J—elx = HJ%(x) D.7.21)
' V itx V itx '

4.8 Asymptotic Expansions
209
and
D.7.22)
References to the work of Bessel, Poisson, Gegenbauer, and Hankel can be
found in Watson [1944, Chapters 2, 3, and 6].
4.8 Asymptotic Expansions
Set t = 1 + iu/x in D.7.19). The integral becomes
/•@+)
/ ,
Joe
e-u{_u)a-l/2
Ш
a-1/2
du. D.8.1)
Compare this with D.3.5) and D.3.6) to obtain the asymptotic expansion
Hankel introduced the notation
it!
Da2 - I2)Da2 - 32) • ¦ • Da2 - Bk - IJ)
Then D.8.2) can be written as
+
A similar argument gives
Since
.7=0
and yeW = ^
2j
we have from D.8.3) and D.8.4)
Ja(x)
an 7Г
cos x —
— sin jc —
атг тг
D.8.2)
D.8.3)
D.8.4)
D.8.5)

210 4 Bessel Functions and Confluent Hypergeometric Functions
and
Ya(x)~J —
ait
ait „ , % ,. ..,v~,-, ¦ ^ D86)
when |argjc| < it. Note that D.6.11) and D.6.12) are special cases of D.8.5).
4.9 Fourier Transforms and Bessel Functions
Many special functions arise in the study of Fourier transforms. In Chapter 6, we
shall see a connection between Hermite polynomials, which were mentioned in
Section 4.4, and Fourier transforms in one variable. Here we consider a connection
with Bessel functions and as a byproduct obtain a generating function for Jn(x).
Start with the Fourier transform in two dimensions. We have
1 ГСС /«00
F(u, u) = — / / f(x,y)ei{xu+yv)dxdy. D.9.1)
2?r J J
/-OO rill
/
Introduce polar coordinates in both (jc, y) and (и, v) by
jt=rcos0, y=rsin6; m =
Then
1 /-OO ri
F(u,v)=— / /
2it Jo Jo
Expand / as a Fourier series in в,
00
f {r cosd, r sin6) = J2 f"ir)eine,
to get
/-2л-
oo /-oo г i /-2л-
F(u,v)=Y / Mr)r\— е1пве"К
D.9.2)
The relation with Bessel functions comes from the inner integral. Since the inte-
integrand is periodic (of period lit) it is sufficient to consider the integral
Fn(x) = — [ eixcoseeinede. D.9.3)
2л- Jo

4.9 Fourier Transforms and Bessel Functions 211
Expand the exponential and integrate term by term to get
^ OO .? ? „27Г
Fn(x) - — Y^ '—?- / cos* BeinBdd. D.9.4)
^ k=o ¦ Jo
Now
2k cos* в = (ew + e-w)k = em -
So, writing к — n + 2m we have
~ in+2mxn+2m {n + 2m
F»(x) = У -,
m=0
= /ИЛ(^) D.9.5)
This relation is interesting as it gives the Fourier expansion of e'xcose:
i"Jn(x)ein9
OO
= J0(x)+2^2 i"Jn(x) cos пв. D.9.6)
n=l
The last equation follows from У_„ (x) = (— 1)" У„ (jc). Equating the real and imag-
imaginary parts gives
00
cosQtcostf) = J0(x) + 2'^2(-l)nJ2n(x)cos2ne D.9.7)
n=l
and
oo
sin(jccos6») = 2E(-1)"-/2n+i(^)cosBn + 1H. D.9.8)
For an interesting special case, take в = л/2 in D.9.7) to get
D.9.9)
n=l
It is worth mentioning Miller's algorithm at this point. The series D.9.9) shows
that for any given x = xo and sufficiently large n, J2n (xo) is small. So take J2n (x0)
to be 0 and 7г(п-1) C*o) = c, which is to be determined. Use the recurrence relation
D.6.5) to compute Л(п-2)(^о) and so on down to Уг(^о) and Jo(*o) as multiples
of c. Since D.9.9) can be approximated by
n
k=\

212 4 Bessel Functions and Confluent Hypergeometric Functions
we obtain an approximate value of с and hence also of the values of the Bessel
functions J2k(xo). This is an example of Miller's algorithm. See Gautschi [1967,
p. 46].
There is another way of looking at D.9.6). Put t = ie'9. Then
oo
exp(jc(r - 1/0/2) = JZ •/»(JC)f"- D.9.10)
n=—oo
Thus,exp(jc(? — 1/0/2) is the generating function for Bessel functions of integer
order.
Replace 0 with | — 0 in D.9.3); then use D.9.5) and the periodicity of the
integrand to get Bessel's formula
1 Г
Jn{x) — — / exp(—inB +ix smB)dB
2тг J_n
1 Г \ Г
— — / exp(—inB + ix sin 0)^0 H / exp(/n0 — ix
2л Jo 2jr Jo
= -/ cos(«0 -x sin6)dB. D.9.11)
In Section 4.7, we obtained this from Poisson's integral formula. We can go in the
opposite direction and derive
(x/2)" Г
*A/2)B Jo
¦71
:_2n,
Jn(x) = ; ' I cos(jccos6»)sin/n вd6 D.9.12)
from D.9.11) by using Jacobi's formula given in Exercise 2.21,
d"'1 sin2" в (-1)"
dy"~
-2"(l/2)nsinn0, y=
In D.9.12), и is a nonnegative integer. But this restriction can be removed.
First multiply both sides of D.9.12) by B/х)"Г(п + 1). Then both sides are
bounded analytic functions of n for Re n > —1/2. By Carlson's theorem, we can
conclude that D.9.12) is true for these values of и.
We end this section with a proof of the inequalities:
Forx real, |/o(*)l < 1, and \Jm(x)\ < 1/V2 form = 1, 2, 3,....
D.9.13)
The first inequality follows from D.9.12), but we have another proof which verifies
them all at once. Change t to —t in the generating function D.9.10) to get
oo
exp(-*(f - 1/0/2) = Y, (-1)"•/»(*)*". D.9.14)

4.10 Addition Theorems 213
Multiply D.9.10) by D.9.14) to get
n~—oo m=—oo
Equate the coefficients of the powers of t and use У_„ (je) = (— 1)" Jn (x) to obtain
00 00
Jn(x) = 1 D-9Л5)
n=—oo n=l
and
oo
(~l)mJm(x)Jn-m(x) =0 forn ^ 0. D.9.16)
The inequalities in D.9.13) follow from D.9.15).
Remark 4.9.1 Observe that, in Bessel's formula D.9.11), n has to be an integer,
for when n = a, a noninteger, the integral on the right side of D.9.11) is no longer a
solution of Bessel's equation D.5.1). However, Poisson's integral formula D.9.12)
holds for all n as long as Re и > —1/2. We also remark that Jacobi obtained the
direct transformation of D.9.12) to D.9.11) by the argument given here in reverse.
For references and details of Jacobi's proof, see Watson [1944, §§2.3-2.32]. Note
also that Jacobi's formula mentioned after D.9.12) is really a consequence of
Rodrigues's formula B.5.13') when applied to Chebyshev polynomials of the
second kind.
4.10 Addition Theorems
In this section, we prove a useful addition theorem of Gegenbauer. First we show
that
OO
Jn(X+y)= J2 Jm{x)Jn-m(y)- D.10.1)
m=—oo
This follows immediately from the fact that
exp((* + y)(t - 1/0/2) = exp(x(t - l/r)/2)exp(y(r - 1/0/2),
for this implies

214 4 Bessel Functions and Confluent Hypergeometric Functions
The result in D.10.1) is obtained by equating the coefficients of t". Observe that
D.9.16) follows from this addition formula.
To state the second addition theorem, suppose a,b, and с are lengths of sides
of a triangle and c2 = a2 + b2 — lab cos в. Then
oo
Mc) = J2 Jm{a)Jm(b)eime. D.10.2)
m=—oo
Set
d = aew - b.
Then c2 — dd, so с and d have the same absolute value. Thus there is a real \jr
such that
с = {aeie - Ь)е'ф.
A short calculation shows that the last relation implies
с sin ф = a sin@ + \jr + ф) - b sin{\jr + ф).
By D.9.11),
2^ Jo
{•27T
2П Jo
Since \jf is independent of ф and the integrand is periodic, by D.9.10),
У„(С) = _L f * еЦапп(в+ф)-Ь*шф]аф
2л" Jo
= У Jm(a)eime— / e"'*™*^»
m=—со °
= V Ут(аУтв— / е'^тфе-п
2л- Jo
oo
= У Jm(a)./mB>yme.
m=—oo
The last equation comes from D.9.11).
Since J-m(x) = (—l)m7m(x),wecan write the addition formula in the following
form:
oo
Me) = Ma)Mb) + 2 J2 Jm(a)Jm(b) cosmB. D.10.3)
m=l

4.10 Addition Theorems 215
Observe that
Id Id
с dc ab sin в d6'
then apply this operator to D.10.3), and use D.6.2) to get
J\ (с) ^л sin тв
^ = 2 V«7(aO00
D.10.4)
/ *v v fit \ / fit V / " /1
m=l
Rewrite this as
Me) _
C m=0
Apply D.10.4) to the last formula; use D.6.2) again to get
m=0
In general, we have the following result for derivatives of ultraspherical polyno-
polynomials:
— C*(cos6>) = -2X. sin вC^{(cos6).
dO
Now apply induction to see that
—— = 2aT(a) / (m +a)~—— m C"(cos6), D.10.5)
m=0
when a — 0, 1, 2, By F.4.11), C?(cos#) is a polynomial in a; hence by the
remarks we made after formula D.9.12) the two sides of D.10.5) are bounded
analytic functions in a right half plane. Carlson's theorem now implies the truth of
D.10.5) for values of a in this half plane. By analytic continuation, D.10.5) is then
true for all a except a = 0, — 1, —2,.... Equation D.10.5) is called Gegenbauer's
addition formula.
We state without proof the following result of Graf:
/a -be~w\a/2 °°
J^ci nf) = Y, Ja+m(a)Jm(b)e""e D.10.6)
V a - be10 I z-^
4 ' m=-oo
when b < a. When a, b, and в are complex, we require that \be±w/a\ < 1 and
с —>¦ a as b —>¦ 0. Graf's formula contains D.10.1) and D.10.2) as special cases.
See Watson [1944, §11.3]. Exercise 29 gives a proof of D.10.6) when a is an
integer.

216 4 Bessel Functions and Confluent Hypergeometric Functions
4.11 Integrals of Bessel Functions
Expand the function F{u,v) in D.9.2) as a Fourier series:
F(u, v) = J2 Fn{R)e
n=—oo
OO 1 fin
/OO 1 fin
/•oo i />2jt oo
= / _L / «,'¦*'»•« V- fn{r)j«9+*>derdr
Jo 27t Jo „fr^
/•oo i r2n °° °o
h 2л- Л mt^o „fr^
Hence
(
[by D.9.6)]
oo г „oo -|
= V j" / Jn(Rr)fn(r)rdr\einf. D.11.1)
/•CO
-i)nFn(R)= fn(r)Jn{Rr)rdr. D.11.2)
The inverse Fourier transform of D.9.1) is
1 ГОО ГОО
f(x,y) = — /
Z7t J-oo J-oo
If a calculation similar to D.11.1) is performed, we obtain
/•OO
fn(r) = (-i)n Fn(R)Jn(Rr)RdR. D.11.3)
Jo
The integral in D.11.2) is called the Hankel transform of order n of the function
fn(r). Then D.11.3) is called the inverse Hankel transform. For a function f(x)
that is smooth enough and vanishes sufficiently fast at infinity, we have more
generally the Hankel pair of order a:
/•00
F(y)= f(x)Ja(yx)xdx D.11.4)
Jo
and
/•00
/(*)= / F(y)Ja(xy)ydy. D.11.5)
Jo

4.11 Integrals of Bessel Functions 217
To obtain an interesting integral, multiply Gegenbauer's formula D.10.5) by
C" (cos в) and use the orthogonality relation which follows from B.5.14), namely
The result is
21~апТ(п + 2а) Ja+n(a)Ja+n(b)
ГJj^-Can
Jo c01
aaba
D.11.6)
where a, b, and с are sides of a triangle, that is, c2 = a2 + b2 — 2abcosO. Rescale
a, b, and с and take n = 0 to arrive at
„2ariJn лТBо!) Ja(ax)Ja(bx)
Уо "
s
J\a
sin
2а~1Г(а) ;
Rewrite this as
fa+b [(a + bJ - c2]a-'2[c2 - (a - bJ]"~l2
Ja(cx)cdc
\a-b\ C"
a + l/2)Ja(ax)Ja(bx)(ab)a
xa
Then, by the Hankel inversion formula, for Re a > —1/2,
f°° ,_ [c2 - (a - ^21«-
/ Ja(ax)Ja(bx)Ja{cx)x adx = —
Jo 2ia
23а-1^/пГ(а+1/2)(аЬс)а
D.11.8)
for \a — b\ < с < a + b. The value of the integral is 0 otherwise. If the formula
for the area of a triangle (denoted by Д) in terms of its sides is used, then the right
side of D.11.8) can be written
2«-1д2а-1
There are important generalizations of integral D.7.6) due to Sonine. These are
contained in the next theorem.
Theorem 4.11.1 For Re \i > -1 and Re v > -1,
J
7M+v+1(x) = 2v^ + J J^x sin0) sin"+1 в cos2v+16Ш D.11.9)
and
D.11.10)

218 4 Bessel Functions and Confluent Hypergeometric Functions
The integrals D.11.9) and D.11.10) are referred to as Sonine's first and second
integrals.
Proof, (i) The proof is simple. Expand J^ (x sin в) as a power series and integrate
term by term. Thus
I 7M(xsin6')sinM
Л
0cos2v+10de.
¦ ifi
m=0 ~ 4r~ '
The last integral is a beta integral equal to
Substitution of this in the above series gives the result.
(ii) In this case expand both J^xsind) and Jv(y cos 6) in power series and
integrate term by term. The details are left to the reader.
Observe that D.11.9) is a special case of D.11.10). Divide both sides of D.11.10)
by yv and let у -> 0. The result is D.11.9). ¦
Corollary 4.11.2 For Re a > -1 /2,
(x/2)a Г ,
J (x) = — / cos (x cos 0) sin 6dd. D.7.6)
Г (a + l/2)V5r Уо
Proof. Take /n = -l/2,v+l/2 = aand recall that
J-1/2KX) = \
V 7ГХ
Sonine's first integral D.11.9) can also be written as
rv+l roo
/ гмA - t'y.Jpixfydt. D.11.11)
^'l^t l) Jo
By Hankel inversion,
2T(v + 1) y J^X) J,Axt)xdx = ^A - t2y+ D.11.12)
for Re pi > -1, Re v > — 1.
We now turn to the computation of the Laplace transform of Bessel functions.
Hankel evaluated the transform of t>J'~1Jct(yt) in terms of a 2^1 function. For
special values of a and д, the 2^1 reduces to more elementary functions. We

4.11 Integrals of Bessel Functions 219
consider this class of integrals next. The simplest integral of this kind was found
by Lipschitz. His result is the following: For Re (x ± iy) > 0,
OO 1
—xt - - - ¦
e-xtJ0(yt)dt = D.11.13)
о V(* + У )
From the asymptotic expansion for Bessel functions D.8.5), it is clear that
Re(x ± iy) > 0 is sufficient for the convergence of the integral. Use D.9.11) to
see that
/OO 1 /•OO /*7Г
e-xtJ0(yt)dt = - / e~xt / eiytcosededt
71 Jo Jo
_ l Г c
dQ_
0 л — iy cos в
1
The more general result, which gives the Laplace transform of t^ 1 Ja{yt)-, is due
toHankel[1875].
Theorem 4.11.3 For Re (a + рь) > 0 and Re (x ± iy) > 0,
f°° _ , _, (^/2х)аГ(а + д)
/ e Ja(yt)t^ dt =
Jo х^Г(а + 1)
a + 1 ' )
Proof. First assume that \y/x | < 1. Substitute the series for Jv (yt) in the integral
to get
m(yqr+2m Г
i-\)m(yq
2m Г
l) Jo
(-l)m(y/2)a+2m Г(« + [i + 2m)
m\T(a + m + 1) xa+n+2m
Since \y/x\ < 1, the final series is absolutely convergent. This justifies the term-
by-term integration. So we have D.11.14) under the restriction \y/x\ < 1. The
complete result follows upon analytic continuation, since both sides of D.11.14)
are analytic functions of у when Re (x ± iy) > 0. This proves the theorem. ¦
When fj, = a +1 or a +2, the 2^1 in D.11.14) reduces to a i^o, which can be sum-
summed by the binomial theorem for \y/x\ < 1. The results are in the next corollary.

220 4 Bessel Functions and Confluent Hypergeometric Functions
Corollary 4.11.4 For Re (x ± iy) > 0,
D.11.15)
and
D.11.16)
Corollary 4.11.5 For Re (x ± iy) > 0,
Г°о r/ 2 , у2ч1/2 _ x-ia
Г e-xtJa(yt)rldt = LV" ' " 7 ^-, whenRea > 0, D.11.17)
Л аУа
and
/•со г/„2 i 2\ 1 /2. v"|<*
/ e~x'Ja{yt)dt = ^ 2 1/2' w^nRea>-l. D.11.18)
Proof. Apply Exercise 3.39. ¦
The formulas in the two corollaries are limits of some formulas for Jacobi
polynomials introduced in Chapter 2. Recall that these are defined by
Theorem 4.11.6 For real a and /J,
lim п-аР<ай( cos-) = lim п~аР^ A Д
^ у л->оо ^ 2пг
Proof. This follows easily from Tannery's theorem (or the Lebesgue dominated
convergence theorem). Suppose a is not a negative integer. Tennwise convergence
is easily checked. Moreover, domination by a convergent series is seen from the
fact that, for large n,
n~a(n + a + p+ l)t(a + 1)» ^ Bn + a + 0)k(a + 1)яи-° ^ С
Л!(и - ifc)!(a + 1L2*и2<: ~ ifc!(a + \)k{2n)kn\ ~ Ща + \)k'
where С is a constant that holds for all k, 0 < к < и.
When a = — ? is a negative integer, use the fact that
to obtain the desired result.

4.11 Integrals of Bessel Functions 221
The integral formulas D.11.15) to D.11.18) are limits of generating-f unction
formulas for Jacobi polynomials, three of which will be proved in Chapter 6. The
corresponding formulas are
) 2
D.11.19)
Bn + a + p + 1)Г(я + a + 0 +
и
n=0
« =2a(l-r+ /?)"", D.11.21)
where Л = A - 2xr + r2I/2, and
(l -r + R)-a(l + r + RyP. D.11.22)
л=0
A proof of a result more general than D.11.21) is sketched in Exercise 7.31. The
other generating-function formulas are proved in Chapter 6 (Section 6.4).
The next theorem gives another infinite integral of a Bessel function due to
Hankel.
Theorem 4.11.7 For Re (д + v) > 0,
/•oo
/ Mat) t^e'^dt --
Jo
Proof. The condition Re (\i + v) > 0 is necessary for convergence at zero. The
asymptotic behavior of Jv(x) given by D.8.5) shows that the integral converges
absolutely. Thus the integral can be evaluated using term by term integration. This
gives
Jv(at)

222 4 Bessel Functions and Confluent Hypergeometric Functions
Since the integral on the right-hand side equals
2
the result follows. ¦
Corollary 4.11.8 For Re (д + v) > 0,
Г Щ)
Л
Л 2pT(v+l) \ v+l '4p2J
D.11.24)
Proof. Apply Kummer's first i Fi transformation D.1.11) to Hankel's formula in
Theorem 4.11.7. ¦
An important particular case we use later is
/ Jv(at)tv+le-pVdt = ——-—e~a2/4p\ Rev > -1. D.11.25)
Jo Bргу+1
See Watson [1944, Chapters 12 and 13] for references.
4.12 The Modified Bessel Functions
The differential equation
where x is real, arises frequently in mathematical physics. It is easily seen that
Ja(ix) is a solution of this equation. Moreover, for x, real e~an'/2Ja(xe7"/2) is a
real function. We then define the modified Bessel function of the first kind as
Ia(x) = e~a7lil2Ja(xenil2) (-it < aigx < тг/2)
^n < argx < tz
= (jt/2)« у ^l . D.12.2)
When a is not an integer, Ia{x) and /_a(x) are two independent solutions of
D.12.1). When a = n is an integer, then
In(x) = /_„(*).

4.13 Nicholson's Integral 223
To deal with this situation, define the modified Bessel function of the second kind:
Ka(x):=—-—[I-a{x) - Ia(x)l D.12.3)
/sinew
It is immediately verified that
hn(x)=\ —sinhx and I-m(x) = \ —coshx. D.12.4)
У И V 71X
Thus
Jx- D-12.5)
We see that Ja(x) corresponds to the sine and cosine functions whereas Ia(x)
corresponds to the exponential function. Perhaps this is why the nineteenth century
British mathematician, George Stokes, took Ia(x), rather than the Bessel function,
as the fundamental function.
The asymptotic expansions for Ia(x) and Ka(x) can be obtained in the same
way as those for Ja{x) and Ya{x). Thus
?
(-Л-/2 < argx < Зтг/2), D.12.7)
and
(-Зтг/2 < argx < Л-/2). D.12.8)
Here (a, n) - (-l)»(o + 1/2)„(-а + 1/2)„/л!.
4.13 Nicholson's Integral
Integral representations for modified Bessel functions can be obtained from those
for Bessel functions. Similarly, there are formulas for integrals of modified Bessel
functions. As one example, take у = i and Rex > 1 in D.11.18) to get
Jo

224 4 Bessel Functions and Confluent Hypergeometric Functions
Set x = cosh /8; then D.13.1) can be written as
/•00
Jo в
sinh/8
Now, since
D.13.2)
Ka{t)~\hre-' asf^oo, D.13.3)
we see from D.13.2), on replacing a with —a, that
/*00
/ л—' cosh fi jf
Jo
when Re (cosh fi) > — 1.
sin got sinh f5
D.13.4)
Let a -> 0 to get
-^—. D.13.5)
o sinh j8
When fS — in 12, we have
n
D.13.6)
Nicholson's formula,
#(*) = — / ?0B* sinhf) coshBaf) dt = J^(x) + Y%(x), Rex > 0,
т Jo
D.13.7)
generalizes the trigonometric identity sin2 x + cos2 x — 1 as can be seen by
taking a = 1/2 and applying D.13.6). We present Wilkins's [1948] verification
of D.13.7). This is done by showing that both sides of D.13.7) satisfy the same
differential equation and then analyzing their asymptotic behavior.
We show first that
N{x) ~ — as x -+ oo, D.13.8)
nx
where N(x) denotes the left side of D.13.7). It is sufficient to prove that
8 f°° 2
lim xN{x) = lim —- / xK0Bx sinht) coshtdt = —. D.13.9)
Jt-i-oo jc->oo я2 Jo П
The second equation follows from D.13.6). For the first equation, we show that
F(x, t) = xK0Bx sinh t) (cosh 2at - coshf)

4.14 Zeros of Bessel Functions 225
converges boundedly to 0. The dominated convergence theorem then implies
D.13.9). From D.13.3) we have
\F(x, f)| < A0(x csch fI/2e-2jcsinh'|cosh2af - coshf|
< Ao(x/t)l/2e-2xsivb'B\a\ + I)f(sinh2|a|f + sinhf).
The second inequality follows upon an application of the mean value theorem to
cosh 2at — cosh t, recalling the fact that csch t < l/t. Let x > 1. Since sinh t > t
and (xt)l/2e~xt is bounded, we have
\F(x, t)\ < A(sinh2|a|f + smht)e~xsiaht
< A(sinh2|a|f+ sinhf)e~sinh'.
This proves D.13.9).
For the next step, check that the product of any two solutions of y"+py'+qy = 0
satisfies the equation у'" + Ъру" + Bp2 + p'+ 4q)y' + Dpq + 2q')y = 0. Apply
this to Bessel's equation to see that {tfvA)(.x)}2, {H^2\x)f and J2(x) + Y2(x) are
independent solutions of
Using differentiation under the integral sign, the reader should verify that
satisfies this differential equation. The differential equation
satisfied by K0(x) is also required in the calculation.
Thus we have
N(x) = A{J2(x) + Y2(x)} + B{H^\x)f + C{HB\x)}\
Let x ->¦ oo and use D.8.3) to D.8.6) to obtain
1 = A + e2'^-!"*-Wb + е-ъ(х-±ал-±я)с + oA)_
Hence В = С = 0, A = 1, and Nicholson's formula is proved.
4.14 Zeros of Bessel Functions
It is easily seen that all nontrivial solutions of the Bessel equation D.5.1) have
simple zeros except possibly at zero. The first derivatives of such solutions also
have simple zeros except possibly at zero and ±a.
From D.8.5) we can conclude that for real a, Ja (x) changes sign infinitely often
as x —> oo. This implies that Ja (x) and J'a (x) have infinitely many positive zeros.
The conclusion for J'a (x) follows from the mean value theorem.

226 4 Bessel Functions and Confluent Hypergeometric Functions
Suppose that jati, ja<2,... are the positive zeros of Ja(x) in ascending order.
Then, fora > — 1,
0 < ja,l < ja+l,l < ja,2 < jc+1,2 < ja,3 < •¦ D.14.1)
From D.6.1) and the mean value theorem it follows that between two zeros of
xa+xJa+\(x) there is a zero of xa+1Ja(x). Similarly, D.6.2) implies that between
two zeros of x~aJa(x) there is a zero of x~a Ja+\{x). This proves D.14.1).
When a < — 1, the zeros of Ja (x) and Ja+i (x) are still interlaced by the above
argument, but the smallest zero of Ja+i (x) is closer to zero than that of Ja (x). It
can also be proved that for —2s < a < — Bs + l),s a positive integer, Ja(x) has
As complex zeros all with nonzero real parts. In contrast, when —Bs + 1) < a <
—2(s + l),sa nonnegative integer, Ja (x) has As + 2 complex zeros, two of which
are purely imaginary. See Watson [1944, p. 483] for a proof.
Theorem 4.11.6 and Theorem 5.4.1 imply the following theorem about Bessel
functions.
Theorem 4.14.1 Letx\n > x2n > • • • be the zeros of P^^(x) in[-\, \\andlet
Xkn = cos#/tn, 0 < вк„ < п. Then for a fixed к,
lim пвкп = ja,k-
n—>oo
In particular, Ja (x) has an infinite number of positive zeros.
In the next chapter, we prove that all zeros of P^a-^(x) lie in (—1, 1) when
a, ft > — 1. This, combined with Hurwitz's theorem, shows that x~a Ja(x) has
only real zeros for a > — 1. For Hurwitz's theorem one may consult Hille [1962,
p. 180].
Another method of obtaining the reality of the zeros of /„ (x) for a > — 1 is to
establish the formula
(b2-a2)[ tJa(at)Ja(bt)dt=x[ja(bx)J'a(ax)-Ja(ax)J'a(bx)}. D.14.2)
To prove this, note that Ja(ax) satisfies the differential equation
Multiply this equation by Ja(bx) and multiply the corresponding equation for
Ja(bx) by Ja(ax); subtract to get
d ( dJa(ax)\ d ( dJa(bx)\ , ,
Ja(bx)—l x "; - Max)—I x  = (b2 - a2)xJa(ax)Mbx)
dx\ dx J dx \ dx J

4.14 Zeros of Bessel Functions 227
or
^-[xJa{bx)J'a(ax) - xJa{ax)J'a{bx)} = (b2 - a2)xJa(ax)Ja(bx).
Formula D.14.2) is simply an integrated form of this. Now if a is a complex zero
of Ja(x), then so is a. Take x = 1, b = a in D.14.2) and note that the integrand
tJa(at)Ja(at) > 0. Hence the left side of D.14.2) is nonzero but the right side is
zero. This contradiction implies that Ja (x) has no complex zeros.
An argument using differential equations can also be given to show that Ja(x)
has an infinity of positive real solutions for real a. This technique goes back to
Sturm. The version of Sturm's comparison theorem given below is due to Watson
[1944, p. 518].
Theorem 4.14.2 Let u\{x) and u2(x) be the solutions of the equations
^ ф{H ^ф() 0
such that when x = a
Mi(fl) = u2(a), Mj(fl) = M2(a).
Lef 0i(jc) an<i 02(*) fee continuous in the interval a < x < b, and u\{x) and
u'2(x) be continuous in the same interval. Then, ifф\{x) > ф2(х) throughout the
interval, \u2(x)\ exceeds J и i (jc ) | as long as x lies between a and the first zero of
ui(x) in the interval. Thus the first zero ofu\{x) in the interval is on the left of the
first zero of u2(x).
Proof. Without loss of generality, we assume that mi(*) and u2(x) are both
positive immediately to the right of x = a. Subtract u2 times the first equation
from mi times the second to get
Mi—— -и2—-г- = (ф\(х) -ф2(х))ихи2 > 0.
dx1 dx1
Integration gives
du2 du\
Mj— u2——
dx dx
Since the expression in the brackets vanishes at a, we have
du2 dux
mi— И2-7— > 0.
dx dx
Hence
d(u2/ui)
dx
>0

228 4 Bessel Functions and Confluent Hypergeometric Functions
or
which implies that u2{x) > u\ (x). This proves the theorem. ¦
Suppose \a\ > \,a real, and take ф\ (x) = 1 - (a2 — 1/4) /x2 and ф2 (x) = 1 —
(a2- l/4)/c2. Thenfor* > c, we have фх (х) > 02(*)-NotethatMi = xl/2Ja(x)
is a solution of u'( + ф\{х)и\ = 0. Denote the general solution by x 1^2Ca(x). It is
clear that u2 = A coscox + В sinwx, where w2 = 1 — (a2 — l/4)/c2. It follows
from Theorem 4.14.2 that if с is any zero of Ca (x) then the next larger zero is at
most с + n/w. When \a\ < 1/2, take фг(x) = w2 < 1. Thus for real a, Ja(x) has
an infinite number of real zeros. Essentially, Sturm's theorem says that the greater
the value of ф, the more rapid are the oscillations of the solutions of the equation
as x increases.
Theorem 4.14.2 can also be used to prove that the forward differences of the
positive zeros of/ff O) are decreasing for \a\ > 1 /2 and increasing for |a\ < 1/2.
Suppose|a| > l/2andlet7Q,n_i < jan < ^n+ibethreesuccessivepositivezeros
of Ja(x). Now set фх(х) = 1 - (a2 - 1/4)Д2 and ф2(х) = ф\{х - k), where
к = 7ffn-7ff,n-i. Now 0i (л) is an increasing function, so 0i (л) > ф2 (х). Consider
the interval [jan, ja,n+\\- At x = jan, u\ = Ja(x) = 0 and u2 = Ja(x - k) — 0.
By Sturm's theorem, u\ oscillates more rapidly and hence y'a.n+i — ja,n < ja,n <
ja,n-i- A similar argument applies to the case |a| < 1/2. It should be clear that
the same argument works for the general solution of Bessel's equation.
We end this section with an infinite product formula for Ja (x), a real. For large
x, the asymptotic formula D.8.5) for Ja{x) suggests that the asymptotic behavior
of the zeros is given by
x ~ (m + Ba + 1)/4)тг. D.14.3)
Since the zeros of /„ (x) are real and simple, one expects that the number of zeros
of x~aJa(x) between the imaginary axis and the line Re x = (m + (a + 1)/4)тг,
for large x, is m. This is true. See Watson [1944, §15.4]. It follows that the entire
function x~a Ja (x) has the product formula
OO
1 - */¦/«.») еЧ>(*//«.»)} П{A + */¦>«¦»> еЧ>(-*//«.»)}- DЛ4.4)
Л=1 П = \
This result continues to hold when a is not real. See Watson [1944, §15.41].

4.15 Monotonicity Properties of Bessel Functions 229
4.15 Monotonicity Properties of Bessel Functions
Sturm's comparison theorem for differential equations stated in the previous sec-
section gave information about the zeros of the solutions of the differential equation
_ 0. ,4.15.1,
Sturm also used his theorem to prove that the second forward differences of the
positive zeros of any nontrivial solution of the above equation are all positive if
|v| < 1/2 and all negative if |v| > 1/2. Lorch and Szego [1963] have greatly
extended this result to higher-order differences. In this section we present one of
their theorems. For further generalizations and other related results, the reader
should see the references section.
Consider the differential equation
y" + f(x)y = 0, D.15.2)
where x e I, an open interval. Let A > — 1 and let y(x) be an arbitrary solution
of D.15.2) with zeros at x\, x2,. ¦. in ascending order. Set
Mk = JXM \y(x)\xdx, k=\,2,.... D.15.3)
Observe that when A = 0, Mk = Axk, the difference of the successive zeros of
y(x). When A = 1, then Mk gives the area under the arch formed by y(x) from xk
We state the theorem of Lorch and Szego without proof. The reader may consult
the original paper for a proof, which is somewhat lengthy. In the statement of the
theorem, the notation An\xk is used to denote the nth forward difference of the
sequence {fj,k}. Thus,
А"цк = An~l/j,k+i — Ап~1цк and А°[л,к = ц,к.
Theorem 4.15.1 Let y\ and уг be two independent solutions of the differential
equation D.15.2) in a closed interval I. Suppose that
(-D"^-f[MW]2 + Ы*)]2} > 0 for n = 0, 1,..., N,
dxn
where the Nth derivative exists in the open interval I, and the lower-order deriva-
derivatives are continuous in I. Then
{-l)nA"Mk>Q forn=0,...,N; к =1,2,....
In particular
(-l)"-lA"xk>0 forn = l,...,N + l; к =1,2,....

230 4 Bessel Functions and Confluent Hypergeometric Functions
Moreover, ify(x) denotes another solution of D.15.2) with zeros at x\, xi,.. ¦ and
ifx\ > x\, then
(-1)иДи(х*-х*)>0 for n = 0,...,N; ?=1,2,....
This theorem yields results on Bessel functions when applied to equation D.15.1).
Two independent solutions of this equation are */xJv(x) and */xYv(x). Let
*/xCv(x) denote the general solution. To apply Theorem 4.15.1, we have to study
the expression
p(x)=x[Jv(x)]2+x[Yv(x)]2, D.15.4)
which can be represented by Nicholson's integral D.13.7).
We need the following formula below:
I-OO
K0(x) = / e~xcoshtdt. D.15.5)
Jo
For a proof, see Exercise 11. Now, by D.13.7),
8 Г°°
= —j / [K0Bx sinht) + 2x smhtK^x sinht)] cosh2vtdt.
л Jo
Integrate the second term by parts to get
Q
p'(x) = — [Ko Bx sinh Otanhf cosh 2vi]g°
8 Г° / d
Л—~ / ABjc sinh t) ( cosh 2vt (tanh t cosh 2vt) ) dt.
л1 Jo
\ dt )
The first term on the right is zero, for by definition D.12.3), it follows that
behaves like log x as x -*¦ 0, while D.12.6) gives the behavior of Ko(x)asx -> oo.
Thus
p'(x) = —j / K0Bx sinh t) tanh t cosh 2vi[tanh t - 2v tanh2vt]dt.
л1 Jo
It is easy to check that the expression in brackets is negative for | v | > 1/2 and the
rest of the integrand is positive. So p'{x) < 0. Similarly, it can be shown that
8 f°°
p(")(x)= / Kg- D Bx sinh f)B sinh r)"~ 'tanh t cosh 2vt
л1 Jo
¦ {tanh t - 2v tanh 2vt}dt.
It is clear from D.15.5) that (-1)" K{Qn) (x) > 0 for x > 0, и = 0, 1, 2,.... Thus,
the conditions of Theorem 4.15.1 hold for Equation D.15.1) when |v| > 1/2. So
we have the following corollary:

4.16 Zero-Free Regions for i F, Functions 231
Corollary 4.15.2 Let cvk, cvk denote the kth positive zeros in ascending order
of any pair of nontrivial solutions of Bessel's equation D.15.1) with |v|>l/2.
Suppose Л > — 1 and set
Mk= [*'Ш xx>2\Cv(x)\xdx.
Then, for к = 1,2,...,
(-1)"Д"М* >0 forn = 0,1,...,
(-l)"-lA"cvk>0 forn = 1,2,...,
t-l)"A"(c,m+t - cvk) > 0 for n = 0,1,...
with m a fixed nonnegative integer, provided that cVitn+\ > cv\.
In particular,
(-I)"'1 A"jvk > 0, (-l)"-1Anyvk>0 forn = 1,2,...
and
(-l)nAn(jvk - yvk) > 0 for n = 0, 1,...,
where jvk, yvk denote the kth positive zeros of Jv(x) and Yv(x) respectively.
Remark 4.15.1 Lorch, Muldoon, and Szego [1970] have extended Theorem
4.15.1 to a study of higher monotonicity properties of
= f Ш W(x)\y(x)\xdx,
Jxk
Mk
Jxk
where W(x) is a function subject to some restrictions. As an example, it is possi-
possible to take W(x) = x~l/1 when у is a solution of D.15.1). This implies the
monotonicity of
(_1)»д» ГШ \Cv(x)\dx,
Jcvk
where the integral is the area contained by an arch of a general Bessel function
instead of xl/2 times a Bessel function. For further extensions of the results see
also Lorch, Muldoon, and Szego [1972].
4.16 Zero-Free Regions for iFi Functions
We end this chapter with some results of Saff and Varga [ 1976] on zero-free regions
for sequences of polynomials satisfying three-term recurrence relations. These
polynomials can be partial sums of i F\ functions, so we may invoke a theorem
of Hurwitz on zeros of an analytic function which is the limit of a sequence of
analytic functions to obtain zero-free regions for i F\ functions.
Saff and Varga's basic theorem is the following:

232 4 Bessel Functions and Confluent Hypergeometric Functions
Theorem 4.16.1 Let {pk(z)}o be a finite sequence of polynomials satisfying
k=l,2,...,n, D.16.1)
h
where p~\{z) := 0, po(z) = po ф 0, and the bk andck are positive real numbers.
Let
a :=min{bk(l -bk-X/ck) : к = 1,2,...,n], b0 = 0. D.16.2)
Г/геи, г/а > 0, f/ге parabolic region
Pa = {z=x+iy :y2 < 4a(x + a), x > -a} D.16.3)
contains no zeros of pk(z), к = 1, 2,..., п.
Proof. Suppose г е Pa is a fixed complex number that is not a zero of any
pk(z), к = l,...,n. Set
:= zpk-i(z)/bkpk(z) fork=l,...,n.
The proof depends on the following two facts:
1. The polynomials pk(z) and pk-\{z) have no zeros in common for each k,
k=l,...,n.
2. Re/u,k <l for/fc= l,...,n.
Assume these results for now and suppose that for some k, pk(z) is zero at a
point zo e Pa. Observe that кф\, for, p\ (z) = po(z + b\)/bi has a zero at -bu
which by D.16.2) is < —a and hence cannot be in Pa. Suppose 2 < к < п. Since
pk(zo) = 0, it follows from D.16.1) that
(zo/bk + l)Pt-i(zo) = (zo/ck)Pk-i(zo)-
Fact 1 implies that pk-i(zo) ф 0 because pk and pk-\ have no common zeros. So
we can divide by pk-\ (zo) to get
Ck . . , , ZOpk-2(Zo)
(zo + h) = -— = /j,k-i(zo). D.16.4)
bk-\bk
The second fact and the continuity of ц, give Re/x^_i(zo) < 1. Then by D.16.4)
Rezo < -bk{\ - bk-i/ck) < -a,
which contradicts the assumption that zo€ Pa- This means that pk(z)(k = 1,..., л)
have no zeros in Pa. It remains only to prove our two assumptions. It is evident
from D.16.1) that none of the polynomials pk(z), к = 0,1,... ,n vanish at 0. So
suppose pk(zo) = Pk-i(zo) = 0, where zo ф 0, for some к > 1. By a repeated
application of D.16.1) we get po(zo) = 0, which contradicts the assumption that
po ф 0. Thus pk(z) and pk~\{z) have no common zeros.

4.16 Zero-Free Regions for i F, Functions 233
We now prove that Re jut(z) < 1 by induction. Clearly,
z
z + bi
Since z e Pa, Re г > —cc > —b\. Thus Re/xi < 1. Now, it follows easily from
D.16.1) that
z
—\
z + h- Ькск
or
where Tk{w) is a fractional linear transformation defined by
z + bk -bkck bk-iw
The function. Tk hajs a pole at the point u>k = (z + bk)/(bkCklbk-i) whose real
part, by D.16.2) and D.16.3), is seen to be >1. So remaps Re w < 1 into abounded
disk with center f^ '= 7\B — п>к), where 2 — wk is the point symmetric to the pole
u>k with respect to the line Re w = 1. By the definition of Tk
Moreover, the point 0 — 7\(oo) lies on the boundary of this disk so its radius is
lit |. Thus the real part of any point in the disk does not exceed
„<_.,<_, Re? + |2|
-h^cp) ~ 2(Rez
where the first inequality follows from D.16.2) and the second from D.16.3). Now,
by the induction hypothesis, Re/^-i < 1; thus Re^i* = Re Tk{iik-i) < 1- This
proves the theorem. ¦
Corollary 4.16.2 For an infinite sequence of polynomials {pkiz)}™ satisfying the
three-term relation D.16.1), suppose that
a- inf \bk(l -bk-!c71)} >0.
k>l l v ''
Then the region Pa defined in the theorem is zero-free for the polynomials Pk(z).
Moreover, if pk(z) —>¦ f(z) ф 0 uniformly on compact subsets of Pa, then f(z) is
also zero-free in Pa.
Proof. The first part is obvious. For the second part use Hurwitz's theorem. ¦
The next corollary applies to the polynomial squence obtained from the partial
sums of a power series.

234 4 Bessel Functions and Confluent Hypergeometric Functions
Corollary 4.16.3 Suppose sk(z) := ?^=o djZj have strictly positive coefficients
and
> > 0, where a-i/ao = 0.
ak ak-\J J
Then the polynomials sk(z),k = 1,2,..., и /?ave ио zeros ш Р^.
Proo/ First observe that
V()
V * = 1, 2,..., n,
ak-i J ak-i
when s-i = 0. Then apply Theorem 4.16.1 to obtain the required result. ¦
Note that a consequence of the above results is that the partial sums sn (z) =
J2o zn/n\ofthe exponential function have the parabolic region, y2 < 4(x+l),x >
— 1, as a zero-free region. This region is sharp, both because of the zero at z = — 1
for s\(z) = 1 + z and asymptotically as n ->• oo in sn(z). The next corollary
concerns the more general i F\ confluent hypergeometric function. The proof is
left to the reader.
Corollary 4.16.4 Suppose sn(z) is the nth partial sum of \F\{c; d; г). Then
$n(z),n =0, 1, 2,... have no zeros in the region
(i) Pd/C, ifO<d< c,
(ii) Puifl <c<d,
(Hi) Pa, a = Bc-d+ cd)/(c2 + с), г/0 < с < 1 and с < d < 2c/A - c).
Moreover, i F\ (c; d; z) has no zeros in the corresponding interior region.
For other applications of Theorem 4.16.1 see de Bruin, Saff, and Varga [1981].
Exercises
1. Show that
1 1V ' ' ' Г(а)Г(с-а)Л v '
when Re с > Re a > 0.
2. Let \Fi(a~) = i_Fi(a — 1; c; jc) and define iFi(a+) etc. in a similar way.
Prove the contiguous relations:
(a) (c-a)lFi(a-) + Ba-c + x)iFi -alFl(a+)=0,
(b) c(c — 1) i F\ (c—) — c(c — 1 + x) i F\ + (c — a)x \ F\ (c+) = 0,
(c) (a-c+l)iFi- a iF,(e+) + (c - 1) iF^c-) = 0,
(d) ciFi-ciF1(e-)-'ifi(c+)=O,
(e) c(a +x)iF\ — (c — a)x \F\(c+) — ас \ F\(a+) = 0,
(f) (a-l+jc)^! +(c-a)iF!(a-)-(c- 1),^(с-) = 0.

Exercises 235
3. Prove the formulas
(a) -—iFi(a;c;x) = —^-iFl(a+n;c+n;x),
dxn (с)„
(b) ^-[e-'iFiia; c; x)] = (-1)" {° ~ a)n e~* ^(a; с + n; x).
dxn (с)„
4. Express the following functions in terms of Whittaker functions:
(a) The sine integral Si(x) — jQx f sin? Jf.
(b) The cosine integral Ci(x) = - /л°° f"' cosrdt.
(c) The Fresnel integrals
rx
C(x) = / f/2 cos tdt/Jbz,
Jo
S(x) = f r1/2sintdt/V2n.
Jo
(d) The exponential integral
/•00 -/
Ei(x)= —dt.
Jx t
5. Use D.3.6) and D.4.4) to derive an asymptotic expansion for erf x.
6. Prove that
,.00
e~sxxc \Fi{a; c; x)iFi(ai; c;
Jo
= T(c)(s- l)~a(s - Xya'sa+ai~c2Fi(a,ai;c;X/[(s - l)(s-k)]).
7. Show that, for the parabolic cylinder function Dn(x) given by D.4.11), the
following properties hold:
(a) Dn{x) = V^V^ViC-wA 1/2; х2/2)/Г(A -п)/2)
2-n)/2\ 3/2;х2/2)/Г(-и/2).
(b) Dn{x) = (-l)nex l4-r-^(e~x212).
8. Prove the formulas D.6.11) and D.6.12) for Jn+i/2(x) and /_n_i/2(x).
9. Show that for Re a > -1/2,
Г (a + l/2)Ja(x) = —=(x/2f / cos(x si
\ln Jo
and
Г (a + 1/2)/Дх) = 4=(x/2f / e~xt(\ - t2)"
л/Л J-\
Deduce that
\JAx)\ < \х/2\аем/ Г (a + 1), where x=u + iv.

236 4 Bessel Functions and Confluent Hypergeometnc Functions
10. Use D.9.11) to obtain Neumann's formulas (Watson [1944, p. 32]):
1 fn 1 Гп
ji(x) = — J2nBxsme)de = — / J0Bx sinO) cos 2n0de.
л Jo n Jo
11. Show that, for |argx| < jt/2,
Deduce that, when Re a > —1/2,
¦'" + ,
Hence
3711") / «""(I - f2)"
Г (о + 1/2) jo
12. Prove that for x > 0 and a > -1/2
2°T(a + l/2) f00 cosxt
13. Show that for a > — 1 and с > О
лA)
2л-г
14. Prove the following result of Sonine and Schafheitlin:
[ Ja.fi(at)Jr-x(bt)
о := -o at
Л
and
S = —-
/a,a-y + l a1
provided the integral is convergent.

Exercises 237
Consider the particular cases (a) p = О, у - a = 1; (b) у — 3/2, a+P =1/2;
(с) у = 1/2, a + P = - 1/2; (d) у = 3/2, a + P = 3/2; (e) у = 1/2, a +
?=1/2.
See Watson [1944, §13.4].
15. Show that when a = b in Exercise 14, the result is
-a-p)T(a)
=
tr-"-» 2ГA-р)Г(у-а)Г(у-р) '
provided that Re a > 0 and Re (y — a - P) > 0.
16. Show that
Г(Р+ 1)Г(а + 1)
y^ (-lyW-w, -a-n;p + l; b2/a2)(ax/2Jn
Deduce that
lJn(x/2)
2"
^ (о + \)Лр + 1)„(а+р
17. Show that for a, ? > 0 and -1 < Rea < 2Re? + 3/2
x"+lJa(bx) g'-W v
^ К
(Note that $™ e-^+^'tPdt = (J}^+, for Re p > -1.)
18. Show that
fr/T\ff /"O
^ io
19. Prove that for a > 0, b > 0, >> > 0, and Rep > -1
ЪЧ \а2 + ЪЛа~*~\
Consider the case a = 1/2, p = 0.
20. Prove the following formula for Airy's integral:
1 f°° -I
Ai(x) := - /
7Г Jo
See Watson [1944, §6.4].

238 4 Bessel Functions and Confluent Hypergeometric Functions
21. Let ф(х) be a positive monotonic function in Cl(a, b) and let y(x) be any
solution of the differential equation
у" + ф(х)у=0.
Show that the relative maxima of |>>|, as x increases from a to b, form an
increasing or decreasing sequence accordingly as ф (х) decreases or increases.
[Hint: For f(x) = {y(x)f + {у'(х)}2/ф(х) show that sgn/'(x) =
—sgn ф'(х).] (Sonine)
22. Suppose that k(x) and ф(х) are positive and belong to С (a, b). If y{x) is a
solution of the equation
then show that the relative maxima of |>>| form an increasing or decreasing
sequence accordingly as к(х)ф{х) is decreasing or increasing. (Butlewski)
23. Show that и = xaja (bxc) satisfies the differential equation
u" + —и' + Lex-1J + a2~a2c2] u = о
r r2
x |_ x j
24. Take 0(x) = 1 + '^. Use D.8.5) and Exercise 21 to prove that
{ s/2/n if-l/2<a < 1/2,
(. finite and > ^/2/n if a > 1/2 .
For Exercises 21, 22, and 24 and the references to Sonine and Butlewski, see
Szego[1975, pp. 166-167].
25. Let a=A— 1/2,0 < Я < 1. Denote the positive zeros of Ja{x) by j\ <
h < h < • • • and the zeros of the ultraspherical polynomial C^(cos#) by
в\ < 02 < • ¦ • < в„. Use Theorem 4.14.2 to show that
+ K), k=\,2,...,n.
[Note that и = (sin#)^C^(cos#) satisfies the equation
I. sin' v
and compare this with the equation satisfied by *f9Ja{(n + X)9}.]
26. Suppose -1/2 < a < 1/2 and mn < x < (m + 1/2)тг, w = 0, 1, 2,.... Show
that Jv(x) is positive for even m and negative for odd m. [Note that when
x = (m + 9/2O1 with 0 < 9 < 1,
2(jt/4)« Z2^ cosGrM/2)
Г (a + 1/2)VttBw + 0)" y0 {Bw
_(

Exercises 239
and show that
sgn Ja(mn + вл/2) = sgn [(-l)m{v'm + (vm - vm.i)
+ (vm-2 ~ Vm-г) H }]
= sgn(-l)m,
where
r2r cos(ttm/2)
and
r _ rl
-!)»„' = f2m+e ^{ли/Ddu
m J2m
12m \\^m -TV) —U-y-
27. Show that
tJl{t)dt = X- [х{У^(х)}2 - /ff(x)^-{x^(x)}l.
/o 2 [ dx a J J
Deduce that /(x) = AJa(x) + BxJ'a{x) ф 0 has no repeated zeros other than
x = 0. (This result is due to Dixon. See Watson [1944, p. 480].)
28. Let f{x) = AJaix) + BxJ'aix) and gix) = CJa(x) + DxJ'aix) with AD -
ВС Ф 0. Prove that the positive zeros of fix) are interlaced with those of
g(x). (Show that 0(x) = fix)/gix) is monotonic.)
29. Prove Graf's formula D.10.6) when a is an integer by using the identity
ea(t-l/t)/2e-b(te-">-l/(te-w))/2 _ ec(tu-l/(tu))/2
where и = (a — be~w)/c.

Orthogonal Polynomials
Although Murphy [1835] first defined orthogonal functions (which he called recip-
reciprocal functions), Chebyshev must be given credit for recognizing their importance.
His work, done from 1855 on, was motivated by the analogy with Fourier series
and by the theory of continued fractions and approximation theory. We start this
chapter with a discussion of the Chebyshev polynomials of the first and second
kinds. Some of their elementary properties suggest areas of study in the general
situation. The rest of this chapter is devoted to the study of the properties of general
orthogonal polynomials.
Orthogonal polynomials satisfy three-term recurrence relations; this illustrates
their connection with continued fractions. We present some consequences of the
three-term relations, such as the Christaffel-Darboux formula and its implications
for the zeros of orthogonal polynomials. We also give Stieltjes's integral represen-
representation for continued fractions which arise from orthogonal polynomials.
In his theory on approximate quadrature, Gauss used polynomials that arise
from the successive convergents of the continued fraction expansion of log( 1 +x)/
A - x). Later, Jacobi [1826] observed that these polynomials are Legendre poly-
polynomials and that their orthogonality played a fundamental role. We devote a sec-
section of this chapter to the Gauss quadrature formula and some of its consequences,
especially for zeros of orthogonal polynomials. We also prove the Markov-Stieltjes
inequalities for the constants that appear in Gauss's formula.
Finally, we employ a little elementary graph theory to find a continued fraction
expansion for the moment-generating function. In the past two decades, combina-
combinatorial methods have been used quite successfully to study orthogonal polynomials.
5.1 Chebyshev Polynomials
We noted earlier that the example of the Chebyshev polynomials should be kept
in mind when studying orthogonal polynomials. The Chebyshev polynomials of
the first and second kinds, denoted respectively by Tn{x) and Un(x), are defined
240

5.1 Chebyshev Polynomials 241
by the formulas
_ B")! T (x) _ B">! g
and
A/2>1/2) Bл+ 2)!
)]
E.1.2)
where x = cos#.
The orthogonality relation satisfied by Tn{x) is given by
/•+1
/ Tn{x)Tm(x){\ - x2)/2Jx = 0, when m ф п.
For x = cos в, this is the elementary result:
Г
cosmdcosnOde—0, when тфп.
Г
/
h
/о
Similarly, the orthogonality for E.1.2) is contained in
/ sin(« + 1H sin(w + 1H d0 — 0, when тфп.
Jo
To motivate our later discussion of orthogonal polynomials, we note a few results
about Chebyshev polynomials. The three-term recurrence relation, for example, is
given by
2хГт(х) = Tm+1(x) + Tm.i(x), E.1.3)
which is equivalent to
2cos#cosm# = cos(w + 1H +cos(w - \H. E.1.4)
The last relation is contained in the linearization formula
2cosm6 cosn0 = cos(w + nH + cos(m — nH E.1.5)
or
Tm(x)Tn(x) = -(Tm+n(x) + Tm.n(x)). E.1.6)
In a more general context, one is interested in the problem of determining the
coefficients a(k, m, n) in
m+n
Pm{x)pn{x) = У?2а(к,т,п)рк(х), E.1.7)
4=0

242
5 Orthogonal Polynomials
where {pn(x)} is a sequence of polynomials with pn(x) of degree n exactly. A
simple but important special case of this is
ml __ v
It is usually difficult to say very much about the coefficients a(k, m, n). Later we
shall see an important example which generalizes E.1.5) and the next formula
about Chebyshev polynomials of the second kind Un{x):
sin(m + 1)9 sin(« + 1)9
sine
sin(w + n + 1 - 2k)9
k=0
where m л п = min(m,n). This formula is easily verified by noting that
sin(m + n+l-2k)9sin9 = ±[cos(m +n -2k)9 -cos(w + n -2k + 2)9]. The
dual of E.1.8) is given by
sin(n + 1)9 sin(n + 1)ф = / sin(n + l)fdf, E.1.9)
2 Je-ф
whereas the dual of E.1.5) is essentially the same formula, that is,
cos«#cos«(/> = -(cos«(# +ф) + cosn(9 — ф)).
In Fourier analysis, one represents a periodic function f(x) by a series of sines
and cosines. This involves analyzing partial sums of the form
1
cosm9 +bm s\nm9),
m=l
where
1 г2ж
am = — / /@)
л Jo
and b
1 Г2ж
= — /
л Jo
Therefore,
1
2a
m=\
¦2n
^ Jo
(cosm9 cosw0 + sinm9 si
m=\
m=\
By using the trigonometric identity
2sin(#/2)cosm# = sin(m + 1/2)9 - sin(w

5.1 Chebyshev Polynomials 243
it can be verified that
1 JU sin (n + Ш
k =¦¦влв)- EЛЛ0)
Thus, the sum inside the last integral is Dn{9 — ф). The function Dn{9) is called
the Dirichlet kernel. Define the Chebyshev polynomials of the third kind by
Vn(x) = 2DnF), where x=cosfl. E.1.11)
It is easy to check that the sequence Vn(x) is orthogonal with respect to A — х)г
A + x)~ 2 on (— 1,1). Polynomials equivalent to Vn (x) were studied by Viete. See
Edwards [1987, p. 8].
A generalization of E.1.10) is given by
Ecos(n + 1N cos«0 — cos ив cos(« + 1H
2cosmflcosm0 = —. E.1.12)
_ cos 6 — cos 0
This is in fact a particular case of the Christoffel-Darboux formula, which holds
for general orthogonal polynomials. Note that the three-term recurrence E.1.4)
implies
2 cos тв cos тф (cos 6 — cos0) = [cos(w + 1N +cos(w — 1N] cos тф
— [cos(m + 1H +cos(w — 1H] cos тв.
E.1.13)
Adding this equation for the various values of m gives E.1.12). To understand the
reason for the factor 2 in the sum E.1.12), observe that
!°rW^!!' E.1.14)
for m = 0.
The normalized function is, therefore, <. -cosmd when тфО and 4= when
у л ' л/7Г
w = 0.
The Poisson kernel for the Chebyshev polynomials defined by cos nx is given
by the sum
oo
1 + yjBcosm# cosm0)rm =: Pr(co&6, cos0). E.1.15)
m=\
When 0 = 0, we have
1-r2
E.1.16)
m=\

244 5 Orthogonal Polynomials
which shows the positivity of the sum Pr(cos9, 1) for \r\ < 1. This implies the
positivity of Pr(cos9, cos0) in \r\ < 1, since
2cosw#cosot0 = cos m (9 + ф) + cosm(9 — ф).
We end this section with some results concerning the zeros of Tm(x), which
can be given explicitly since we are dealing with cos m9. Since cos m9 = 0 when
9 = Bn+ \)n/2m, it follows that Tm{x) has m simple zeros in (—1, 1), given by
cosBn + \)n/2m, for n = 0, 1, 2,..., m — 1. Moreover, the zeros of Tm(x) and
Tm+\(x) mutually separate each other. Also observe that between two successive
zeros of Tm{x), cos[Bfc + Ъ)п/2т] and cos[Bfc + 1Oг/2от], there is a zero of
Tn (x) for n > m. This follows from the fact that we can always find a nonnegative
integer I < n — 1 such that
2k + 1 21 + 1 2k + 3
< < .
2m 2n 2m
These properties of the zeros of Tn (x) have generalizations to general orthogonal
polynomials.
5.2 Recurrence
Let a{x) denote a nondecreasing function with an infinite number of points of
increase in the interval [a, b]. The latter interval may be infinite. We assume that
moments of all orders exist, that is, Ja x"da(x) exists for n = 0, 1, 2,....
Definition 5.2.1 We say that a sequence of polynomials {pn (x)}g°, where pn (x)
has exact degree n, is orthogonal with respect to the distribution da{x) if
Pn{x)Pm(x)da{x) = hn8mn. E.2.1)
/
Ja
In effect the next theorem says that {pn{x)} satisfies a three-term recurrence
relation.
Theorem 5.2.2 A sequence of orthogonal polynomials {р„(х)} satisfies
pn+i (x) = (Anx + Bn)pn (x) - CnPn-\ (x) for n > 0, E.2.2)
where we set p~i(x) = 0. Here An, Bn, and Cn are real constants, n — 0, 1, 2, ...,
and An_\AnCn > 0, n = 1,2, If the highest coefficient of р„(х) is kn > 0,
then
. _ kn+i _ An+\ hn+\
" ~ ~J~' +1 ~ ~A h~'
where hn is given by E.2.1).

5.2 Recurrence 245
Proof. First determine An so that pn+i (x) — Anxpn (x) is a polynomial of degree
n. Then
n
pn+i(x) - А„хр„(х) = ^2/bkPkix) E.2.3)
k=0
for some constants bk. Note that, if Q(x) is a polynomial of degree m < n, then
by E.2.1)
р„(*)б(*)</а(*)=0.
This implies that ^ = 0 for к < n — 1, as can be seen by multiplying both sides of
E.2.3) by Pk(x) and integrating. This proves E.2.2). It is clear that An — kn+\/kn.
To derive the final result, multiply E.2.2) by pn-\ (x) and integrate to get
0 = An pn(x)xpn-i(x)da(x) - Cn / pl_x{x)da{x).
J a J a
Since
Xpn-\(X) = -—pn{x) + 2__
" k=0
we get
A
- Cnhn-X = 0.
An-\
This proves the theorem. ¦
Corollary 5.2.3 hn = (A0/An)ClC2 ¦ ¦ ¦ Cnh0.
This follows from the equation hn = An-iCnhn-i/An by iteration.
Corollary 5.2.3 shows that the L2 norm of р„(х) can be computed from the
recurrence relation. A converse of Theorem 5.2.2 also holds. If a sequence of
polynomials [pn{x)} satisfies E.2.2) then {pn(x)} is orthogonal with respect to a
positive measure. This is usually called Favard's theorem. See Szego [1975, §3.2]
orChihara[1978,p. 21].
Remark 5.2.1 The form of the recurrence relation given in E.2.2) is the most
useful when finding pn+i(x) from pn(x) and pn-i(x). However, there are other
reasons for considering the three-term recurrence relation satisfied by orthogonal
polynomials. Another useful form is
xpn(x) =anpn+\(x) +bnpn(x) + cnpn-i{x),

246 5 Orthogonal Polynomials
where а„,Ь„, and с„ are real. A similar calculation gives the relation
an-\hn = cnhn-\.
This implies that an~\cn > 0, n = 1,2, The L2 norm of р„(х), that is, hn,
now has the form
An important consequence of the recurrence relation in Theorem 5.2.2 is the
following result, called the Christoffel-Darboux formula.
Theorem 5.2.4 Suppose that the р„ (х) are normalized so that
rb
K= Pi
J a
2{x)da{x) = 1.
Then
Е, ч , ч к" Pn+l(x)Pn(y)- Pn + l(y)Pn(x)
Рт (у)Рт (X) = , E.2.4)
т=0 *л+1 X — У
where kn is the highest coefficient of pn(x).
Proof. The recurrence relation E.2.2) implies that
Pn{y)Pn+\{x) = (А„х + В„)рп(х)р„(у) - Cnpn-i(x)pn(y)
and
р„(х)р„+х(у) = (А„у + В„)р„(у)р„(х) - Cnpn-i(y)pn(x).
Subtract and divide by An (x — y) to get
1 Pn (y)Pn+i (x) - р„ (x)pn+i (y)
An x — у
,ч ,Ч| 1 Pn-l(y)Pn(x) - Рп-\(Х)Р„{У)
= Pn (x)Pn (у) + ¦ E.2.5)
We have used the fact that С„ = An/An-\, since hn = 1. Repeated application of
E.2.5) gives the required result when we observe that An = kn+i/kn. ¦
Remark 5.2.2 \ihn ф 1, then E.2.4) takes the form
V^ Pm (y)Pm (X) к„ pn+i (X)pn (y) - pn+l (у)р„ (х)
?^0 hm kn+i (x - y)hn
The following theorem gives the confluent form of E.2.4), that is, when x — y.

5.2 Recurrence 247
Theorem 5.2.5 When hn = 1, then
" к
J>2(x) = T-MK+iW^W - Pn+i(x)p'n(x)). E.2.6)
Proof. Write the right side of E.2.4) as
К (р„+\ (х) - pn+i (y))pn (y) - (р„ (x) - pn (y))pn+i (y)
K+\ (x - y)
and let у —> x. The result follows. ¦
Corollary 5.2.6 p'n+l(x)pn (x) - pn+x (x)p'n (x) > 0 for all x.
To conclude this section, we show how the three-term recurrence relation for
Jacobi polynomials can be found. Earlier, we derived this formula from a contigu-
contiguous relation, but this is hardly a practical idea. The methods given below can be
extended to other hypergeometric orthogonal polynomials.
Consider the polynomials
n — 0,1, 2, Write the recurrence relation as
A -x)pn(x) = Anpn+i(x) + В„р„(х) + Cnpn-i(x), n =0, 1,...,
where p-\(x) = 0. To obtain An, equate the coefficients of A — x)n+l. It remains
to find Bn and С„. Take x = 1 to get
0 = An + Bn + Cn
or
Bn = -{An + Cn).
From Remark 5.2.1, we see that
Cn = An_ihn/hn_u
where hn is the L2 norm of pn (x) and its value follows from the L2 norm for Jacobi
polynomials given in B.5.14). Thus the recurrence relation is obtained.
The next method simultaneously yields the recurrence and the orthogonality of
Jacobi polynomials. It is clear that
1 — x\
' р„(х) = An+ipn+i(x) + Anpn(x) + An-lpn-i(x)+ ¦¦¦ +Aopo{x).
E.2.8)

248 5 Orthogonal Polynomials
Set x = 1 to get
An = -(An+i + А„_! + А„_2 + • • • + Ao). E.2.9)
This implies that
' 1
- An _,(/?„ (x)
— pn-i (x)) + other terms.
A short calculation shows that
Therefore,
1-лЛ
' 2
An_12/V
1 2 a + 1 "»-^"V « + 2 ' 2
+ other terms.
Equating the highest power of ^- gives
"+1 ~ B
The next highest power gives
A n(n
Now one finds that with these values of An and А„_! the sum of the first two terms
on the right equals the polynomial on the left. Thus An_2 = А„_3 = • • • = Ao = 0,
and we have the three-term recurrence relation. Moreover, we have also proved
the orthogonality of these polynomials, since by Favard's theorem such recurrence
relations are satisfied only by polynomials orthogonal with respect to some positive
measure.
5.3 Gauss Quadrature
The need to approximate an integral that cannot be evaluated exactly has existed
since the start of calculus. Newton used the method of interpolating a function at
n points and then integrating the interpolating function. He used polynomials to
do the interpolation.

5.3 Gauss Quadrature 249
We use Lagrange interpolation polynomials here. Suppose x\ < X2 < ¦ ¦ ¦ < xn
в a set of n numbers in an increasing sequence and y\,y2,...,ynisan arbitrary
set of numbers.
Definition 5.3.1 The Lagrange interpolation polynomial is a polynomial of de-
decree n — 1 that takes the value yt at x,- for i = 1, ..., n. This polynomial is given
by
j^ P'(Xj)(X - Xj)
*here
P(x) = (x - xi) ¦ ¦ ¦ (x -xn).
We write
It is clear that
Thus if f(x) is a continuous function whose values /(x,) are known at the points
t, . i = 1, 2,..., n in an interval [a, b], then
(x)f(xj) E.3.3)
is a polynomial of degree < n — 1 which interpolates the function / in [a, b].
Formula E.3.3) can be applied to approximate integration. We have
г
f(x)da(x)
'а
where
fb
Xj := / ?j(x)da(x). E.3.5)
Ja
It is evident that E.3.4) is exact if /(x) is a polynomial of degree < n - 1. For,
in this case Ln(x) = /(x). There are a number of ways to measure how well the
quadrature method approximates the integral. The most obvious is to see how large
the difference is. There is another way that has been very fruitful: Require that the
quadrature method be exact for as large a class of functions as possible.
For the interpolation method above, there are 2n parameters, A.* and xk. When
the xk are given in advance, it is easy to determine kk so that there is equality in

250 5 Orthogonal Polynomials
E.3.4) for all functions /(x) that are polynomials of degree at most n — 1. One
simply requires that the approximating polynomial agree with / at n points, so
that the two are identically equal when / is a polynomial of degree at most n — 1.
This is as far as one can go by means of Lagrange interpolation at fixed points.
However, if one does not require that the points xk be fixed, there is a possibility
of increasing the degree of the polynomial by one for each хь which is allowed
to vary. The maximum degree should be In — 1 when all the x^s are allowed to
vary.
This appears to be a difficult nonlinear problem, for we seem to need to solve
In equations that are linear in the A^s but nonlinear in the x*s:
pb
pb
l xjda(x), j =
The solution is contained in the next theorem, known as the Gauss quadrature for-
formula. Before stating it we introduce some notation. Suppose \Pn (x)} is a sequence
of polynomials orthogonal with respect to the distribution da(x), that is,
?
b
Pn(x)Pm(x)da{x) = 0 for m ф п. E.3.6)
Let Xj = Xjn = Xj<n, j = 1, 2,..., n, denote the zeros of Pn(x). We prove in the
next section that these zeros are simple and lie in the interval [a, b] used in E.3.6).
We saw an example of this in the Cheybshev polynomials of the first kind, Tn (x),
which had n simple zeros in [—1, 1].
Theorem 5.3.2 There are positive numbers X\, Яг,..., Я„ such that for every
polynomial f(x) of degree at most 2n — 1
/b
E.3.7)
where Xj, j = 1, ..., n are as defined after E.3.6), and Xj = Xjn = Я7,п.
Proof. Let /(x) be an arbitrary polynomial of any degree. Then by the division
algorithm,
f(x) = Pn(x)Q(x) + R{x),
where Pn(x) is as in E.3.6) and deg R < n — 1.
Since Xj are the zeros of Pn (x), we have
f(Xj) = R(xj) for./= 1,2,..., и

5.3 Gauss Quadrature 251
and
ь л «
f(x)da(x) = Pn{x)Q(x)da(x) + Y,*-jf(xj), E-3.8)
where Xj is defined in E.3.5). Now E.3.7) is exact if
jf
Pn(x)Q(x)da(x)=0. E.3.9)
Since E.3.8) is true for deg Q{x) < n — 1, it follows that E.3.7) is exact for poly-
polynomials f(x) of degree < In — 1. We have only to show the positivity of Xj. For
this, note that tj — tj is a polynomial of degree In — 2 which vanishes at xk,
к = 1,2, ...,n. So,
where deg Q < n - 2. Thus
pb pb
/ (t) - lj)da(x) = / Pn(x)Q(x)da(x) = 0
J a J a.
and
/"* , fb ¦>
Xj = I lj(x)da(x)= / lj(x)da(x) > 0.
This proves the theorem. ¦
Now, if /(x) is not a polynomial of degree < 2n — 1, then E.3.7) is not exact, but
we can use the right side as an approximation for the left side. Here the question of
the error involved is of great importance. We do not go into this question in depth
but merely prove that the right side of E.3.7) tends to the left side as n ->• oo if
/(x) is a continuous function.
Theorem 5.3.3 If /(x) is continuous on a finite interval [a, b], then
n ^ rb
Hm 2jA>/(xjn) = / f(x)da(x),
"^°° j=l ¦>«
where Xjn andx]n are as in Theorem. 5.3.2.
Proof. First note that by Weierstrass's approximation theorem (see Exercise 1.40),
for every e > 0, there is a polynomial p(x) such that
|/(x) - p(x)\ < e/BS) for all x € [a, b].

252 5 Orthogonal Polynomials
Here
S = 2^knk= / da(x).
For notational convenience, denote
g{x)da(x) and In(g) = V^jfi;,)
7=1
where g is any continuous function in [a, b]. Then
\ПЛ-1(Р)\< I \f(x)-p(x)\da(x)<e/2
Ja
and
\Ш) - In(p)\ < ^Я7„|/(х7„) - p(xjn)\ < e/2,
7 = 1
so that
\Ш) - Hf)\ < |/и(Л - /Я(РI + |/h(p) - Up)\ + \Hp)
Take 2n - 1 > deg p(x) so that In(p) — I(p) and
The conclusion of the theorem follows immediately from this inequality. ¦
Remark 5.3.1 Gauss considered the case where da(x) = dx in Theorem 5.3.2.
The orthogonal polynomials are then Legendre polynomials given by
_ p@,0)/ ч _
for the interval [— 1, 1].
Remark 5.3.2 When da(x) = dx/*J\ — x2, we get the Chebyshev polynomials
of the first kind. In this case, one can prove that
A-i = A.2 = ¦ • • = A.,,
and E.3.7) reduces to
COS— 7Г ,
2n
when / is a polynomial of degree < 2n - 1. A converse of this result is also true.
See Natanson [1965] for this and for Exercises 3-10.

5.4 Zeros of Orthogonal Polynomials 253
5.4 Zeros of Orthogonal Polynomials
We have seen that the Chebyshev polynomial Tn (x) has n simple zeros in [ — 1, 1 ].
More generally, one can prove the same about Jacobi polynomials by using the
representation C(l-x)~a(l+x)^?^{(l-x)"+a(l+x)"+^} and Rolle's theorem
together with induction. The next theorem shows that a similar result is true for
orthogonal polynomials in general.
Theorem 5.4.1 Suppose that {Р„(х)} is a sequence of orthogonal polynomials
with respect to the distribution da(x) on the interval [a,b]. Then Pn(x) has n
simple zeros in [a, b].
Proof. Suppose Pn(x) has m distinct zeros хь хг,..., xm in [a, b] that are of
odd order. In that case
Q(x) = Pn(x)(x - xO(x - x2) ¦ • ¦ (x - xm) > 0 E.4.1)
for all x in [a, b]. If m < n, then by orthogonality
I
¦b
Q{x)dx = 0. E.4.2)
However, the inequality in E.4.1) implies that the integral in E.4.2) should be
strictly positive. This contradiction implies that m = n and that the zeros are
simple, yielding our result. ¦
For the next theorem we denote the zeros of Pn (x) in increasing order by x\n <
X2n < " ' • < Xnn.
Theorem 5.4.2 The zeros of Pn (x) and Pn+\ (x) separate each other.
Proof. From Corollary 5.2.6,
Pn+l{x)P'n(x) - Pn(x)P'n+x(x) < 0.
Since Xjt,n+i is a zero of Pn+i(x), we get
l(xk,n+l) > 0.
The simplicity of the zeros implies that P^+l(xkin+\) and P^+x{xk+\in+\) have
different signs. It follows that Pn(xk,n+i) and Pn(xk+iill+1) have different signs.
By the continuity of Р„, we know it has a zero between Xk,n+i and x^+i,n+\ for
к = 1,2,... ,n, and our result follows. ¦
We can obtain an extension of Theorem 5.4.2 by using the Gauss quadrature
formula.
Theorem 5.4.3 Let m < n. Between any two zeros of Pm(x) there is a zero of
Pn{x).

254 5 Orthogonal Polynomials
Proof. Suppose there is no zero of Pn(x) between xkm and xk+i<m. Consider the
polynomial
Pm(x)
g(x) =
(x - xkm){x - xk+Um)
It is clear that g(x)Pm(x) > 0 for x g (xkm,xk+\_m). By the Gauss quadrature
formula
b "
g(x)Pm(x)da(x) = y^g(xjn)Pm(xjn).
" 7 = 1
Since g(xJn)Pm(xjn) > 0 and cannot vanish for all j = 1,..., n, and Xj > 0 for
all j, we see that the sum is positive. The integral, however, is zero by orthogonality.
This contradiction proves the result. ¦
We conclude this section with the Markov-Stieltjes inequalities for the sums
Z)t=i ^ь where j < n. Once again we let xj, j = 1,2,... ,n, denote the zeros of
Pn (x) in increasing order.
Theorem 5.4.4 The Markov-Stieltjes inequalities
j
da(x) < У^Хк
= 1 "" k=\
hold for j = 1,2, ...,n.
Proof. The Gauss quadrature formula is
fb ТГ-
/ f(x)da(x) = 2,^kf(xk), E.4.3)
J" k=i
where / is a polynomial of degree < 2л — 1. We have already noted that f(x) = 1
gives
da(x).
If E.4.3) were exact for the step function
k=i Ja
fjix)-\0, x>Xj,
we would have the value of Хд=1 ^ as f*' da(x). But fj(x) is not a polynomial
of degree < 2n - 1, so we use the following idea. Define polynomials фп(х, j)

5.4 Zeros of Orthogonal Polynomials
255
a x1
Figure 5.1
and Фп(х, j) (see Fig. 5.1) of degree <2n —2 such that
0, k = j,...,n;
E.4.4)
О, fc = j + !,...,«;
and
E.4.5)
E.4.6)
We first assume the existence of these polynomials to derive the Markov-
Stieltjes inequalities; we then prove that they exist. Use ф„(х, j), a polynomial of
degree < In — 1, in the Gauss quadrature formula E.4.3), to get
7-1 л
$^л* = / фп(х, j)du{x)
k=l ¦'"
< / fj(x)da(x)= f '
J a J a
Use of Ф„(х, У) gives
к=\
da(x).
These inequalities may be written as
J (X
and we have the Markov-Stieltjes inequalities.
To show the existence of фп(х, j), note that the conditions E.4.4) together with
ф'н(хк, j) = 0, кф],
E.4.7)

256 5 Orthogonal Polynomials
determine a polynomial of degree < 2n — 2. It is now sufficient to prove that
Фп(х, j) touches but does not cross the line у = 1 and crosses у = 0 at xj and
does not cross it again after that. Observe that, if фп(х, j) crosses у = 1 at Vi
points and у — 0 at v2 + 1 points, then by E.4.4) and E.4.7), ф„(х, j) - 1 has at
least 2G — 1) + vi zeros for x < Xj and ф„(х, j) has at least 2(n - j) + v2 + 1
zeros for x > Xj counting multiplicity. Thus ф'п (x, j) has at least 2(j — 1) + vi —
1 + 2(и — j) + V2 = 2n — 3 + vi+V2 zeros. Since ф'п is of degree In — 3, vi and v2
must be zero. This proves one part of the inequality E.4.6). The other part is done
in the same way and the theorem is proved. The second part is also a consequence
of the first part; replace xk with b - xk and ф with 1 — ф. ш
5.5 Continued Fractions
Continued fractions of a certain type are closely connected with orthogonal poly-
polynomials. This connection has been extensively studied. We shall merely touch on
this topic and prove an interesting result of Stieltjes.
Suppose {an}f and {bn}8° are sequences of complex numbers. One notation for
an infinite continued fraction is
We shall denote the nth convergent of this continued fraction by Cn. So
Co = b0 =: —, Ci = bo + — = =: —,
? b tS
a\ a2 a\ bo{b\b2 + a2) + a\b2
t2 = Do + -.—- j- = t>0 +
0+0
T;7Г =
t>l+a2/t>2
Definition 5.5.1 We say that the continued fraction E.5.1) converges, if at most
a finite number of Cn are undefined and limn^oo Cn exists. (In the above example,
C2 is undefined if B2 = b\b2 + a2 = 0.)
The sequences {An} and {Bn} defined above satisfy the three-term recurrence
relations given in the next lemma.
Lemma 5.5.2 Forn > 1,
An = ЪпАп-Х + а„Ап_2, A_i = 1 E.5.2)
and
Bn = bnBn-i + anBn_2, B-i = 0. E.5.3)
Proof. Since
a\ a2 an
C b +

5.5 Continued Fractions 257
and
„ , , a\ a2 an an+l
Ln+i =t>o+ -— -—— ¦ ¦ ¦ t—
t>+t>+ O+
t>2+ On+On+l
we see that Cn+\ is obtained from Cn by replacing an with anbn+\ and bn with
bnbn+\ + an+i. Suppose the result of the lemma is true up to n (it is clearly true
for и = 1); we have
An+l = (bnbn+i +an+i)An-i + anbn+lAn-2
= bn+i(bnAn-i +anAn-2)+an+lAn-i
= bn+\An + an+iAn-i.
This proves the result for An by induction; the proof for the sequence {Bn} is
similar. The lemma is proved. ¦
We prove the following lemma by a method similar to the proof of the Christoffel-
Darboux identity.
Lemma 5.5.3
Proof. Multiply E.5.2) by Bn-X and E.5.3) by An_i and subtract to get
AnBn-i - BnAn_Y = -an(An_iBn-.2 - Bn-.xAn-2).
Now iterate to get the result. ¦
The recurrence relation E.2.2) satisfied by a sequence of orthogonal polynomials
{Р„{х)} when compared with E.5.3) suggests the consideration of the continued
fraction
Ao C\ C2
E.5.4)
Aox + BQ- Aix + #i- A2x + B2-
In this case the nth convergent is a rational function whose denominator is Pn(x).
We denote the numerator by P*(x). The sequence {P*(x)} satisfies the same
recursion, namely
CiW = (A"x + Bn)P*{x) - CnP*_x{x), n > 1, E.5.5)
but
P*(x) = 0, P?(x) = Ao-
Suppose that the sequence {Pn (x)} is orthogonal with respect to the distribution
da(x) on [a, b]. The next result relates P*(x) to Pn{x).

258 5 Orthogonal Polynomials
Theorem 5.5.4 With Pn(x) and Р„{х) as defined above, we have
P*n{x) = § f" Pnix) ~_Pn{t) da{t), n > 0, E.5.6)
Ja X t
where S is a constant.
Proof. The result holds for n = 0 because in that case Pq(x) = constant. For
n = 1, P\{x) = k\x-\- constant. Thus the result holds in this case as well, if we
adjust S.
If n > 2, then denote the right side of E.5.6) by Rn{x) and observe that
Rn(x) - (AB_i* + Bn^)Rn^{x) + Cn
b Pn(x) - Pn(t) - (An-ix + fi
X - t
Cn-x{Pn-2{x) - Pn-2i,t))
H da(t)
x -1
X — t
(A,.,* + Bn-X)Pn-j{t)
fb
= 8An-i /
Ja
This means that /?n(x), which is the right side of E.5.6), satisfies the same recur-
recurrence relation as P* with identical initial values. This proves the theorem. ¦
The next result is an application of the Gauss quadrature formula. Let jc* ,k = 1,
2,..., n, denote the zeros of Pn(x).
Theorem 5.5.5 Using the notation of Theorems 5.3.2 and 5.3.3, we have
P^1=S±^^, E.5.7)
where S is the constant appearing in E.5.6).
Proof. The rational function P*(x)/Pn(x), expressed as a partial fraction, is
PnW fr[ Р!г(Хк)(х - Xk)
(Note that the degree of P* (x) is less than the degree of Pn (x).) By Theorem 5.5.4,
it follows that
P (t)
nK da(t)=SXk. E.5.8)

5.6 Kernel Polynomials 259
The last equality follows from the Gauss quadrature formula in Theorem 5.3.2.
The result is proved. ¦
Theorem 5.5.5 is due to Stieltjes [1993, paper LXXXI]. The next theorem was
presented by Markov [1895, p. 89].
Theorem 5.5.6 Let [a, b] be a finite interval. For any x g [a, b]
E.5.9)
Proof. For any x g [a,b], the function ^ is a continuous function of t in
[a, b]. This observation taken together with Theorems 5.3.3 and 5.5.5 imply the
result. ¦
Remark 5.5.1 Since Xk > 0, it is an immediate consequence of E.5.8) that the
zeros of P* and Pn alternate.
Remark 5.5.2 In Theorem 5.5.6, we may take x to be a complex number that does
not lie in [a, b]. If we denote the right side of E.5.9) by F{x), then the inversion
formula of Stieltjes is given by
1 fd
a(c) - a(d) = lim / Im{F(M + iv)}du.
Ж v->0+ Jc
Thus, the distribution can be recovered from F.
5.6 Kernel Polynomials
In Section 5.1 we saw that the partial sum of the Fourier series of a function when
expressed as an integral gave us the Chebyshev polynomials of the third kind:
Vn(x) — sin(n + 1/2N/sin(в/2), where x — cos6. More generally, we get the
kernel polynomials when we study partial sums involving orthogonal polynomials.
Let {pn(x)} be a sequence of polynomials orthogonal with respect to the dis-
distribution da(t) on an interval [a, b]. As before, — oo < a < b < oo. Let / be a
function such that fa f(t)pa(t)da(t) exists for all n.
The series corresponding to the Fourier series is given by
aoPo(x) + aipi(x) H h anpn(x) -\ , E.6.1)
where
fl»=/ f(t)pn(t)da(t) / f {pn(t)}2da(t). E.6.2)

260 5 Orthogonal Polynomials
In this section we assume that the denominator of an is one, that is, the sequence
{pn(x)} is orthonormal. Then the nth partial sum Sn(x) is given by
f(t)pk(t)da(t)
= / f(t)Kn(t,x)da(t), E.6.3)
Ja
where
n
J2 E.6.4)
*=0
Definition 5.6.1 For a sequence of orthonormal polynomials {pn(x)}, the se-
sequence {Kn(x0, x)}, where
n
Kn(xo,x) = у ^ Pk(xp)pk(x),
k=0
is called the kernel polynomial sequence.
Lemma 5.6.2 If Q(x) is a polynomial of degree < n, then
Q{x)= / Kn{t,x)Q{t)d<x{t).
Ja
Proof. Clearly,
k=0
for some constants ak. Multiply both sides by pj(x) and integrate. Orthogonality
gives us
/ Q{t)Pj(t)da{t)=aj.
Ja
The lemma follows immediately. ¦
Theorem 5.6.3 Suppose xq < a are both finite. The sequence {Kn(xQ, x)} is
orthogonal with respect to the distribution (t — Xo)da(t).
Proof. In Lemma 5.6.2, let Q(t) = (t — x0)Qn-\{t), where Qn-\ is an arbitrary
polynomial of degree n — 1. The theorem follows. ¦
Remark 5.6.1 A similar result is obtained when b < xq are both finite.

5.6 Kernel Polynomials 261
Remark5.6.2 In the case ofthe Chebyshev polynomials Tn(x) with xo = a = —1,
we see that for x = cos в,
Kn(-l,cosd) = cos#+ cos2#- ••• + (-l)"cosn#
cos(n+1/2H
1 ' cos(#/2) "
The polynomials
cos (n + k)Q
Wn(x) = i—^-,x=cos<?
cos I
are the Chebyshev polynomials ofthe fourth kind and Theorem 5.6.3 implies that
the sequence {W^x)} is orthogonal with respect to the weight function л/yzf ¦ If
we choose xq = b = 1, then we get the Chebyshev polynomials of the third kind,
Vn(x), which are orthogonal with respect to the weight [A — x)/(l + x)]1/2.
Remark 5.6.3 Another straightforward consequence of Theorem 5.6.3 is that if
{Pn(x)} is orthogonal on [a, b], then
pn{x) pn+l(x) _
г~г ГТ" — An4n\X)\x a),
pn(a) pn+\{a)
where {qn{x)} is orthogonal with respect to (t — a)da{t), for we have
Pk(x)Pk(a)
k=0 *
Anpn(a)pn+i(a) \pn+i(x) pn{x)
{x-a)hn [pn+i(a) pn(a)
- ИпЧп(х),
for a constant fxn. The last equation also implies that
, л , . р„(х)рп(а)
пп
These results will be used in Chapter 6, Section 6.4.
The Christoffel-Darboux formula (Theorem 5.2.4) gives a compact expression
for the kernel polynomials:
kn pn+i(x)pn(x0) - pn+\(xo)pn(x) /CzrC4
^„(xo,x) = . E.6.5)
K-n+l X — Xq
If we choose x0 to be the kth root of pn(x), that is, x0 = xk, then we can write
„ , , kn pn+i(xk)Pn(x) ,„ , ,.
К„(хк,х) = — . E.6.6)
«71+1 X — Xk
This expression for Kn suggests a connection with the Gauss quadrature formula.
In fact, we have the following theorem:

262 5 Orthogonal Polynomials
Theorem 5.6.4 The numbers Xk (or Xkn) occurring in the Gauss quadrature
formula are given by
X k^\
Xk = --^ \ ;
kn pn+\(xk)p'n(xk)
their reciprocal is
E.6.7)
\/Xk = Kn(xk, xk) = ^(pk(xk)f. E.6.8)
k=o
Proof. The expression for Xk in Gauss's formula is
/b pn(t)da(t)
p'n{xk){t -xk)'
By E.6.6) and Lemma 5.6.2,
f Kn{
n(Xk) Ja
kn Pn+\(Xk)P\
_ kn+i 1
kn рп+\{хк)р'п{хкУ
This proves E.6.7). To derive E.6.8), let x ->• xk in E.6.6). Thus,
К
K(xk,Xk) — Pn+l(Xk)Pn(Xk)-
kn+i
This proves E.6.8). ¦
The kernel polynomials also have a maximum property, as contained in the next
theorem.
Theorem 5.6.5 Let xq be any real number and Q{x) an arbitrary polynomial of
degree < n, normalized by the condition
rb
\ (Q{t)fda{t) = \.
Ja
The maximum value of(Q(xo)J is given by the polynomial
Q(x) = ±Kn(x0,x)/^/Kn(x0,x0)
and the maximum itself is Kn(xo, xq).
Proof. Since Q(x) is of degree < n, we have
Q(x) = aopo(x) + aipi(x) -\ \- а„р„(х).

5.7 Parseval's Formula 263
The normalization condition gives
al + a\-{ bfl?= 1.
By the Cauchy-Schwartz inequality
n
J2 = *«(x°' Xo)"
k=0
Equality holds when ak = Apk(xo), where A is determined by
k=0
This proves the theorem. ¦
5.7 Parseval's Formula
Let L?(a, b) denote the class of functions / such that
rb
/ \f\"da(x) < (X).
Ja
As always, we assume that f x"d<x(x) < oo for n > 0. In this section, we are
interested in the space L2a(a, b). By the Cauchy-Schwartz inequality, we infer the
existence of
/ f(x)x"da(x),
Ja
for n > 0.
Theorem 5.7.1 Suppose f € L2a(a, b). Let Q(x) be a polynomial of degree n,
such that
n
Q(x) = ^akPk{x),
k=0
where {pn(x)} is the orthonormal sequence of polynomials for da. The integral
I U(x) - Q(x)]2da(x) E.7.1)
Ja
becomes a minimum when
fb
ak= f{x)pk{x)da{x). E.7.2)
Ja

264 5 Orthogonal Polynomials
Moreover, with ak as in E.7.2),
Proof. Let с* = fa f (x)pk{x)da(x). By the orthonormality of {pn(x)}, we get
/¦
0< /
[/(x) -
/•ft « n
= / [f(x)]2da(x) - 2 J] fltct + ^ a,2
^a k=0 k=0
= f [f(x)fda(x) -J24+ J2(ak - ckf.
The last expression assumes its least value when ak = ck. This proves both parts
of the theorem. ¦
Corollary 5.7.2 For f e L2a{a, b) and ak as in E.7.3), we have
Х^«„2< / U(x)]2da(x). E.7.4)
Proof. The sequence of partial sums sn — YH=oak is increasing and bounded. ¦
The inequality E.7.4) is called Bessel's inequality. We now seek the situation
where equality holds. Assume that [a, b] is a finite interval. We shall use the
following result from the theory of integration.
Lemma 5.7.3 For f e L2a{a, b), and a given € > 0, there exists a continuous
function g such that
fb
[f(x)-g(x)]2da(x)<€.
/
Ja
Theorem 5.7.4 Let [a, b] be a finite interval. With the notation of Theorem 5.7.1,
we have Parseval's formula:
oo „b
al= U(x)]2da(x). E.7.5)
k=o Ja
Proof. Suppose g is as in Lemma 5.7.3. By Weierstrass's approximation theorem,
for a given e > 0 there exists a polynomial Qn (x) such that
/
Ja
\g(x) - Qn(x)]2da(x) < €.

5.7 Parseval's Formula 265
This implies
rb
[/(*) - Qn(x)]2da(x) < 4e. E.7.6)
/
Ja
By Theorem 5.7.1, we may choose Qn(x) = J2l=o <*kPkM> where a* is given by
E.7.2). As in the proof of Theorem 5.7.1, it follows from E.7.6) that
Since € is arbitrary, we have proved the theorem. ¦
Corollary 5.7.5 Suppose f e L2a(a, b), where [a, b] is a finite interval. If
I
ь
f(x)xnda(x) = 0 for all integers n > 0, E.7.7)
then / = 0 almost everywhere.
Proof Since ak = 0 for all к, it follows from Parseval's formula E.7.5) that
[ [f(x)]2da(x)=O.
Ja
This implies the result.
Exercises 24-28 give results similar to Corollary 5.7.5 for infinite intervals.
Stone [1962] contains proofs of these results.
Remark 5.7.1 We could also argue as follows: With Qn (x) as in E.7.6) it follows
from E.7.7) and the Cauchy-Schwartz inequality that
[f(x)]2da(x)\ = (j /MUM ~ Qn(x)]da(x)
f [f(x)]2da(x) f [f(x) - Qnix)]2daix).
Ja Ja
So
and the result follows.
jf
Corollary 5.7.6 With f as in Corollary 5.7.5 andsnix) = J2l=o akPkix), where
ak is given by E.7.2), it follows that
rb
\\sn(x)-f(x)\\22= / [sn(x)-f(x)]2da(x)^0
Ja

266 5 Orthogonal Polynomials
Proof. This result is contained in the proof of Theorem 5.7.4. ¦
The results of Theorem 5.7.4 and its corollaries are in general false when [a, b]
is not finite. As an example (see Exercise 1.20), one may take
da(x) = exp(—xM cosixn)dx, f(x) = sin^^ sin/Kjr), 0 < ц, < 1/2.
There are, however, important examples of orthogonal polynomials over infinite
intervals, such as the Laguerre polynomials on @, oo) and Hermite polynomials
on (—oo, oo). We prove in the next chapter that Theorem 5.7.4 continues to hold
in these cases.
5.8 The Moment-Generating Function
In this section we obtain a continued fraction expansion for the moment-generating
function J2n>o /J-nX", where
Mn = (l,f«) = / t"da(t). E.8.1)
Ja
The treatment here follows Godsil [1993] and the reader should consult this book
for further information on the methods of algebraic combinatorics in the theory
of orthogonal polynomials. We assume a minimal knowledge of graph theoretical
terminology.
Let G be a graph with n vertices. The adjacency matrix A = A(G) is the n x n
matrix defined as follows: If the j'th vertex is adjacent to the 7 th vertex, Ац = 1;
otherwise it is zero. An edge {г, j} in G is considered to be composed of two arcs,
(г, j) and (j, i). A walk in a graph is an alternating sequence of vertices and arcs
where each arc joins the vertices before and after it in the sequence. If the first
vertex is the same as the last one in the sequence, then it is a closed walk. The
number of arcs in a walk is called the length of the walk. The following result is
easily checked by induction: The number of walks in G from vertex г to vertex j
of length m is given by (Am),7, that is, the entry in the jth row and 7th column of
the matrix Am.
We shall have to consider graphs with weighted arcs so that the entry (A)^ is
the weight of the arc (г, j). We continue to denote this matrix by A = A(G). Let
0(G,x) = det(x/-
and
n>0

5.8 The Moment-Generating Function
267
Thus Wjj (G, x) is the generating function for the set of all walks in G from vertex
i to vertex j. Let W(G, x) be the matrix whose entries are Wtj(G, x). Then
n>0
From the fact that A adj (A) = det( A) /, where adj (A) is the adjoint of A, it follows
that
,-K-l
W(G,x)=x-l4>(G,x~l)
This implies that
Wu(G,x)=x-l4>(G\i,x~l)/4>{G,x-1),
E.8.2)
E.8.3)
where G\i is the graph obtained from G by removing vertex i.
The connection of the above discussion with orthogonal polynomials is obtained
as follows: Suppose {pn(x)} is an orthogonal polynomial sequence satisfying the
three-term recurrence
pn+i(x) = (x- an)pn(x) - bnpn-x{x), и > 1. E.8.4)
It is assumed that the polynomials are monic. Let A denote the matrix
Ъ\ \
a\ b2
1 a2 Ъъ
\
I
where the rows and columns of the matrix are indexed by the nonnegative integers.
Let An denote the square matrix obtained from A but taking the first n rows and
columns. Observe that when det(xl — An) is expanded about the last row, we get
det(x/ - An) - (x -an_!
= pn(x).
Thus pn(x) is the characteristic polynomial of An.
Observe that matrix A is the adjacency matrix of a particular weighted directed
graph G whose vertex set is indexed by the nonnegative integers. If only the first
n vertices of G are taken, then the adjacency matrix of the subgraph is An. Denote
this subgraph by Gn.
We need the next lemma to derive the continued-fraction expression for the
moment-generating function 5Zn>o(^' ?")x"> which is understood in the sense of
a formal power series. We assume that /xq = A, 1) = 1.

268 5 Orthogonal Polynomials
Lemma 5.8.1 For nonnegative integers n,
Proof. First note that (Ak)oo = (A*)oo for ifc < 2л + 1, because no closed walk
starting at 0 and of length < 2n +1 can include a vertex beyond the nth vertex. This
implies(pn(A))oo = (pn(An))oo.Wehavealreadynotedthatforn > 1, pn(x)is\he
characteristic polynomial of An. By the Cayley-Hamilton theorem, pn(An) = 0,
and hence
(I, pn) = (Pn(A))oo
for n > 1 and for n = 0 by definition. Now x" is a linear combination of po, pi, ¦ ¦¦,
pn, so the result follows. ¦
Theorem 5.8.2 With an and bn as in E.8.4),
1 x2b\ x2b2
, tn)xn =
1 — xuq— 1 — xa\ — 1 — xaz—
Proof. Let An^ be the matrix obtained from An by removing the first к rows and
columns. Set
qn-k{x) =det(I -хАпЛ).
Observe that
0(Gn, x) = det(xl - An) = x" det(/ - дс"^^) = ^«„(дГ1)
and
0(Gn\O, x) = det(xl - АпЛ) = xn~l det(/ - x~l АпЛ) = ^-^„-itc).
By E.8.3), we can conclude that
x Woo(Cjn,x ) = — — x ——. E.8.5)
<p(Gn,x) qnix-1)
Expansion of det(/ — xAn) about the first row gives
qn{x) = A -xao)qn-i(x) - x2biqn-2(x)
or
qn-\(x)
qn(x) 1 - xa0 - x2biqn-2(x)/qn-.i(x)'
E.8.6)

Exercises 269
By Lemma 5.8.1,
V У2 = WooiCx) = lim ^zl^l.
This combined with E.8.6) proves the theorem.
Exercises
1. (a) Prove the positivity of the Poisson kernel for Tn(x) in the interval — 1 <
r < 1 by showing that
1 -r2
rm =
r.
1 - 2r cos в + r2
m = \
(b) Compute
00
sin(m + 1N sin(m + 1H m
sin6> sin0
m=0
which is the Poisson kernel for Un(x). Observe that it is positive in the
interval — 1 < r < 1.
(c) Show that the Poisson kernel for sin(n + 1 /2N is
r" sin(n + \/2)в sin(n + 1/2H
A - r) sin@/2) sin@/2)[(l -rJ +4r(l - cos@+.p))/2cos@ - 0)/2]
+ 0)/2+r2][l -2r cos@ -0)/2+r2]
2. Suppose / has continuous derivatives up to order n in [a, b] and x\
• ¦ ¦ < xn are points in this interval. Prove the following Lagrange interpolation
formula with remainder:
fix) = Ln{x) + J—^(x-x,)(x-x2) ¦ ¦ ¦ (x-xn),
where a < min(x, x\, хг,..., xn) < | < max(x, x\,..., х„) < b. Here
Ln (x) is the Lagrange interpolation polynomial (denned by E.3.1)) that takes
the value /(*,) at д:,-, i = 1,2,... ,n.
A discussion of the results in Exercises 3-10 can be found in Natanson
[1965]. This book also contains the references to the works of Hermite and
Fejer mentioned in the exercises.
3. With the notation of Exercise 2, suppose that
n
Ln(x) = ^2Ak(x - x\)(x - x2) ¦ • ¦ (x - xk).
k=0

270 5 Orthogonal Polynomials
Show that
Ак-г = '
Xj X\) • • • \Xj Xj — \){Xj X}+\) ' ' ' \Xj Xk)
J
Now let Xj¦ = a + (j — \)h for j = 1, 2,..., n. Show that
Ak~l = /^
where
and
Aef(Xj) =
4. Let tj(x) be defined by E.3.2) with P(x) = (x - xi)(x - x2) ¦ ¦ ¦ (x - xn).
Check that t'j(xj) = P"(xj)/2P'(Xj). Now show that the function H(x)
defined by
H(x) = E yj [l - |^(x - *,)] t){x) + XJy'jix ~ Xj)l){x)
(where y\, уг,..., yn, y[,..., y'n is a given set of 2n real numbers) satisfies
H(xj) = yj and H'(xj) = y'j, where H' denotes the derivative of H. Prove
also that if / is as in the previous problem with derivatives of order 2n, then
where yj = f(Xj) and y'j = f'(Xj) in the definition of H. Again f lies in the
same interval as before.
5. Apply Gauss quadrature to the formula for f(x) in the previous problem to
obtain
rb n fQ-n), ч rb
/ f(x)da(x) = J2 hf(xk) + n / P^(x)da(x), a<r]<b.
Ja ?_[ l^"). Ja
Here {Pn(x)} is a sequence of polynomials orthogonal with respect to da(x)
on [a, b] and the leading coefficient of Pn(x) is one.
6. (a) Prove that, for the Legendre polynomials,
2"n! dxn

Exercises 271
(b) Use the Christoffel-Darboux formula to prove that
1 PH(t)PH-i(x)-PH(x)PH-i(t) 2
dt = —.
i t — x n
(c) Deduce from (b) that if xk, к = 1,..., n are the zeros of Pn(x), then
p"(r)-* = —2
1-х t - xk
(d) Use Exercise 5 and the above to obtain the formula
rl — 2f(xk) , 22и+1(п!L/Bи)A)
J-x f^nPn-l
where — 1 < § < 1.
7. Use Chebyshev polynomials to prove the following formula, which is similar
to Exercise 6{d):
dx -lff (co, Bk-Dn\ n
with — 1 < § < 1. Note that all the Xk equal ^ in this case. (Hermite)
8. (a) Show that the roots of the Chebyshev polynomials Un (x) are cos -^,
к = 1, 2,..., n and those of Vn(x) are cos ^j, k=l,2,... ,n. [Recall
rn(cos#) = cosnd and Un(cos6) = si
(b) Prove the quadrature formulas
fl I г л v^ . , кл (
I \/\-x2f{x)dx = > sm2 /cos
кл
' /r\ \ in?»_1_ 1 J \S/'
Bn)!22"+1
and
1-х 4л" ^—^ 2 кж ( 2кл \
f(x)dx = > sin f cos
1 + x J 2n + 1 f^ In + Г V 2n + \)
' Bn)!22«J Vb/'
where —1 < ^ < 1. Obviously the various f are not necessarily the
same.

272
9. Prove that
5 Orthogonal Polynomials
Tn(x)dx
-i т;(Хк)(х-хк)
>o,
B1с)ж
where xk = cos
Hint: Write the integral as
sinflt Г
n Jo
cos пв
cos 0 — cos 6k
sin Odd,
where #? = B*2n1>7r • Apply the Christoffel-Darboux formula and integrate
term by term. (Fejer)
10. Prove that
-хк)
where Un(xk) = 0.
11. Suppose {Pn(x)} is an orthogonal polynomial sequence. Let*b k = 1,2,...,
n, denote the zeros of Pn(x). Suppose that
Pn-i(x)
ak
is apartial fraction decomposition of Pn-\{x)/Pn{x). Prove thata^ > 0.
12. With the notation used in Lemma 9.5.2, show that
provided bt ф 0, В, фО A < i < n).
13. Show that if {Р„ (*)} satisfies
andF_!(x) = 0, then
Bo 1 0
0 C2 A2x + B2
Pn(x) =
for и
0
0
1
0
0
0
Cn-\

Exercises
273
14. Suppose An = 1 for all n in Exercise 13 and let Cn = \dn |2 = dndn. Then the
zeros of Pn (x) are the eigenvalues of the matrix:
Bo di
J D
*1 — " 1
0 J2
0
0
0 •••
d2 •••
-B2 rf3
0 0
0
dn-2
0
0
0
0
-Bn-2
dn-\
0
0
0
: dn-\
—Bn-\
15. Prove the following recurrence relations for Laguerre and Hermite polynomi-
polynomials respectively:
(а) (и
(b)
\)Lan{x) - (n +a)Lan_l(x),
n =0, 1,2, 3,...
= 2xHn(x)-2nHn-1(x), и =0,1,2,...,
= 1, Я_!(х)=0.
16. Let {pn(x)}§° be an orthonormal sequence of polynomials with respect to the
distribution da(x). Let
rb
i«= x"
J a
da(x), и = 0, 1,2,
Show that
pn(x) =Cn
tin
¦ ¦ ¦ Mn+1
1 X X1
where Cn is a constant given by Cn = (Dn-\Dn) 1/2, when Dn is the positive-
valued determinant [/X/t+mb,m=o,i H.
17. With the notation of Exercise 16, prove that
pn(x) = Cn
ц,ох - Mi Mi* - M2
- М2 Ц-2Х - M3
finx — Mn+i
fln-lX — fln Ц,пХ — Mn+1 ••• M2n-2* — M2n-1

21A 5 Orthogonal Polynomials
18. With the notation of Exercise 16, prove that
i=0
П (Xi - XjJda(xo)da(xi)---da(xn-i)
and
¦da(xn).
1 f^ f^
D" = , , 14, / • ¦ • / П (Xi - x,-)
(For the reference to Heine, see Szego [1975, p. 27].) (Heine)
19. Let 1 > x\(a, P) > хг(а, /J) ¦ ¦ ¦ > xn(a, fi) > -1 be the roots of the Jacobi
polynomial P^a-^(x). Show that, fora > -1 andjS > — 1,
да Эуб
Proceed as follows:
(a) Take a = -\,b = 1, and da{x) = A - x)"(l + x)fi in the Gauss
quadrature formula (Theorem 5.3.2). Take the derivative with respect to
a to get
fi
Лу-VI r1aC1 -I- r\P 1пяП rWr
-1
(b) Take /(x) = {P^a-fi)(x)}2/(x - xk) to show that
-f k x - xk
дхк
Н
a \ dx
Now observe that the expression in curly braces and x — xk have opposite
signs. Prove Щ > 0 in a similar way. (Stieltjes)
20. This result generalizes Exercise 19. Let <w(x, r) be a weight function depen-
dependent on a parameter r such that <w(x, r) is positive and continuous for a <
x < b, t\ < т < T2- Assume that the continuity of the partial derivative Щ
fora < x < b, x\ < т < T2 and the convergence of the integrals
I
dt
occur uniformly in every closed subinterval r' < г < г" of (ть гг). If the

Exercises 275
zeros of Pn(x)=Pn(x, r) (the polynomials orthogonal with respect to
w(x, r)) arexi(r)>X2(r)> • • • >xn(t), then the kth zero xk(x) is an increas-
increasing function of т provided that ff/w is an increasing function of x, a<x<b.
(Markov)
21. Let w(x) and W(x) be two positive, continuous weight functions on [a, b].
Let W(x)/w(x) be increasing. Prove that if {xk} and {Xk} denote the zeros
of the corresponding orthogonal polynomials of degree n in decreasing order,
then
xk < Xk, к = 1,..., и.
Hint: Take w(x, r) = A - x)w(x) + xW(x) in Exercise 20.
See Szego [1975, §§ 6.12 and 6.21] for Exercises 19-21 and for references.
22. Use the result of Exercise 20 to show that if the parameters a and /3 of the
Jacobi polynomials lie in [-1/2, 1/2], then the zeros of P%*'p)(x) satisfy
2k-\ 2k ,
7Г < xk < п, к = 1, 2,.... п.
2и + 1 ~ 2и + 1
23. Suppose {?>п(*)}о° is an orthonormal sequence of polynomials with respect
to the distribution da(x). Let
where ck = f(x)pk(x)da(x).
k=o Ja
Prove that
rb
f(x0) - sn(x0) = / [/(xo) - f(x)]Kn(x, xo)da(x).
Ja
24. Suppose / : [0, oo) —> Л is continuous and lim^^oo /(x) = 0. Show that
/ can be uniformly approximated by functions of the form e~ax p(x) where
p(x) is a polynomial, when a is a fixed positive number.
25. Prove that if / e Lp@, oo), p > 1 (or / is a bounded measurable function),
and (for a given a > 0)
Г
Jo
f(x)e-axxndx=O for и = 0,1,2,...,
о
then/(x) =0а.е.
26. Suppose / : (—00, 00) —> R is continuous and lim^^ioo /(x) = 0. Prove
that / can be uniformly approximated by functions of the form e~a x p(x)
where p(x) is a polynomial.
27. Show that if / e Lp{~oo, 00), p > 1, and
2jr2xVx = 0, и =0,1,2,...,
then /(x) = 0 a.e.

276 5 Orthogonal Polynomials
28. Show that
k=0
(п + 2)Г(п+а+р
x
x-y
29. (a) Show that the Legendre polynomial Pn(x) is a solution of
A - x2)y" - 2xy' + n(n + 1)у = 0.
(b) Show that Qn(x) = \ /ij ^fzj-dt, x g [—1,1] is another solution of the
differential equation in (a).
(c) Show that Qn(x) = Pn(x)Q0(x) - Wn-ito, where Wn-{(x) is a poly-
polynomial of degree n — 1 given by
„. , , 1 /"' Pn(x)-Pn(t)
Wn-\(x) = - dt.
2j-i x-t
For -1 < x < 1, define Qn(x) = Pn(x)Q0(x) - Wn-i(x) with Goto =
(d) Prove the following recurrence relations:
Bn + \)xPn{x) = (n + l)Pn+l(x)+nPn-i(x), л = 0, 1, • • •,
Bи + l)xG«to = (и + l)Gn+iCO + «Gn-i(x), и = 1, 2,....
(e) Prove that
n
i=0 " J
| 1)[6«+itoG«(>')-6«(xNn+i(>')"
and
1 "
1- V^r?i--4- ШГ>,ГгЛ12
Bk+l)[Qk(x)]2
(f) Show that Gn(^) has n + 1 zeros in — 1 < x < 1.
See Frobenius [1871].

Special Orthogonal Polynomials
Special orthogonal polynomials began appearing in mathematics before the signi-
significance of such a concept became clear. Thus, Laplace used Hermite polynomials
in his studies in probability while Legendre and Laplace utilized Legendre polyno-
polynomials in celestial mechanics. We devote most of this chapter to Hermite, Laguerre,
and Jacobi polynomials because these are the most extensively studied and have
the longest history.
We reproduce Wilson's amazing derivation of the hypergeometric representation
of Jacobi polynomials from the Gram determinant. This chapter also contains
the derivation of the generating function of Jacobi polynomials, by two distinct
methods. One method, due to Jacobi, uses Lagrange inversion. The other employs
Hermite's beautiful idea on the form of the integral of the product of the generating
function and a polynomial. This generating function is then used to obtain the
behavior of the Jacobi polynomial Fn(" ^'(x) for large n. We quote a theorem of
Nevai to show how the asymptotic behavior of P^"'^ (x) gives its weight function.
It is important to remember that the classical orthogonal polynomials are
hypergeometric. We apply Bateman's fractional integral formula for hypergeo-
hypergeometric functions, developed in Chapter 2, to derive integral representations of
Jacobi polynomials. These are useful in proving positivity results about sums of
Jacobi polynomials. We then use Whipple's transformation to obtain the lineariza-
linearization formula for the product of two ultraspherical polynomials. This clever idea is
due to Bailey. One of the simplest examples of linearization is the formula
cosmd cosnO = -[cos(m +п)в + cos(m — п)в].
We observe that a linearization formula for a set of orthogonal polynomials is
equivalent to the formula for an integral of the product of three of these poly-
polynomials.
We briefly discuss the connection between combinatorics and orthogonal poly-
polynomials. In recentyears, this topic has been studied extensively by Viennot, Godsil,
277

278 6 Special Orthogonal Polynomials
and many others. We content ourselves with two combinatorial evaluations of an
integral of a product of three Hermite polynomials.
This chapter concludes with a brief introduction to ^-ultraspherical polynomials.
This discussion is motivated by a question raised and also answered by Feldheim
and Lanzewizky: Suppose f(z) is analytic and \f(rew)\2 is a generating function
for a sequence of polynomials pn(cos6). Do the pn(cos0) produce an orthog-
orthogonal polynomial sequence other than the ultraspherical polynomials? The answer
involves an interesting nonlinear difference equation that can be neatly solved.
6.1 Hermite Polynomials
The normal integral f^°ooe~x2dx, which plays an important role in probability
theory and other areas of mathematics, was computed in Chapter 1. The integrand
e~x has several interesting properties. For instance, it is essentially its own Fourier
transform. In fact,
e~x = — / e-'e2ixtdt. F.1.1)
This can be proved in several ways. (See Exercise 1.) The integral is uniformly
convergent in any disk \x \ < r and is majorized in that region by the convergent
integral
-= / e-'2e2r'dt.
Thus the integral can be repeatedly differentiated with respect to x, and we have
^— = ЩИ f e-t2tne2ixtdt. F.1.2)
dxn Vя J-oo
The polynomials orthogonal with respect to the normal distribution e~x are the
Hermite polynomials. They can be defined by the formula
Hn(x) = (-l)"e*2 * . F.1.3)
It is easy to check that Hn (x) is a polynomial of degree n.
By F.1.2), it is seen that
(-2i)nex2 f°° i ,.
Hn(x) = =— / e ' tne2lx'dt. F.1.4)
Let us first prove the orthogonality property of Hn(x), namely
/ e-x2Hn(x)Hm(x)dx = 2nn\*Jn8mn. F.1.5)
J — OO

6.1 Hermite Polynomials 279
Consequent from the definition F.1.3), we can write this integral as
Suppose n > m and integrate by parts n times. This shows that the integral is zero.
The m = n case will be considered a little later.
The Hermite polynomials have a simple generating function. Observe that the
term tn in the integrand suggests that we consider
Fл 6)
^i-
The integral can be computed by F.1.1). The result is the generating function for
Hn(x):
F17)
This generating function is useful for deriving several properties of Hermite
polynomials. For example, we have the following expression for these polynomials:
This can be obtained by writing
^ p\ ^-^ Bq)\
and equating the coefficient of r11 on each side.
From F.1.8), it follows that
2»!.
dxn
We can now complete the proof of F.1.5). Integration by parts shows that
F.1.9)
where F.1.9) is used in the final step.
We learned in the previous chapter that orthogonal polynomials satisfy three-
term recurrence relations. To find this relation for Hermite polynomials, note that
F(x,r)= e2xr^2

280 6 Special Orthogonal Polynomials
satisfies
dF
— -Bx-2r)F=0.
dr
Substitute the series in F.1.7) for F to get
Hn+i(x)-2xHn(x) + 2nHn_1(x)=0, и = 1,2,.... F.1.10)
This is the recurrence relation for Hermite polynomials. Another recurrence rela-
relation comes from
dF
Тх-2гР-°-
This implies
H'n{x) = 2nHn_x(x), и = 1,2,.... F.1.11)
Eliminate #n_i(x) from F.1.10) and F.1.11) to obtain
Яп+1(х)-2хЯ„(х)+Я»=0.
Differentiate this equation and use F.1.11) again to get
Щ(х) - 2xH'n(x) + 2nHn(x) =0, и = 0, 1, 2,....
Thus the Hermite polynomials Hn(x) satisfy the second-order linear differential
equation
u" - 2xu' + 2nu = 0. F.1.12)
It is also worth noting here that the function
V(x)=e'x2/2Hn(x)
satisfies the differential equation
V" + Bn + l -x2)V =0.
As another application of the integral representation F.1.4) for Hn(x), we derive
a closed expression for the Poisson kernel for Hermite polynomials, namely
^Lrn = A _ r2rl/2eP4<r-UJ+>V]/U-'2). FJ.13)
By F.1.4),
Hn(y) = -

6.1 Hermite Polynomials 281
So for \r\ < 1, the left-hand side of F.1.13) becomes
e*2+y2 г00 г00
x +y r°o roo
/ / -s2-t2+2iys+2itx-2str
Л J-oo J-oo
Now use the formula
/•00
f
J —
e
—oo
twice in the double integral and F.1.13) follows. The various formal processes
can be justified by the absolute convergence of the integrals involved.
A different approach to the proof of F.1.13) using the three-term recurrence is
as follows: Denote the series on the left of F.1.13) by K(r, x, y). By F.1.10) and
F.1.11),
дх
2»-i(n- 1)! '
y, Hn(x)Hn(y)
*-( 2nn\
n=0
— = 2ry*
Эх
00
и=0
n=l
э#
2»-
(х)Яп+1
2"и!
:*)д,-1<
Thus
and by the symmetry in x and j
ал ол
=2гхК -г .
ду Эх
The last two equations imply that
1 ЪК _ 2ry - 2r2x
К Эх I — г2
and so
2rxy — r2x2
log К 1- g(y, r)
or
A" = h(y, r)e{2rxy~r x )/A-r \
Again by the symmetry in x and y, we can conclude that
K _ с(гче[2д:у/--(дг2+У2)г2]/A-г2)

282 6 Special Orthogonal Polynomials
To find c(r), set x = у = 0 to get
n=0
From F.1.8),
- and Я2п+1@)=0. F.1.14)
Therefore,
n=0
,
^
This proves F.1.13) once again.
Remark 6.1.1 An interesting formula for Fourier transforms can be formally
obtained from formula F.1.13). Multiply both sides of F.1.13) by Hn (y)e~y2 and
integrate over (—oo, oo) to get
e-y2+[2xyr-(x2+y2)r2]/(l~r2)
== Hn{y)dy = Hn(x)r".
The validity of this formula for \r\ < 1 can be proven. Let r —> i and we have, at
least formally,
L Г е1ХУе-У212Hn{y)dy = ine~x2/2Hn(x). F.1.15)
27Т J-oo
This equation embodies the self-reciprocity of Hermite polynomials. It gives
e-x /2 ^ ^ as an gjgenfunction of the Fourier transform with eigenvalue /". There
are various ways of proving F.1.15). See Exercise 11 for one method. Also see
Exercise 12, which reproduces de Bruin's [1967] proof of Heisenberg's inequality
using F.1.15).
6.2 Laguerre Polynomials
The Laguerre polynomials are orthogonal with respect to the gamma distribution
e~xxadx, where a > — 1. The definition of the Hermite polynomials and the proof
of their orthogonality given by F.1.5) suggest consideration of the polynomial
dxn

6.2 Laguerre Polynomials 283
An application of Leibniz's rule for derivatives shows that this expression is a
polynomial of degree n. The Laguerre polynomial L" (x) is defined by the formula
Lan(x):=^f~(e-xxn+a), forn>0. F.2.1)
It is easy to check that
KM = J1—rL 2J / Г „ = ^r^ i^i(-«;« +1; *)• F-2-2)
The orthogonality relation for the Laguerre polynomials is contained in
[°° a о a -г Г(а+и + 1)
/ Lm(x)Ln(x)x e ax = ; omn, a > —1. F.2.3)
Jo
The integral on the left is
dn
(e~xxn+a)La (x)dx
n\J0 dx"{ )Lml)
Suppose n > m and integrate by parts n times to see that its value is 0. For n = m,
first observe that, by F.2.2),
so that n integration by parts gives F.2.3) after an evaluation of the gamma integral
at the last step.
The generating-function formula for L" (x) is given by
(-n)kxk
h h
00 , 4t 00 • , -i ч и
(a+k+\)nrn
(-*#•)*
= A - r)-" exp(-xr/(l - r)). F.2.4)
Denote the generating function in F.2.4) by F(x,r). It is readily verified that
dF
A - r2)— + [x - A + a)(l - r)]F = 0.
or

284 6 Special Orthogonal Polynomials
This gives the three-term recurrence relation
(n + \)Lan+l{x) + (x-a-2n- \)Lan{x) + (n + a)Lan_,{x) = 0, F.2.5)
where n = 1, 2, 3,
Before deriving the differential equation for Lan{x), we obtain an interesting
formula for the derivative of Lan (x). The formula is
dLa(x)
xdhT = nL"(x) ~ (n+ff)L"-iw' for" - L F-2-6)
This arises from the identity
dF
—
aX
which implies
dL°{x) dLan_x(x)
dx dx
Eliminate Lan_x (x) from F.2.5) and F.2.7) to get
+ Lan_l(x)=0, forn>l. F.2.7)
+0, + I)
dx dx
for n > 0. Replace n by n — 1 in this equation and eliminate (d/dx)L"_l(x) by
means of F.2.7) to get F.2.6), the required result.
Now differentiate F.2.6) and then apply F.2.6) and F.2.7) to arrive at
d2La(x) dLa(x)
x /У+(a+l~x)—^-+nLan(x) = 0, for«>0, F.2.8)
dx1 dx
Thus и = L" (x) satisfies the second-order linear differential equation
xu" + (a + 1 - x)u' + nu=0. F.2.9)
Because the normal integral is a particular case of the gamma integral, it should
be possible to express Hermite polynomials in terms of Laguerre polynomials.
Such a relationship exists and is given by
H2m(x) = (~l)m22mrn\L-l/2(x2) F.2.10)
and
H2m+l(x) = (-l)m22m+]mlxL]J2(x2). F.2.11)

6.2 Laguerre Polynomials 285
To prove that #2m(x) = CL~1/2(x2) for some constant C, it is sufficient to show
that, for any polynomial q(x) of degree < 2m — 1,
/•0
/
J—
A general polynomial is the sum of an even and an odd polynomial. When q is odd,
the integral is obviously zero. When q is even, it can be written as q(x) — r(x2),
where л is a polynomial of degree < m — 1. Then for у = x2, the above integral
becomes
/•O
/
Jo
) F.2.12)
a ) J
/о
by the orthogonality of L~1/2(y). The value of С can be found by setting x = 0.
Relation F.2.11) can be proved in the same way, or by differentiating F.2.10).
There is another way in which the normal integral is related to the gamma
integral. The normal integral is a limit of the gamma integral. This gives another
connection between Laguerre and Hermite polynomials.
By Stirling's formula,
r°° /r\a Иг r°°
/ (?) e-u-«) " _^^г= / e-**dx asa^oo.
Jo \a) V2^ i-oo
A change of variables x = a + t/*/2a gives
/•oo
У e
The orthogonality relation for L"(x) implies
Г° а 2 a -x Г(
Jo L"(X)X € dX = ~ n\
Setx = a + \p2au. Then, by Stirling's formula,
\ „--Jbiu
lim (-) L«(a + V2^x) = (-1)"^^. F.2.14)
^o \a / n\
n!
A comparison of F.2.12) and F.2.13) suggests that

286 6 Special Orthogonal Polynomials
This may be verified by using the generating functions for the Laguerre and Hermite
polynomials and is left to the reader. Recurrence relations or the definitions F.2.1)
and F.1.3) can also be used to give easy derivations.
We started the treatment of Hermite polynomials by expressing e~x as an
integral. A similar approach can be taken for Laguerre polynomials. It follows
from D.11.25) that
e x"
*n+" = /
Jo
By D.6.1), this leads to the integral representation of Lan{x) given by
Lan(x) = — / t"+0l/2JaB*Jxt)e-'dt, F.2.15)
n\ Jo
fora > — 1.
The reader should verify that the generating function for Laguerre polynomials
follows easily from this. Also, since
/2 /2
Jm(x) = \ —sinx and J-m(x) = \—cosx,
' V ТГХ ' V ПХ
<2„х poo
L-l/2(x) = —— / e-'2t2ncosBVx~t)dt
one gets
and
e-t {2n+l sjj
Compare these with F.1.4) to get alternative proofs of F.2.10) and F.2.11).
We note a few elementary formulas that are useful:
—L"(x) = -L"+|(x), F.2.16)
dx n
?[x«^(x)]=(n+a)x«-1L«-1W, - F.2.17)
^ [e-xLan{x)} = -e-xLan+l{x), F.2.18)
= (n + X)x<*-le-xLan-\(x). F.2.19)

6.2 Laguerre Polynomials 287
We now prove F.2.19); the others can be proved in a similar way. By F.2.2)
and Kummer's transformation D.1.11), we have
dxl J n\ dx
)nd F a +
dx
и! dx
(а + 1)„ ^ (л + а + !)*(-!)*(* +«)**4в-1
a (a + 1)» „_,
±1ха-1е-, F (_„ _ ! }
n\
Formulas F.2.17) and F.2.18) can be written as integrals and then extended as
fractional integrals. Formula F.2.17) extends to
fx-tr-^-\F,{a;b;t)dt, F.2.20)
o
for Re д > 1. (See B.2.4).) Write this as
when ft > a. Formula F.2.18) extends in a slightly different way. The extension
is given by
1 f°°
/ (t -
1 f
e~xLan(x) = — / (t - xf-^e-'L^dt, F.2.22)
when ft > a. We give a proof below that uses F.2.21) and the orthogonality and
the completeness of Lan (x). A proof of completeness is in Section 6.5. Observe that
/•OO
/
Jo
- г,Лг<1^'+1) Г *«г[
F.2.23)

288 6 Special Orthogonal Polynomials
The orthogonality relation F.2.3) applied to F.2.23) implies
By F.2.3), F.2.23), and F.2.24)
for n = 0,1, Now the completeness of L" (t) gives F.2.22).
The formula for the Poisson kernel for Laguerre polynomials is given by
F.2.25)
when \r\ < 1, a > —1, and /„ is the modified Bessel function of order a. A
simple proof of F.2.25) is obtained by using the generating function F.2.4) and
the derivative formula F.2.19). Write the left side of F.2.25) as
k{x)r". F.2.26)
To find a closed form for the inner sum, start with the generating-function
formula
OO
^L«+*(x)r" = е~хг/а-г)/A - r)a+k+l.
я=0
Multiply both sides by xk+ae~x, take the &th derivative, and apply F.2.19). The
result is
я=0
1 dk
Г k+a -x/d-г
i i \n dk rn+k+a
( l> dxkX
A -r)nTl\
+ « +1;« +1; -*/(i - »•))• F-2.27)

6.2 Laguerre Polynomials 289
Apply Kummer's transformation D.1.11) to the 1F1 to see that the expression
F.2.27) is equal to
k\Lak(x/(l-r))e-x/(l-r\
Use this for the inner sum in F.2.26) to get
n\Lan{x)Lan{y)r"
00
E
n=0
The sum in the last expression can be written as
i-r
~Jxyr
r)).
The last equation follows from the series expansion for the modified Bessel
function Ia(x). This proves F.2.25).
We can obtain the Hankel transform and its inverse from F.2.25). The argument
given here can be made rigorous. See Wiener [1933, pp. 64-70] where the Fourier
inversion formula is derived from the Poisson kernel for Hermite polynomials.
Let
Then F.2.25) can be written as
H(x,y,t) := —
1 -t
F.2.29)
я=0

290 6 Special Orthogonal Polynomials
Let /(x) be a sufficiently smooth function that dies away at infinity. Then /(x)
has the Fourier-Laguerre expansion
я=0
Multiply F.2.29) by /(y) and integrate to get
/oo °°
/(у)Я(х,у,г)^ = У>„(*
r.
/л
я=0
Let t —>¦ e~ni to arrive at
/oo
f{y)emcIa{y/xye-*i'2)dy =
я=0
=: g(x). F.2.30)
Thus,
/•oo
«W=/ f(y)Ja{y/xJ)dy. F.2.31)
Jo
Now g(x) has a Fourier-Laguerre expansion that, by the definition of g(x), implies
that
poo /*oo
/ g(x№n(x)dx = (-1)" / f(x)irn(x)dx.
Jo Jo
By a derivation similar to that of F.2.31), we have
/ g{x)Ja(y/xy)dx = y2(-l)n
Jo n=0 Jo
n=0
n=0
= /(x). F.2.32)
We can write F.2.31) and F.2.32) as the Hankel pair
/•OO
/ f(y2)JAxy)ydy = g(x2),
J° F.2.33)
/oo v '
g(x2)Ja(xy)xdx = f(y2).
This may be the place to point out that F.2.21) contains Sonine's first integral
D.11.11) as a limiting case. This follows from the fact that
lim n-aLan(x/n) = x-ff/2yaBVx). F.2.34)

6.2 Laguerre Polynomials 291
If in the Hankel inverse of Sonine's first integral D.11.12), we change x to x/t
and, in the formula thus obtained, change t to \/s, we get
/O
1
when Л > a > — 1. The Hankel inverse of this is
Г(Л-
for д < Л < 2д + 3/2. Write it as
1 ' -Pi,„2 iyL-M-1,
= — /
Г(Л - /л) Jx
for-1 < /л <Л <2д+3/2. This is theanalog of F.2.22).Two analogs ofSonine's
second integral are
+ y) = Y^Lak(x)Lfin_k{y) F.2.35)
and
L«+n@) Г (a - /3)
F.2.36)
Formula F.2.36) is due to Feldheim [1943]. Formula F.2.35) is an immediate
consequence of the generating function F.2.4), and F.2.36) is proven by using
the series representations of Laguerre polynomials, the value of the beta integral,
and the Chu-Vandermonde sum. When у = 0, F.2.35) is equivalent to
= у Г(п-к + р-а)
This is an easy consequence of the generating-function formula F.2.4). The details
are given in Section 7.1. This formula is equivalent to
/ Li{x)Lak{x)xae-xdx
Jo
t~"~uil F.2.38)

292 6 Special Orthogonal Polynomials
This can be used to show that the Fourier-Laguerre expansion of xa~^Lf(x) in
terms of L& (x) is given by the formula
xae-xLak{x)
+ 1) „ p _x
F.2.39)
for a > (P — 1 )/2. To understand this condition on a and ft, needed for convergence,
see Theorem 6.5.3 and the remark after that. Note that F.2.39) is the inversion
of F.2.37). Just as the Sonine integral and its inversion can be used to solve dual
integral equations, F.2.37) and F.2.39) help to solve a dual sequence equation
involving Laguerre polynomials.
Theorem 6.2.1 Let a, X, с be given, such that с > (Л — 2a — l)/2, а, Л > —1.
Then ifan, bn are given (and are small enough) and if
pec
an= xcf(x)Lan(x)xae-xdx, n=O,l,...,N,
Jo
/•OO
bn= f(x)L\{x)xxe-xdx, n = N + 1, N + 2,...,
Jo
and iffi — a + c, then
^ T{n + \)
Proof. By F.2.37), we have
¦a) f°
1 - a) Jo
However, by F.2.39) we have
Г (A: - n + Л - Р)Г(п + ft + 1)Г(к + 1)
Г (к - п + 1)Г(А, - Р)Г(п + 1)Г(к + k + l) k
Jo
Now use the Fourier-Laguerre expansion of f(x) to get F.2.40). This makes more
than merely formal sense if the bn s are small enough. This proves the theorem. ¦

6.3 Jacobi Polynomials and Gram Determinants
293
Similarly, F.2.21) and F.2.22) can be used to solve dual-series equations in-
involving Laguerre polynomials.
Theorem 6.2.2 Let a > 8 > — 1, and let a < min(<5 + 1, 28 + 1). Suppose that
/(*) =
and
Then
n\
an =
Г(п+8
g(x) =
n=0
-а
x < у < oo.
a(x - t)s-adt]Lsn(x)e-*dx
J
/OO 1 Г /-0O -I
/ g(t)e-'(t-xr-s-4t\Lsn(x)xsdx
Г (а -8) \_JX J
6.3 Jacobi Polynomials and Gram Determinants
If
rb
Cn= Xn
J a
da{x)
are the moments with respect to a given distribution da(x), then the polynomials
Pn(x) =
CO
Cl
Cn-1
1
Cl
C2
Cn
X
C3
Cn+1
X2
с„
• ¦ • cn+i
¦ • • c2n-i
x"
F.3.1)
are orthogonal with respect to da(x). It appears at first sight that it would be difficult
to obtain a more useful representation of the polynomials from the determinant.
Wilson [1978, 1991] showed that, in many interesting cases, it is possible and
quite easy to get a hypergeometric representation of pn (x) from the determinant.
In this section, we give his derivation of the hypergeometric form of the Jacobi
polynomials.
We start with the following lemma, which generalizes the result contained in
F.3.1).

294
6 Special Orthogonal Polynomials
Lemma 6.3.1 Suppose {</>n}o° is a sequence of independent functions. The se-
sequence of functions {pn(x)\ given by
pn(x) = Cn
AH.o
Дл-1,0 Дл-1,1 ¦¦¦ Дл-1,л
F.3.2)
where
,-,_,- = / (j>i(x)(l>j(x)da(x),
Ja
k=0
and where Cn is a constant, satisfies the relation
/ pn(x№m(x)da(x) = 0, form<n.
Ja
(Here a(x) need not be a positive measure but all the integrals are assumed to
exist.)
Proof. Expand the determinant F.3.2) as
pn(x) = Cn
Then
fb "
F.3.3)
If от < n — 1, then the right-hand side of F.3.3) represents a determinant in which
two rows are identical; the value of such a determinant is zero. This proves the
lemma. ¦
Corollary 6.3.2 Suppose {</>n}o° and {^п}о° are two sequences. Define pn(x) as
in F.3.2) with
rb
jda(x).
lU — / &Ф]С
Ja
Then
f"
/ Pn
Ja
(x)^m(x)da(x) = 0, form <n — 1.
We now specialize to the situation where the weight function is a'(x) = A — x)a
A + x)P over the interval (—1, 1). Our aim is to find pn(x). For this we choose

6.3 Jacobi Polynomials and Gram Determinants
295
= A— x)k andi/aW = A +x)k. Other choices of ф^ and ^ as polynomials
of exact degree к would work, but these make the calculations simpler. In this case
l
(
f"\Cl
V{a
F.3.4)
Theorem 6.3.3 The polynomial F.3.2) with /x,j given by F.3.4) is a constant
multiple of the Jacobi polynomial
n\
-n,n +a + ft + 1 1-х
a + l ' 2
F.3.5)
Proof. From each row, j = 0,..., и — 1, pull out the first factor in F.3.4) and
absorb it in the constant Cn. We will continue to call it С„, so it may differ
from one line to the next. Then factor out 2J (a + 1)_,- from each of the columns,
j = 0,1,... ,n. The resulting determinant is
Cn
Мод
Ai.o Ai.i
An-1,0 An-1,1
MO.n
Аи-1,и
where
and <j>j(x) =
a-xy
2J(\+a)j'
Now expand about the last row to obtain the expression
1 -x
F.3.6)
Set A = a + p + 2. The problem now is to compute the determinant
1
A(A,n,k) =det -
, A(A,0,0) = l.
Observe that the first column in this determinant consists of Is when к ф 0. We
shall consider this case first. Subtract row n — 1 from row n, then row и — 2 from
row и — 1, and so on. This makes the first column zero except for the first entry,

296 6 Special Orthogonal Polynomials
which is one. Expand the determinant along the first column to get
Д(А,иД) = det
But since
1
1
1
(A + i +
i (A + i)j+i (A + i)j+2
-O' + D = -I/ + 1)
(A + i)j+2 (A + /)(A + i + 1)(A + i + 2),'
[(A + j) - (A + i + j + 1)]
then
A(A,n,k) = det
-0" + 1)
(A + i)(A + i + 1)(A + i + 2)j J o<.-<n-2
0<j<n-l
Now —(j + 1) is a common factor in the yth column and 1/f (A + /)(A + i + 1)] is
common in the /th row and so these factors can be taken outside the determinant.
Thus
A(A,n,k) =
n-\
c-d-1 По-
y=0
ln-2
i=0
¦2,и-1Д-2)
"\\ "I, А(А + 2,п-1Д-1).
Repeat this process /c times to get
А(А,иД)
k-l
s=0
(A
2ik, и-ik,0). F.3.7)
Recall that we had assumed к ф 0. The above equations holds trivially for к = О,
provided we assume that the product in the braces is one. In the determinant
A (A + 2k, n — k, 0), the column index j is not 0 so that j goes from 1 to n — k.
Therefore,
=det
= det
1
(A + 2/c + /)(A + 2k + i
A(A + 2k+ l,n-k,n-k)
(A + 2k)n-k '

6.4 Generating Functions for Jacobi Polynomials 297
Combine this with F.3.7). The result after some rearrangement is
'"'
k\(n-k)\
The product depends only on n so it can be absorbed in С„. Substitute this value
of the determinant in F.3.6) to arrive at
« + Dt k\{n-k)\
Choose Cn = (a + 1)„. This gives the hypergeometric representation of P^a'^(x)
and the theorem is proved. ¦
An immediate consequence of the hypergeometric representation F.3.5) of
Jacobi polynomials is the formula for the derivative:
{P^(X)} (л + a + P + 1)Р(х). F.3.8)
We have seen in Chapter 3 that iFiia, b\ c; x) is a solution of
d2y dy
x(l - x)-l + {c - (a + b + \)x}-f- - aby = 0.
dxz dx
Therefore, и = Р^а^\х) is a solution of the differential equation
A - x2)u" + {/3-a-(a + l3+ 2)x\u' + n(n + a + fi + \)u = 0. F.3.9)
Another straightforward result from F.3.5) is that the coefficient of xn in
is
2"n\
F.3.10)
6.4 Generating Functions for Jacobi Polynomials
Generating functions are of great importance in the theory of orthogonal polynomi-
polynomials. We have already used them to study Hermite and Laguerre polynomials. In this
chapter, we derive Jacobi's form of the generating function by two methods. One is
due to Jacobi and uses Lagrange inversion; the other is due to Hermite. A different
generating function, particularly useful in studying ultraspherical polynomials, is
also derived.
One way of finding the generating function for Jacobi polynomials is to use
Lagrange inversion. We use the following lemma. Its derivation and some appli-
applications are given in Appendix E.

298 6 Special Orthogonal Polynomials
Lemma 6.4.1 Suppose that ф (у) is analytic in a neighborhood ofy = x,
oo
r — = У^an(y - x)" with ax ф 0, F.4.1)
п=1
and f is analytic in a neighborhood of y = x. Then f(y) can be expanded in
powers of r:
00 r" d"~l
fiy) = fix) + Х^^Т(ЛХ)«КХ))Я)- F.4.2)
n=l
Theorem 6.4.2 The generating function for the Jacobi polynomials P^a^\x) is
given by
F(x, r) = 2a+fiR-l(l -r + R)-a(l +r + R)-0, F.4.3)
when
tf = (l -2xr + r2)l/2.
Proof. Take ф(у) = (у2 - l)/2 in F.4.1). Then
1 A-2ХГ + Г2I/2 1 R
y = = .
r r r r
The derivative of F.4.2) with respect to x is
For f'{x) — A - x)a{\ + x)P this becomes
R
00 rn d"
+ Е-7УТ«1 -^d +^(^2- D72").
^—' и! ax"
n=l
Use Rodrigues's formula B.5.13'),
A -x)a{\ +xfP^\x) - (-^-?-{(l -x)n+a(\ +x)"+P},
to arrive at
A short calculation shows that
\-y 2
and
1-х \~r + R l+x \+r + R
This completes the proof of the theorem. (See Jacobi [1859].) ¦

6.4 Generating Functions for Jacobi Polynomials 299
If one were to take the definition of Pjf •^) (x) as the polynomials that satisfy
F(x,r) = 2a+liR-l{\-r + R)~a(\ + r
, F-4-4)
then one would face the problem of proving orthogonality. When a = fi = 0, we
have the Legendre polynomials Р„(х). For these polynomials Legendre observed
that
1
л/rs I — *Jrs
-L
dx
- 2xr + r2v 1 - 2xs + s2
Pn(x)Pm(x)rnsmdx.
This implies orthogonality. Chebyshev [1870] applied the same method to the
general Jacobi polynomials. His proof is a tour de force and shows his skill in
handling formulas. A simpler proof was given by Hermite [1890]. He treated the
Legendre case, but his argument extends in an obvious way to Jacobi polyno-
polynomials.
Theorem 6.4.3 If{P^a^}(x)} is given by F.4.4), then the sequence is orthogonal
with respect to the weight function A — x)a{\ + x)& and conversely.
Proof. Consider the integral
/•l
/m = / xmF(x,r){\ -x)a{\+xfdx
" / xmP^\x)(\-x)a(\+xfdx.
n=0
Set
or
A -2xr +r2I/2= 1 -ry
Then
У +
r(l-y2)

300 6 Special Orthogonal Polynomials
Clearly, Im is a polynomial in r of degree m. So we must have
/•i
/ хтР^\х)(\-х)а{\+х)Ых = 0, п = т + 1,т + 2,
This is equivalent to
-x)a(\ +xfdx = 0, тфп.
-l
For the converse, consider lm again. The same change of variables gives
-r + l-/
This is clearly a polynomial of degree m in r if
F(x, r) = C(l - r + 1 - r>)-a(l + r + 1 - ry)^(l - ry)~\
where С is a constant. To find C, take x = 1 to get
n=0 n=0
= A - r)-"'1 = C2-a~P{\ - r)-a~\
огС = 2а+Р. m
Remark 6.4.1 The last theorem gives another way of finding the polynomials
orthogonal with respect to the beta distribution A — x)a(\ +x)$dx. These polyno-
polynomials must have F(x, r) in F.4.4) as their generating function. Then by Lagrange
inversion the polynomials must be
-x)~a{\ +x)-^{(l -x)n+a(l
The ultraspherical polynomials are the important subclass of the Jacobi poly-
polynomials р(а'Р)(х), when a = /J. These are defined by a different normalization
than P(a'a\x), giving a simpler generating function than F.4.4) when a = fi.
The new generating function is useful in obtaining properties of ultraspherical
polynomials. To motivate this generating function consider the Poisson kernel
oo
? P^p\x)P^0)(y)rk/hk (a, ft > -1), F.4.5)
k=0

6.4 Generating Functions for Jacobi Polynomials 301
where
= I [P^\x)f{\ - x)aA + xfdx
1)Г(к
Bk + a + p + 1)Г(к + a + p + 1)Г(к + 1)
Take у = 1 in F.4.5) to get the sum
00 ">k +a + в + l)(a + В + I)
Observe that the last series, except for a factor, is obtained from
by differentiation. Rewrite this series with the hypergeometric expression for
P%"'P)(.x) in powers of A + x)/2. We have
(n-k)\k\(l+p)k
_ v^ A +<* + P),
/1
(A + « + /3)/2)t(B + « + P)/2)k2k(x + \frk
We used the binomial theorem in this computation, along with the fact that
(aJk = 22k(a/2)k((a
The final result is
+ 1}«
F.4.7)

302 6 Special Orthogonal Polynomials
This is the other generating function we were looking for. When a = /3, this gives
Ba + l)nP^a\x)rn ^ 1 / 2(l+x)r
A+J
r2y°'-L2. F.4.8)
It is reasonable to define polynomials
C«W := а 12поъ pWW-^Oc) F.4-9)
with the generating function
oo
A -2xr + г2Гх = ]Pc*(Jt)r". F.4.10)
n=0
When к = 0, C\ (x) = 0. So the case к = 0 has to be considered as к -*¦ 0 in
the formulas involving C%(x). The polynomials C^(x) are called ultraspherical
polynomials or Gegenbauer polynomials. They are orthogonal with respect to the
weight function A - х2)л"A/2) when к > -A/2).
An important expression for С„(х) follows immediately from the generating
function F.4.10). Let x = cos в, factor the left-hand side using
1 -2rcos0+r2 = A ~rem)(\ -re~w),
expand by the binomial theorem, and equate the coefficients of r". The result is
The second equation is true because С„ (cos 0) is real and the real part of e'("
is cos(n - 2k)9. When к > 0, F.4.11) implies
Another hypergeometric representation for C*(x) is obtained by taking a different
factorization:
1+r2
(A.)* Bxr)k
k\ (l+r2)*+*

6.4 Generating Functions for Jacobi Polynomials 303
ifc! и!
n,k
n,k
oo
Thus,
F.4.12)
Even though ultraspherical polynomials are special cases of Jacobi polynomials,
their great importance compels us to note and prove some of their properties. When
к —> 0, we get the Chebyshev polynomials of the first kind:
1, n=0,
F.4.13)
2Г„(х), и = 1,2,...,
Ck(x\
When к ->¦ oo, we have
The Rodrigues formula B.5.13') takes the form
к^-'г)^"г- FА14)
and the formulas for the derivative and the three-term recurrence relation are
—— Cn(x) = 2kCn^i (x) F.4.15)
and
n&n{x) = 2(n+k- \)xCxn_y{x) - (л + 2k - 2)Ckn_2(x) F.4.16)
for и > 2 and C#(jt) = 1, C\(x) = 2kx.
It should also be noted that uF) — (sin#)xC^(cos#) satisfies the differential
equation
= 0. F.4.17)
sin

304 6 Special Orthogonal Polynomials
A proof of F.4.15) from the generating function for С„{х) is as follows: The
derivative of F.4.10) with respect to x gives
oo .
У —Cx(x)rn = 2Xr(l - 2xr + r2y^+»
^—' ax
n=0
n=0
and hence F.4.15). To get F.4.16) take the derivative of F.4.10) with respect to
r. The result is
oo
2X(jc - r)(l - 2xr + r2Yk = A - 2xr + r2) ^2 пС^(х)г"'\
n=0
The left side is 2X(x - r) Y^Lo Cnix)r"- Relation F.4.16) follows on equating
the coefficients of rn on each side.
We now state a property of the relative extrema of ultraspherical polynomials.
Let y^l, к = 1,..., n — 1 denote the zeros of the derivative of Р^а'а)(х). Order
the zeros so that ук,„(а) < Ук-\Аа) and set Уо,п(а) = Ь Уп,п(а) = -1- Define
)A), * = 0,1,...,я. F.4.18)
These numbers satisfy the inequality
fiktn(a) </jLktll-i(a), a > -1/2, к = 1, 2,..., n - 1, и = 1,2,...,
F.4.19)
but the inequality is reversed for — 1 < a < —1/2. For a = —1/2,1лкп(а) = 1.
For а = 0, F.4.19) was observed by Todd [1950] after studying graphs of Legendre
polynomials. Todd's conjecture was proved by Szego [1950]. We prove F.4.19)
with a = 0 by an argument that generalizes to give F.4.19). It is left to the reader
to prove the general case.
We begin by stating some necessary results about Jacobi polynomials. The
following two identities follow directly from Remark 5.6.3:
(n + a + 1)Р„("'«(х) - (л
Bn+a + 6+2)(l -x)
F.4.20)
- (n + р)Р?Хш(х). F.4.21)
For the reader's convenience, we again note that
(r/'+1) F.4.22)

6.4 Generating Functions for Jacobi Polynomials 305
and
pW\-x) = (-l)nP^a)(x). F.4.23)
Following convention, we denote the Legendre polynomial Pn@0) (jc) by Pn (jc). By
F.4.20), F.4.23), and then F.4.21), we have
[Pn(x)]2 - [Pn+i(x)]2 = [Pn(x) - Рй+1(х)][Р„(х) + P
= (l-jc)PnA'O)(Jc)(l+Jc)Pn(O-1)(x)
2
2 (n
2(и
2(л
where the last step follows from F.4.22).
Thus
(I — x2) \ d ~[2
f + ' \
/(x) := [Рп(х)У + j—~^ \—Pn(x)
(и + IJ
The zeros of /'(•«) contain the zeros of P^(x) = Pn(i'11)(x)/2 as well as the zeros
of Pj,+i(x) = Р^Л)(х)/2. Since /(x) is of degree In, we must have
By comparing coefficients of highest powers, we see that
Because of the separation of the zeros of PnA1)(x) and P^ (x), they take opposite
signs for ykttl <x < Ук,п+1- Thus /(x) decreases in this interval and F.4.19) is
proved for a = 0.
It is also easy to prove that
й,»<й-1,- * = l,2,...,lii/2J, F.4.24)
where ixk n = /x,t,n@). Consider the function /(x) defined in Exercise 40 with
a = p = 0. Then
, A -x2)[P'(x)l2
/(x) = [Pn(x)]2 + П " П . F.4.25)
n(n + 1)

306 6 Special Orthogonal Polynomials
Compute f'(x) and check that /(x) is increasing for 0 < x < 1. This implies
F.4.24).
Szasz [1950] proved F.4.19). His argument also works for the case — 1 < a <
— 1/2, though he did not make this observation. Later he proved a similar result
for Hermite functions. See Szasz [1951]. Exercise 10 contains a statement of his
result.
We close this section with the observation that the Hermite and Laguerre poly-
polynomials are limits of Jacobi polynomials. There are a number of ways to obtain
these limits. One uses a generating function. Observe that
and we conclude that
lim X-"/2C^(x/X) = Hn{x)/n\. F.4.26)
A->0O
Another method is to use the hypergeometric representations
ш ц± ± (
/3->oo " - " ' ?->oo n\ ^ (a -
F.4.27)
This means that it is possible to derive the properties of Laguerre and Hermite
polynomials from those of Jacobi polynomials. However, it is usually easier to
deal with these polynomials directly as we did in Sections 6.1 and 6.2.
6.5 Completeness of Orthogonal Polynomials
The problem of expanding an arbitrary function in terms of orthogonal polynomials
was briefly considered in the previous chapter. Here we consider expansion by
means of Jacobi, Laguerre, and Hermite polynomials. The latter two are the most
interesting as they involve integration over infinite intervals. Just as in the case of
Fourier series, the result can be nicely stated for functions in L2(a, b), that is, the
Hilbert space of square integrable functions. We reproduce Hewitt's [ 1954] proof of
the completeness of the Hermite and Laguerre polynomials. This proof depends on
the uniqueness of Fourier transforms of integrable functions. For completeness,
we give a complex variables proof of the latter result due to Bak and Newman
[1982, p. 228].
Theorem 6.5.1 If f is integrable on (—oo, oo) and if
/•OO
fix) = / f(t)eix'dt = 0,
J —oo
then / = 0 almost everywhere.

6.5 Completeness of Orthogonal Polynomials 307
Proof. Clearly,
f f(t)eix('-a)dt = 0.
J — 00
So, if a is real, then
ra /*oo
/ f{t)eix{'-a)dt = - / f{t)eixi'-a)dt. F.5.1)
J — oo </a
Define two functions of z — x + ry:
L(z) = / f(t)e^'-a)dt, R(z) = - / f(t)eU('-a)dt.
J —oo J a
It is clear that L(z) exists for Im z < 0 and is analytic in Im z < 0. Similarly,
R(z) exists for Im z > 0 and is analytic in Im z > 0. Moreover, by the dominated
convergence theorem and F.5.1), we have
lim L(x + iy) = I f(t)eix('~a)dt = lim R(x + iy).
У-*0 J-oo У^°
This implies that
|
ltf(z), Imz>0,
is a bounded entire function. By Liouville's theorem, F(z) is a constant. Again,
by the dominated convergence theorem,
/•OO
lim F(iy) = lim - / f(t)e~y('~a)dt = 0.
¦y^OO J^OO _/a
Thus F(z) = 0 and, in particular, F@) = 0. This means that for all real a (since
a is arbitrary)
/ f(t)dt = O.
J-oo
This implies that / = 0 almost everywhere and the theorem is proved. ¦
Let p{t) denote a square integrable function that dies out exponentially at infi-
infinity, that is,
p(t) = 0(e~aW) for some a > 0 as \t \ ->¦ oo. F.5.2)
Theorem 6.5.2 Let -oo < a < b < oo. Let p(t) € L2(a, b), with p(t) different
from zero almost everywhere, and let p{t) satisfy F.5.2), if a = —oo orb — oo.
Iffe L2(a,b)and
/ tnf(t)p(t)dt = O forn = 0,1,2,...
Ja
then / = 0 almost everywhere.

308 6 Special Orthogonal Polynomials
Proof. Let z = x + iy and define
rb
F{z) = / elzlp(t)f(t)dt.
Ja
If —oo < a < b < oo, then F is an entire function; otherwise, F is analytic in
—a < у < a. Therefore,
F<">(Z) = i» f e^tnp(t)f(t)dt.
Ja
By hypothesis F(n)@) = 0 for n = 0, 1, 2,.... This implies F(z) s 0 in -a <
у < a. In particular,
fb ¦
F{x) = / eMp{t)f(t)dt = 0.
Ja
Since p(t)f(t) is integrable on (a, b), the uniqueness of the Fourier transform
gives p(t)f(t) = 0. Since p(t) is different from zero almost everywhere, f(t) = 0
almost everywhere. This proves the theorem. ¦
For the next theorem, let da(x) denote either xae~xdx or e~*2dx. In the former
case (a,b) = @, oo) and in the latter (a, b) = {—oo, oo).Let0n denote either the
nth Laguerre or Hermite polynomial, normalized so that
f" 2dax -1
Ja
For any function / such that
fb
/ f(xJda(x) < oo,
Ja
set
rb
c« = /
Ja
Theorem 6.5.3 Suppose sn — Y?k=o сьФк- Then
b oo
/b
n=0
and
fb
2
lim / [f(x)-sn(x)]2da(x) = 0.

6.5 Completeness of Orthogonal Polynomials 309
Proof. For n > m, it is clear that
m+l / m+1
Also,
implies that
k=0
\ck\z < f f(x)zda(x). F.5.4)
Ja
It follows from F.5.3) and F.5.4) that {.?„(.*)} is a Cauchy sequence in L2a(a, b).
(Here L2a is the set of all square integrable functions with respect to the measure
da.) There is therefore a function g e L2a such that
lim /
b
2
[g(x)-sn(x)fda(x)=0. F.5.5)
Now for n > k,
rb
1
Ja
g(xL>k(x)dot(x) - ck
= f [gW - sn(x)](t>k(x)da(x) < / [g(x)-sn(x)fda(x),
J a J a
by the Cauchy-Schwartz inequality and the fact that the norm of фк is 1. Let
n ->• oo to get
f"
ck
f"
=
J a
By Theorem 6.5.2, it follows that f = g almost everywhere. By F.5.5), we arrive
at
/b rb
s2n(x)da{x) = / f(xJda(x),
J a
so that
rb
\2,
2_j\Cn\ = I f(xYda{x).
n=0
This proves the theorem.

310 6 Special Orthogonal Polynomials
Remark In the last theorem, the series ^o° с"Фп converged to / in the L2 sense.
Pointwise convergence can be obtained, for example, by assuming that / is smooth
or piecewise smooth. In the latter case the series converges to ^[f(x + 0) +
f(x — 0)], when x is a point of discontinuity.
We can use Theorem 6.5.2 to prove the following result: Suppose {pn(x)} is a
sequence of polynomials orthogonal with respect to the weight function w(x) =
O(e~c^) on @, oo). Then the sequence {pn(x)} is complete.
We can prove this by taking another sequence {qn(x)} orthogonal with respect
to xw(x) on @, oo). Define a sequence {rn(x)} by
Г2п{х) = pn(x2),
f2n+i(x) = xqn(x ).
This sequence is orthogonal with respect to |jc|i«(jc2) on (—oo, oo). This weight
function satisfies the conditions for Theorem 6.5.2. Hence the sequence {rn (x)} is
complete, which implies the completeness of {r2n(x)} for even polynomials. Thus
the result is proved.
Compare this result with the comments after Corollary 5.7.6.
6.6 Asymptotic Behavior of P^ix) for Large n
Suppose f(r) = YlT=oa"r" *s an analytic function in a neighborhood of zero
with only a finite number of singularities on the circle of convergence. Suppose,
for convenience, that the radius of convergence is one. We would like to have
an estimate of an for large n. Let us see how an can be found approximately by
knowing the singularities of /. Assume that the singularities are poles that we take
to be of order one for simplicity. Let the singular part of / on the unit circle be
a\ ot2 oik
S= I-far + I-far +'"+ I-far'
These functions can be expanded by the binomial theorem, so we know the coeffi-
coefficient of r", which we call bn. Now / — S = ?(а„ — bn)r" has a larger radius of
convergence than /. This means that
(a, - bn)r" = o(l)
for some r > 1, or
an-bn= o{r~n).
Since bn is known, we have an estimate of an. This is the idea behind what is known
as Darboux's method for finding the asymptotic behavior of an. The application
to orthogonal polynomials is possible because the generating function has the nth

6.6 Asymptotic Behavior of p("-P> (x) for Large n 311
polynomial as a coefficient of rn. In this case, there is also a need to consider
algebraic singularities. Take as an example A — 2xr + r2)~1/2, the generating
function for the Legendre polynomials. With x = cos в, we have
fir) = A - 2xr + r2)-1'2 = A - rewr1/2il - re-wr1/2.
The singularities are at r = ew and r = e~'e. In the neighborhood of r = e'e, the
behavior of fir) is like
8 = A - е2>в.
In this case fir) — gir) = /г(г)A — re~w)l/2, where/г (r) is continuous at ew. So
f — g still has an algebraic singularity, but it is now continuous. Thus it is possible
to say something about an — bn. The precise result is contained in the theorem
below. Before stating it, we consider the case
If X > 1, then subtracting a term like g does not make f — g continuous. More
terms have to be subtracted. These can be determined by expanding A — re'e)~x
in powers of A — re~w) and vice versa. We have, about r = e'e,
f{r) - A -
If for an integer n, n — к > 0, we can take
Theorem 6.6.1 Let /(z) = X^o° anz" ^e anafytic in \z\ < r, r < oo, and have
a finite number of singularities on \z\ = r. Assume that giz) = ^o° bnz" is also
analytic in\z\ < r and that f — g is continuous on\z\ = r. Then an—bn=o(y~n)
as n —> oo.
Proof. By Cauchy's theorem and the hypothesis on / - g,
1 /* fGЛ — рGЛ 1 С
2ni Jk]=r zn+l 2nrn Jo
The Riemann-Lebesgue lemma for Fourier series implies that the last integral
tends to zero as n ->• 00. This proves the theorem. ¦

312 6 Special Orthogonal Polynomials
In fact, in the above theorem it is not necessary to assume continuity of / — g
on \z\ = r. The same conclusion can be obtained by assuming that / — g has a
finite number of singularities on \z\ =r and at each singularity Zj, say,
where <ry is a positive constant. For these refinements and further examples, see
Olver [1974, §8.9] or Szego [1975, §8.4].
The generating function for Pjf-^ix) is
-r + \/l-2rx+r2)-a(l +r + v/l-2rjc+r2)-/i(l - 2rx + r2y1/2.
Take x = cos в. The above function has singularities at r = е±1в. In the neighbor-
neighborhood of r = ew, the generating function behaves as
шГа-
= 2^A - ешГа-Ц1 +e
This implies that
Write
where
Therefore,
pic
\A\
\ri\
'¦«(c
A
2
:os0)
= 0
+
— I
l)/2
[2a
the
v<
conjugate] ¦
1
l-jcr+J(
A/2)
n\
n
"I =|/
Observe that the denominator of \A\2 is A — jc)"A + x)&, except for the factor
2"+^+1(l — x2I/2. This is exactly the weight function for Jacobi polynomials.
This is not a coincidence. To state the theorem in a form due to Nevai [1979, pp.
141-143], suppose
Pn+i(x) = (Anx + Bn)Pn{x) - CnPn_i(x), n = 0,1,...,
P_1=0, P0 = l, and An^AnCn > 0 forn = 1,2,....

6.7 Integral Representations of Jacobi Polynomials 313
Theorem 6.6.2 If the series
Bn , ( Cn+i \ 8 I
— + -— -- > converges,
i A.. \ A.. A.. , 1 / 7 I
n=0
then dxjf can be expressed in the form
Here ir'(x) is continuous and positive in (—8, 8), supp {\jr') = (—<5, 8), and i/fj (jc)
is a step function constant in (—<5, S). Furthermore, the limiting relation
2
holds almost everywhere in supp(di/r).
The singularity in the generating function for Laguerre polynomials is more
complex. This makes it difficult to apply Darboux's method. Fejer, however, has
shown how this can be done. See Szego [1975, §§8.2-8.3].
6.7 Integral Representations of Jacobi Polynomials
Integral representations for hypergeometric functions imply the existence of such
representations for Jacobi polynomials. A few important and useful integral repre-
representations are given in this section.
Recall Bateman's [1909] fractional integral formula:
) Г(с)Г(д) Jo
\ + ) Г()Г(д) J \ с )
where Re с > 0, Re /x > 0, and |jc| < 1, if the series is infinite. This formula is a
particular case of B.2.4) and we use it to prove the next theorem, which is called
the Dirichlet-Mehler formula.
Theorem 6.7.1 For 0 < в < л, the nth Legendre polynomial is given by
2 f cos(n + iH
_ 2 Г
sin(n + \)ф
BCOS0-2COS0I/2
Proof Takea = -n,b = n + \,c = /x = 1/2, д: = sin2 (в /2), and t = sin2@/2)
in Bateman's formula. This gives
Р„(СО5в)
_ 1 fe
^ Jo
(sin2@/2) - sin2@/2))!/2

314 6 Special Orthogonal Polynomials
The 2 F\ in the integral is the hypergeometric form of the fourth Chebyshev poly-
polynomial given by
cos(n + i)</>
cos(</>/2)
To get the other form of the integral change в to n — в and ф to л — ф. Then use
the fact that Pn{-x) = {-\)nPn{x). ¦
One way to use this theorem is to show, as Fejer did, that the sum of Legendre
polynomials, X^=o Pk(x), is greater than or equal to 0 for 0 < x < 1, since
sin(w + l/2H
; - •
PtA) s
For the reference to Fejer and other related results, see Askey [1975, Lecture 3].
Theorem 6.7.2 For yc > 0, -1 < x < 1,
(c)
(d)
Г(а + 1)Г(д) Л A + у)п+а+1Л+1
х{у-хУ~хс1у, a>-\;
Г* A+
/_! A - у)«
> -1.
Proof. To obtain (a) use the hypergeometric representation of the Jacobi poly-
polynomials and apply Bateman's formula with an appropriate change of variables,
(b) now follows from (a). Apply Р^^{-х) = (-l)nP^'a)(x).

6.7 Integral Representations of Jacobi Polynomials 315
Finally, (c) and (d) are derived from (a) and (b) respectively by an application
of the Pfaff transformation:
Note that when this is applied to Bateman's formula we get
The theorem is proved. We also note that Bateman's formula and the results in the
theorem are all particular cases of Theorem 2.9.1. ¦
An important integral of Feldheim [1963] and Vilenkin [1958] can be obtained
from (c) by using the quadratic transformation
and
and taking /3 = ±1/2. The result is
Cvn(cos9) sin2"-1 в
Cvn{\)
2r(v + i) Гв 2, [cosV-cos2^-^1 C^(cos0)
Ф'
7Г 1
< -, V > > --.
A change of variables gives
Corollary 6.7.3 Forv > k> -1/2, 0 < в < ж
CvJcos0) 2f(v + i) fn'2 ,. , ,. , 99
-!L^ = —, ^-—^ / sin21 ^cos2"-2^1 ф[\ - sin2 в cos2 ф]п
Cvn{\) T(k + \)T{v-k) Jo
C^(cos6»(l -sin2 в cos2 </>Г1/2)

316 6 Special Orthogonal Polynomials
Here
Cn°(cos0) C*(cos0)
2= lim —^ = cosn6>.
с*A)
The final theorem of this section is the Laplace integral representation for ultra-
spherical polynomials. It is due to Gegenbauer [1875].
Theorem 6.7.4 For X > 0,
C*(cos 6>) = 22X_]_,/W,442 / [cos в + i sin в cos ф]п sir"
Proof. Recall that
Rewrite this using the beta integral. Then
k=0
Set у = sin2 i/f to get
2Г(п + 2к) Г'2
/
i,w,442 / fc°s 0 + i sin 0(cos2 ^ - sin2
n-\\ W) Jo
x sin21 f cos21 fdf.
Now let ф = 2i/f to get the result in the theorem. ¦
6.8 Linearization of Products of Orthogonal Polynomials
The addition theorem for cosines implies the formula
cos тв cos пв = \ cos(n + т)в + \ cos(n — т)в.
In the previous chapter, we noted that this result pertains to Chebyshev polynomi-
polynomials of the first kind, /)(~1/2'~1/2)(x), where x = cos#. This is called a linearization
formula because it gives a product of two polynomials as a linear combination of

6.8 Linearization of Products of Orthogonal Polynomials 317
other polynomials of the same kind. More generally, given a sequence of polyno-
polynomials {pn{x)} one would like to know something about the coefficients a{k, m, n)
in
m+n
Pm(x)pn(x) = ^2a(k,m,n)pk(x). F.8.1)
/t=0
If the pn(x) are orthogonal with respect to a distribution da(x), then
1 r
a{k,m,ri) = — pm(x)pn(x)pk(x)da(x). F.8.2)
"k Ji
Thus the problem of the evaluation of the integral of the product of three orthogonal
polynomials of the same kind is equivalent to the linearization problem.
As another example of a linerization formula, recall the identity
sin(n + 1N» sin(w + 1N» m^)
sin# sin 6* -^ sin 6*
k=0
This comes from the addition formula for sines. The addition formula is contained
in an important special case of F.8.1), that is,
m n m+n
Л л — А
whenx = e'°.
One way of obtaining linearization formulas would be to look for those poly-
polynomials for which the integral F.8.2) can be computed. A simpler integral would
involve only the product of two polynomials; but this would yield the orthogonality
relation. As we have seen, using the generating function is one way of obtaining
orthogonality in some cases. The simplest generating function is for a Hermite
polynomial, since it involves only the exponential function, which can be multi-
multiplied by itself without resulting in something very complicated. For example, to
get orthogonality, note that
~ dx
_oo-'—' mini
00 m,n
roo
= 1 e2"-r2+2*°-
J — oo
[¦OO
J — CO
S2 2
x dx
-oo
V
= V^r2"
Therefore,
2
Hm(x)Hn(x)e-x dx =

318 6 Special Orthogonal Polynomials
Similarly, to find the integral of the product of three Hermite polynomials,
consider
t\m\n\
/•CO
_ / 2xr-r2+2xs-s2+2xt-t
J — oo
-'2-*2dx
COO
e-(x-r-s-tJ
<ya+b+c ~a+b fb+c „a+c
u а\Ъ\с\ '
Й,О,С
This shows that
He(x)Hm(x)Hn(x)e~x2dx = (e+~_n, (m+n-e'\]~G+i-m\ ' F-8-3)
when ?+m + n is even and the sum of any two of ?, m, n is not less than the third.
In all other cases the integral is zero.
Theorem 6.8.1
min(m,rt)
Hm(x)Hn(x)= J2
Proof. This follows from the integral formula F.8.3) and the orthogonality rela-
relation for Hermite polynomials. ¦
Remark 6.8.1 An important feature of the coefficients in the linearization formula
is their positivity. This property is shared by the integral F.8.3).
At this point we have linearization formulas for Р^/2л/2)(х), .Pn(~1/2~1/2H0,
and the Hermite polynomials. Symmetric Jacobi polynomials are ultraspherical or
Gegenbauer polynomials and the Hermite polynomials are limiting cases, since
= lim
This suggests the possibility of the existence of a linearization formula for the
Gegenbauer polynomials. Of course, there are linearization formulas for f^1/2.-*/2)
x(x) and pt~l/2'l/2){x), which are essentially the Chebyshev polynomials of the
third and fourth kind, but these are related to P^I2'll2\x) and Р^^~^2){х) by
quadratic transformations.
The idea of using the generating function A — 2xr+r2)~x to obtain a lineariza-
linearization formula for Gegenbauer polynomials does not appear very hopeful, because

6.8 Linearization of Products of Orthogonal Polynomials 319
the product of three of these does not seem tractable. One useful idea, due to Bailey
[1933], is to express Cx(x) as a hypergeometric series and then use the Whipple
transformation for a very well poised yFe to handle the product C^(x)Cx(x). The
resulting linearization formula is due to Dougall [1919]. It is interesting to note
that a more general result was known to Rogers [1895]. See formula A0.11.10).
Theorem 6.8.2
min(m,n)
Ckm(x)Cx(x) = J2 a(k,m,n)Cxm+n_2k{x), F.8.4)
*=o
when
(m + n + X- 2k){X)k{X)m_k(X)n-k{2X)m+n-k
a(k, от, n) =
(от + n + X - k)k\(m - k)\{n - k)\{X)m+n.k
(m+n-2k)\
BX)m+n-2k
Proof. Recall that
^() F.8.5)
n\ \\ -n - X
Apply Euler's transformation (Theorem 2.2.5) to get
~ *, 1 " 2A. - Л!
e(ley2FA
от! \ \—X — m
Multiply the two equations above to obtain
. F.8.6)
\X)m(X)n ^i(jn+nH ^ -2i0\l
m\n\
(-n)k(X)k 2Ш ^ A-Я)Д1 -2X-m)j 2jW
\ ^ !AЯ) '
f^ A -X-n)kk\
When s = j + k, the two sums can be rewritten as
[-2k- m)s _2sW ^ (~n)k(X)k(m +X-s)k
(l-X-m)s в f^k\(l-X-n)k(X-s)kBX
E(l -A),(l - lk-m)s
j>0 (\-X-m)s 4 3Vl -X-n,X-s,2X
-n,X,-s,m+X-s ^\ _2sW
F.8.8)

320 6 Special Orthogonal Polynomials
This 4F3 is balanced. Recall Whipple's formula (Theorem 2.4.5), which transforms
a balanced 4F$ to a very well poised jF^:
a + l-b-c,d,e,-s \ (a + 1 - d)s(a + 1 - e)s
l-c,d+e-a-s' ) (a + l)s(a + 1 - d - e)s
a,l+a/2,b,c,d,e,-s
Л
a/2, a + \ -b,a + l -c,a + l-d,a + l -e, a + l+s' )'
when s — 0, 1, 2, Take a — —k—m—n,b = -m, с = l—2X — m — n + s,
d — k, and e = —n. Then F.8.8) is transformed to
- 2k - m - n)se
~1M
s\{l-k-m-n)s
{-k-m-n)k{\-{k + m+n)/2)k{-m)k
f k\(-(k+m+n)/2)k(l-k-n)k
A-2А.-Ш-И+ s)k(k)k(-n)k(-s)k
(k — s)k(l —2k — m — n)k{\ — к — m)k{\ — к — m — n + s)k
Reverse the order of summation, set s = к + ?, and simplify. Put this in F.8.7) and
use F.8.6) for the inner sum to get an identity that reduces to Dougall's identity
F.8.4). ¦
The limit к —> 0 gives the identity for cos тв cos пв and к = 1 is the identity
for sin(w + 1)9 sin(n + 1)9. When к -> оо, Dougall's identity reduces to
The next corollary was given by Ferrers [1877] and Adams [1877].
Corollary 6.8.3 For Legendre polynomials Pn (x),
min(m,n) - _ , .,
x-^ 2m +2n + 1 — 4-k
) g
Proof. Take к = 1/2 in Dougall's identity. ¦
Remark 6.8.2 The coefficients a(k,m,n) in F.8.4) are positive for к > 0.
Moreover, Theorem 6.8.2 implies a terminating form of Clausen's formula. See
Exercise 3.17(d) for the statement of Clausen's formula.

6.8 Linearization of Products of Orthogonal Polynomials 321
Corollary6.8.4 ForX > -1 /2andX^O
1
Cxt(x)Cxm{x)Cxn{x){\ -x2)
„ VJ I r,, F810)
(s - ?)\(s - m)\(s - n)\(k)s [T(\)]2s\(s+\)
when ? + m + n = 2s is even and the sum of any two of?,m,n is not less than
the third. The integral is zero in all other cases.
This is straightforward from Dougall's identity. It contains F.8.3) as a limiting
case.
Integrals involving products of some orthogonal polynomials also have com-
combinatorial interpretations. In Section 6.9, we show how F.8.3) can be computed
combinatorially.
The coefficients a{k, m, n) in F.8.1) can also be computed in terms of gamma
functions when pn(x) = P^a^\x) and a, /3 differ by one. This once again covers
the cases of the third and fourth Chebyshev polynomials.
Hsu [1938] showed how to use the result in Theorem 5.11.6 to go from F.8.10)
to the corresponding integral of Bessel functions:
Ja(at)Ja(bt)Ja(ct)t1-adt
( 0 if a, b, с are not sides of a triangle,
i of a triangle of area A.
This integral was evaluated by a different method in Section 4.11.
It is possible to define a second solution of the differential equation for ultra-
spherical polynomials that converges to the second solution Ya(x) of the Bessel
equation. In fact, there is an analog of the second solution for general orthogonal
polynomials {pn{x)}. For simplicity, suppose that the orthogonality measure (or
distribution) dce(t) has support in a finite interval [a, b] and that, on [a, b],
da(t) = co(t)dt,
where со(t) is continuously differentiable and square integrable. Define the function
of the second kind qn outside [a, b] by
qn(z)= [ —a>(t)dt, zeC,z?[a,b], F.8.11)
J Zt
and on the cut (a, b) by
qn(x) = lim \(qn(x + iy) + qn(x - iy)) = / ^-co(t)dt, F.8.12)
3-^0+ 2 Ja X -t

322 6 Special Orthogonal Polynomials
a < x < b. Note that, on the cut, qn is the finite Hilbert transform on (a, b) of the
function copn. The ultraspherical function of the second kind, Dx (x), is defined by
A - x2f-l'2D"n{x) = -qn(x) = - /' ^A - t2f-dt. F.8.13)
It can be shown that
^) ^^^^^ F-8.14)
Askey, Koornwinder, and Rahman [1986] have considered the integral
I Охп(х)С^(х)С^(х)A-х2Jх-Ых. F.8.15)
It vanishes if the parity of ? + m + n is even. It also vanishes when I + m + n is
odd and there is a triangle with sides I, m, n. In the other cases its value is
1/2)]2BА)яBЩ2А)т
\)]2n\l\m\
M(n+m+e+i)/2((n +m -I - l)/2)!(-A)(n-m-?+i)/2((n -m -I + l)/2)<
They also evaluate a more general integral in which Dx(x) is replaced by a func-
function of a different order, D^{x). The proof of these results used Whipple's 7F6
transformation. Corresponding integrals for Bessel functions are also derived. For
these results and for references, the reader should see their paper. This work arose
from a special case studied by Din [1981].
For general Jacobi polynomials, P^a^\x), the linearization coefficients can
not be found as products. Hylleraas [1962] found a three-term recurrence rela-
relation for these coefficients from a differential equation satisfied by the product
p(a'P\x)P^P\x). In addition to the case a = ft, Hylleraas showed that when
a = p + 1, the linearization coefficients are products. For many problems, only the
nonnegativity of these coefficients is necessary. Gasper used Hylleraas's recurrence
relation to determine the values of (a, /3) with all of the linearization coefficients
nonnegative. For many years, the best representation of these coefficients was as a
double sum. Finally, Rahman [1981] showed that these coefficients can be written
as a very well poised 2-balanced gFg. These series satisfy three-term contiguous
relations and comprise the most general class of hypergeometric series that satisfy
a three-term recurrence relation. Although Rahman's result was unexpected, in
retrospect it would have been natural to expect such a result.
If Jacobi polynomials are normalized to be positive at x = 1, as they are when
a > — 1, then the linearization coefficients are nonnegative when a>p> — \,
whena+ye>0, -1 < P < 1/2, and for some a < -ySwhen-1 <p< -1/2. For

6.9 Matching Polynomials
323
the first two of these regions, there is a general maximum principle for hyperbolic
difference equations that implies nonnegativity. See Szwarc [1992].
6.9 Matching Polynomials
Orthogonal polynomials have connections with some combinatorial objects. These
connections have been studied intensively in recent years. Here we shall define
matching polynomials of graphs and show their relation to Hermite polynomials.
This relationship will then be used to evaluate integrals of products of Hermite
polynomials; in particular, it will give their orthogonality. We begin with some
definitions from graph theory.
Let G be a graph. We can think of G as an ordered pair (V, E), where У is a set
of vertices (points) and ? is a set of edges that join pairs of vertices. We denote
the number of vertices by | v |. A graph G is complete if every pair of vertices is
joined by an edge. We denote a complete graph on от = \v\ vertices by Km. The
complement of a graph G is another graph G that has the same set of vertices as G
but those and only those edges of Кщ that are not in G. See Figure 6.1. A k-match
on G is a set of k disjoint edges of G. By disjoint we mean that no two edges meet
at the same vertex.
Example
The set of edges {{1, 2), {3,4)} in Figure 6.2 is a 2-match in G. There is no 3-match
in this example.
Let p(G,k) denote the number of &-matches in G. We take p(G,0) = 1 and
p(G, — 1) = 0. It is clear that p(G, 1) is the number of edges in G and that for
G
Figure 6.1
Figure 6.2

324 6 Special Orthogonal Polynomials
к > [m/2] (the greatest integer in m/2 = \v\/2), p(G, k) = 0. A match that uses
every vertex of G is called a complete match and is denoted by pm{G).
For a graph G, we define the matching polynomial of G, a(G), by
[m/2]
a{G) = a{G, x) = ^ (-l)kp(G, k)xm~2k, F.9.1)
k=0
where m = |u|.
Theorem 6.9.1 a(Km,x) = 2"m/2#m(x/V2).
First Proof. We proved earlier that
ft—0
It is therefore enough to show that
1 m\
p(Km, k) = -r
2*
r.
2* k\(m — 2k)!
We have to find the number of к matchings in a complete graph with m vertices.
The number of ways of choosing 2k vertices for the к matching from m vertices is
(^). From a given set of 2k vertices any particular vertex can be joined to 2k — 1
vertices to give a match. Any vertex of the remaining 2k — 2 unmatched vertices
can be matched up with 2k — 3 vertices and so on. This implies that
p(Km,k) = (™\Bk - 1)B* - 3) ¦ • • 3 • 1
1 m\
2kk\(m-2k)\
This proves the theorem. ¦
Second Proof. A more interesting approach is to show that a (Km ,k) satisfies the
same recurrence relation and initial conditions as Hem(x) := 2~m/2Hm(x/^/2).
From the recurrence relation F.1.10) for Hm(x), we find that
Hem+i(x) =xHem{x) -mHem_i(x), Heo(x) = 1, Hei(x) = x.
We first prove that
p(Km+l,k) = p{Km, к) + тр{Кт_и к - 1).
Take a vertex и e Km+\. This vertex can be a part of a fc-match in m ways, and
the number of ways to complete each &-match is p(Km~i, к — 1). If v is not in

6.9 Matching Polynomials 325
a fc-match of Km+u then the number of such ^-matches is p{Km,k). This proves
the recurrence relation for p. Now
= V (-l
a(Km+i,x)=
k=0
Km+l)/2]
= ? (-1
k=0
Jfc=0
If [(m + l)/2] > frn/2] in the first sum, then p(Km, [{m + l)/2]) = 0. So the first
sum is
[m/2]
k=0
In the second sum, change ktok + l.lt becomes
Km-l)/2]
This proves the theorem. ¦
Our next objective is to give a combinatorial evaluation of the integral
/•OO
I{nun2,...,nk)= / Hni(x)Hn2(x)-Hnt(x)e-x2dx.
J — oo
For the case к = 2, its evaluation will give the orthogonality of the Hermite poly-
polynomials. A related integral is
/»OO
J(ni,n2,.-.,nk)= / Heni(x)Hen2(x) ¦ ¦ ¦ Hent(x)e~x2/2dx,
J-oo
where Hem(x) = 2~m/2#m(x/V2). A change of variables shows that
/(«!, «2, . ... Л*) = 2(+-+"'-1)/27(l!b «2, • • • , Л*).
It is convenient to shorten the notation by writing Я = (n\, m, ¦ ¦ ¦, и/t). Also, let
.., л,-_ь и, - 1, и,+ь ..., л*).
Here the Jth parameter и, is reduced by 1. Similarly, J~'lj) will mean that the ith
and 7th parameters are reduced by 1. The next lemma gives a recurrence relation

326 6 Special Orthogonal Polynomials
Lemma 6.9.2 J^ = Y?i=2"< Jn"l) and J° = л^1"'•
Proof. First observe that the Rodrigues-type formula F.1.3) for Hermite poly-
polynomials gives
Hem(x) = (-!)"
dxm
Also, F.1.11) implies
H'em = —-Hem(x) = mHem-x{x).
dx
Applying integration by parts, we get
f°° d
h= \ (-1Г ~П^е~Х /2HemW ¦ ¦ ¦ Hemk(x)dx
J—CO ^
= (-1)--1 /
J—с
(=2
к
i=2
Since Jq is the normal integral,
ЛОО
/
J-c
the lemma is proved.
A combinatorial object that satisfies the same functional relation as in Lemma
6.9.2 is obtained as follows: Let Vi, V2, ¦ ¦ ¦, Vk be a disjoint set of vertices. Let
V = Vi U V2 U • • • U Vk. Let | Vt\ = n, so that | V| = Ya=i "<"• Construct a graph
G from V by putting an edge between every pair of vertices that does not belong
to the same V,¦. G is called the complete fc-partite graph on Vi U V2 u • • • U Vk. Let
P(ni,ri2,. ¦ ¦ ,nk) = Pn denote the number of complete matches on G, that is, the
number of matches that use all the vertices of G. We set Pg = 1, in accordance with
the earlier convention. It is clear that if X^=i ni is an 0<ld number, then Р„ = О.
We also define P%'J) similarly to J%J).

6.9 Matching Polynomials 327
Lemma 6.9.3 Рц = Y^i=2niP^'l) and ^o = !•
Proof. Choose a specific vertex in V\. This vertex can be matched with any of the
n, vertices in V,-, i ф 1. Once one such match is made, the rest can be completed
in pi1(> ways. This implies that
к
рп = ^2щР^и\
i=2
and the proof of the lemma is done. ¦
Since Jn and Pn satisfy the same recurrence relation we have the following:
Theorem 6.9.4 h = л/ЪгРц, h = B"'++ ¦+Я*яI/2РЯ.
Theorem 6.9.5 P(m, n) = m\Smn.
Proof. In this case V = V{ U V2, \Vi\ = m, and \V2\ = n. If m ф n, then
the vertices of V\ cannot be matched with the vertices of V2 to give a complete
matching. So P(m, n) = 0 for m ф и. If m = n, then the number of complete
matchings is m! and the theorem is proved. ¦
This theorem implies the orthogonality of the Hermite polynomials, for we have
the well-known result
о
Hm(x)Hn(x)e~x*dx = 2mm\-y/nSmn.
Now suppose V = V\ U V2 U V3, where Vj, V2, V3 have ?,m,n elements respec-
respectively. If ?+m+n is odd, or if ? > ш+и, then it is easy to see that P{?,m,n) —0.
The next theorem considers the other situations.
Theorem 6.9.6 Suppose l+m+n is even ands = (?+m+ n)/2. Suppose also
that the sum of any two of ?, m,n is greater than or equal to the third. Then
?\m\n\
P(?,m,n) =
(s - ?)\(s - m)\(s - n)\
Proof. Without loss of generality, we assume m > n. After all vertices in Vi are
matched with vertices in V2 and V3, the same number of vertices must be left over
in V2 and V3 for a complete matching to be possible. This means that there are
m — n more matchings of Vi into V2 than V\ into V3 in a given complete match.
So if x denotes the number of V\, V2 pairs and у the V\, V3 pairs, then x + у = ?
and x — у = m — n. Therefore, x = s — n and у — s — m. This implies that there
are (s — ?) V2, V3 pairs. There are (s_n) ways of choosing elements in V\ to pair
with elements in V2. The remaining elements in V\ then pair with elements in V3.
Moreover, for any given 2(s — n) elements, taking s — n from Vi and s — n from

328 6 Special Orthogonal Polynomials
V2, there are (s — и)! ways of doing the pairing. All this means that
P(l,m,n)=( l )( m \( П )(s-nV.(s-l)\(s-m)\
\s — n) \s — I) \s — mj
t\m\n\
(s - l)\(s - m)\(s - n)\'
The theorem is proved. ¦
The theorem implies that
Г Ht(x)Hm(x)Hn(x)e-x2dx = 2<«+'"+")/277-
J-oo (— ){ )( 2 )
when I + m + n is even and the sum of any two of I, m, n is not smaller than the
third. Otherwise the above integral is zero. It is possible to compute P(k, I, m, n)
as well, but the result is a single series rather than a product. The reader should
read the paper of Azor, Gillis, and Victor [1982] for this and other results. For
further results on matching polynomials see Godsil [1981].
We now give a different approach to the theorem that 7я = s/brPn- First observe
that the matching polynomial of G, a(G, x), can be written as
where a runs through all the matchings of G and \a\ — the number of edges in
the matching a. Let the disjoint union of two graphs G\ and G2 be denoted by
Gi U G2. See Figure 6.3.
Lemma 6.9.7 a(Gi U G2) = a(Gi)a(G2).
Proof. Suppose Gi has m vertices and G2 has n vertices. Then

6.9 Matching Polynomials 329
The last relation follows because every matching у breaks up uniquely into a
matching a of G\ and fi of G2. The lemma is proved. ¦
Let ф be a linear operator on polynomials defined by
1
ф(х") = = /
\lLit J-00
If и is odd, then ф{хп) = 0. When n is even, say n = 2m, then
We have seen that this is also the number of perfect matchings on Km. We denote
this quantity by pm(Kn).
Let Vt (i = 1,2,..., k) have и, vertices and У = V\ U V2 U • • • U Vk be their
disjoint union. Let Kv be the complete graph on V. An edge of Kv is called
homogeneous if it joins two vertices in the same V|; otherwise it is inhomogeneous.
With this terminology, P% is the number of perfect matchings of Kn with no
homogeneous edges.
Lemma 6.9.8
where a runs over all matchings ofG = KVl U KVl U ¦ • • U KVt. Here n = X)*=i
Proof. By the previous lemma and the above remarks
L~n = ф(Нещ(х)НеП2(х) ¦ ¦ ¦ Hent(x))
= ф(а(Кщ)а(КП2)---а(КПк))
This proves the lemma. ¦
The expression Х)„(—1)'а'/мп(?п-2|а|) can also be written as
where a, is a matching in KVt. Finally, we can rewrite this as
«Ь —.«*,}'

330 6 Special Orthogonal Polynomials
where у runs through all the complete matchings of Kn-2\a\ with \a\ = \cti\-\ h
\ak\. The matchings a\, a2,..., ak, y, taken together, give a complete matching
ofKv.
To complete the final step, we need one more lemma that uses the concept of
a colored complete matching of Kv. For each matching ct\, a2,..., ak, y, color
the edges in each a; red and the edges in у blue. Thus all the red edges are
homogeneous and the blue edges are either homogeneous or inhomogeneous. The
set of all matchings ct\,..., ak, у in the summation is the set X of all matchings
of Kv in which only the homogeneous edges are red. Let Y c. X, where Y is the
set of all matchings in which there are no red edges and all the blue edges are
inhomogeneous. These are the complete matchings in the к -partite subgraph in
Ky. If r(a) denotes the number of red edges in a, then by Lemma 6.9.8 we have
shown that
Also, by definition
a&Y
The next lemma will complete the proof of P^ = Ь„.
Lemma 6.9.9
]T (-iy«"> = o.
a&X-Y
Proof. First define an involution в on X — Y. Number the edges of К у arbitrarily.
For any a e X — Y, consider the set of all homogeneous edges of a. This set is
nonempty. Consider the smallest edge in this set and change its color from red to
blue or from blue to red. This gives a new matching a' = в (a) in X — Y. Clearly
в (в(a)) = a. It is also clear that (-l)r<a> + (-l)r<0(«)) = o. This proves the
lemma and the theorem. ¦
The above proof follows DeSainte-Catherine and Viennot [1983]. Also see
Viennot[1983].
6.10 The Hypergeometric Orthogonal Polynomials
The hypergeometric representations of the Jacobi, Laguerre, and Hermite poly-
polynomials, which we have extensively studied in this chapter, are respectively

6.10 The Hypergeometric Orthogonal Polynomials 331
given by
n,n+a + p + l l-x
n\ * V a + l 2 ) n\
and
-n/2,-(n-l)/2 _)_
— ' x2
In Chapter 3, the Wilson polynomials were introduced. These polynomials can be
represented as 4F3 hypergeometric functions:
Wn(x2;a,b, c,d)
(a + b)n(a + с)„(а + d)n
f-n,n+a + b + c+d-l,a + ix,a-ix \ ,,,.,,
= 4^3 . , , , , ; 1 • F.10.1)
\ a+b,a+c,a+d )
We saw that Jacobi polynomials are limiting cases of Wilson polynomials and in
turn the Laguerre and Hermite polynomials are limits of Jacobi polynomials. A
question arises as to whether there are hypergeometric orthogonal polynomials at
the 3F2 level. In fact there are such polynomials. A few are treated in this section
and others are given in the exercises. For a more complete treatment the reader
should see Koekoek and Swarttouw [1998].
It is easily seen that
Wn(x2;a,b,c,d) ( -n,a + ix,a — ix
lim ' (a + b)(a + c)F( ' ;
rf^oo (a + d)n V a+b,a+c
=: Sn(x2;a,b,c) F.10.2)
and
Wn((x + tf; a - it, b - it, с + it, d + it)
t^So (-2)" л!
,n(a + c)n(a +d)n f-n,n+a+b + c + d -l,a+ix
= 1 3Г2 I ; 1
n\ \ a + c,a + d
=:pn(x;a,b,c,d). F.10.3)
The polynomials Sn{x2; a, b, c) and pn(x; a, b, c, d) are called the continuous
dual Hahn and continuous Hahn polynomials respectively. Their orthogonality and
recurrence relations can be obtained from those of Wilson polynomials, which were
derived in Chapter 3. We restate them here for convenience. When Re(a, b, c, d)
> 0 and the nonreal parameters occur in conjugate pairs, the orthogonality is

332
given by
1 Г
6 Special Orthogonal Polynomials
Г (a + ix)T(b + ix)T(c + ix)T(d + ix)
ГB/дс)
¦Wm(x2; a, b, с, d)Wn(x2; a, b, с, d)dx
— (n+a + b + c+d- l)nn\
T{n+a+b)T(n+a + c)T(n+a + d)T(n+b+c)T(n+b+d)T{n+c+d)
rBn+a+b +
The recurrence relation is
-(a2 + x2)Wn(x2) = AnWn+{(x2) - (An + Cn)Wn(x2)
where
F.10.4)
F.10.5)
An =
(a+b)n(a+c)n(a+d)n'
(n+a + b + c + d- 1)(л + a+ b)(n +a+ c)(n + a + d)
and
Cn =
{2n + a + b + с + d - 1)Bл + a + b + с + d)
n(n + b + c- 1)(л + b + d - 1)(л +
Bn+a+b + c + d- 2) Bл + a + b +
These polynomials also satisfy a difference equation that is a dual of the recurrence
relation. This is given by
n{n + a+b + c + d- l)y(x) = B{x)y(x + i) - [B(x) + D(x)]y(x)
where
and
B(x) =
D(x) =
+ D(x)y(x-i),
= Wn(x2;a,b,c,d),
(a — ix)(b — ix)(c — ix)(d — ix)
F.10.6)
2ixBix - 1)
(a + ix)(b + ix)(c + ix)(d + ix)
2ixBix+ 1)
The corresponding results for the continuous dual Hahn polynomials are
Г (a + ix)T(b + ix)T(c + ix)
Sm(x2)Sn(x2)dx
T{2ix)
= Г (л + a + b)Y(n + a+ с)Г(п +b + c)n\Smn.
F.10.7)

6.10 The Hypergeometric Orthogonal Polynomials 333
Here Sn(x2) = Sn(x2; a, b, c) and a, b, с are either all positive or one is positive
and the other two are complex conjugates with positive real parts,
-(a2 + x2)\{x2) = AnSn+l(x2) - (An + Cn)Sn(x2) + C~Sn-\(x2), F.10.8)
where
~Sn(x2) = Sn{x2)/[(a + b)n(a+c)n],
An = (n + a + b)(n +a + c),
and
Cn =n(n + b + c- 1);
ny{x) = B{x)y(x + i) - [B{x) + D(x)]y(x) + D{x)y{x - i), F.10.9)
where
y(x) = Sn{x2),
{a- ix){b - ix)(c - ix)
2ix{2ix —
and
=
2ix{2ix + 1)
In the case of the continuous Hahn polynomials, the results are
1 f°°
— Г(а + ix)T(b + ix)T(c - ix)T(d - ix)pm(x)pn(x)dx
2n J_oo
Г(и + a + с)Г(и + a + d)T(n + b + с)Г(п +b + d)
= omn, (o.lO.lO)
Bи + a + b + с + d - \)T(n + a + b + с + d - 1)
when Re(a, b, c,d) > 0, с = a, and d — b;
{a + ix)pn{x) = Anpn+i{x) - (An + Cn)pn(x) + Cnpn-\(x), F.10.11)
where
n\
Pn(x) = ———— —Pn(x; a, b, c, d),
in(a+c)n(a + d)n
(n + a + b + c + d — \)(n + a + c)(n + a + d)
Cn =
+ a + b + c+d- 1)Bл + a + b +
n(n + b + с - 1)(и + b + d - 1)
Bn+a + b + c + d- 2){2n +a + b + c

334 6 Special Orthogonal Polynomials
and
n(n+a+b + c + d- l)y(x)
= B(x)y(x + i) - [B(x) + D(x)]y(x) + D(x)y(x - i), F.10.12)
where
У(х) = pn(x;a,b,c,d),
B(x) = (c - ix)(d - ix),
and
D(x) = (a + ix)(b
We observed earlier that Jacobi polynomials are limits of Wilson polynomials.
In this case, however, the difference equation for the Wilson polynomials becomes
the differential equation for Jacobi polynomials. See the exercises for other ex-
examples of hypergeometric orthogonal polynomials. For recent developments on
some polynomials considered here and their extensions and applications, see Nevai
[1990].
6.11 An Extension of the Ultraspherical Polynomials
The generating function of the ultraspherical polynomials is given by the product
of A — re'e)~x and its conjugate. More generally, Fejer [1925] studied a sequence
of polynomials defined as follows.
Let f(z) = YlT=o anZn be a function that is analytic in a neighborhood of z = 0,
with real coefficients. The generalized Legendre polynomials or the Legendre-
Fejer polynomials are defined by
n=0 k=0
n=0 k=0
F.11.1)
л=0
Feldheim [1941a] and Lanzewizky [1941] independently asked whether the
pn(cose) give rise to orthogonal polynomials other than the Gegenbauer poly-
polynomials. We know that if the pn{x) are orthogonal with respect to some positive
measure, then they must satisfy
xpn(x) = Anpn+l(x) +Bnpn(x) + Cnpn-i(x), n =0,1,2,... F.11.2)

6.11 An Extension of the Ultraspherical Polynomials 335
with AnCn+\ >0 and A, Bn, Cn+\ real. We can normalize to take p-i(x) = 0,
Pq{x) = 1. The converse is also true, although we did not prove it. So, to find
polynomials pn (x) that are orthogonal, it is enough to derive those that satisfy the
three-term recurrence relation.
Note that if
n
pn(cosd) = 2_\пкап-к cos(n — 2к)в,
then by в -> в + л we obtain
Therefore, if р„(х) satisfies F.11.2), it must, in fact, satisfy
2xpn(x) = Anpn+i(x) + Cnpn-\{x),
with An, Cn real, AnCn+i > 0, n = 0,1, 2, This implies that
n n+1
2COS# 2_,ak0n-k COS(n — 2к)в = An ^^OkOn+l-k COS(n + 1 — 2к)в
k=0 t=0
n-1
+ Cn ^atan_i_tcos(n - 1 - 2к)в.
F.11.3)
Now use the trigonometric identity
2 cos6» cos(n - 2к)в = cos(n + 1 - 2к)в + cos(n - 1 - 2к)в
to write the left side of F.11.3) as
пп-к cos(n + 1 — 2kH + 2^ пкпп-к cos(n — 1 — 2k) в,
i=0 k=0
Substitute this in F.11.3) and equate the coefficient of cos(« + 1H to get
or
The coefficient of cos (n — 1 — 2к)в gives
пп—к On Qk+l On—к
an-k-\ an+\ ak an-t-i

336
6 Special Orthogonal Polynomials
Take k = 0 and 1 to obtain an equation for the variables a. To simplify this
equation, set sn = an/an-i.We obtain the nonlinear difference equation
Sn+1
Sa-i -
or
~ Sn-i + S\- S2) = S\Sn ~ S2Sn-\.
For further simplification, set sa = tn + s\. The equation becomes
tn+i(tn - /„_! - t2) = -t2tn-uh = 0.
Write tn = t2un; then
un+\(un - и„_1 - 1) = -м„_ь u\ = 0.
For linear difference equations we get polynomial solutions ^ Anq". Such a so-
solution is not possible here and as there is no general method for solving nonlinear
equations, we try the simplest rational expression as a possible solution, keeping
in mind that и \ = 0. Set
U"=
withal < l,foiun(q,A,B) = un(q~\ A/(Bq), B~l). Then
A(\-gn)
1 - Bq"+l
A(l - q"'1) - A - Bq")
1 - Bq"
- qn) - A - Bqn+l)
1 — Bqn+l 1 — Bq"'1
For this to be true, we must have В = 1 and
A - q"-l)(A - 1 - (A - q)q"-1) = A - Чп~2){А - 1 - (A
This is identically true for A — 1 = q. Thus
A + q){\ - qn~])
So
l-q"
(l+eXl-?11-1)^-*!)
F.11.4)
1 -q"
which shows that for some a and /
sn -
1 -^r"

6.11 An Extension of the Ultraspherical Polynomials 337
This gives
1 - qn+l
A" = ^T^ FJL5)
and after some simplification
Cn = n . F.11.6)
The recurrence relation for pn (x) is given by
2xa(l - Pqn)Pn = A - qn+l)pn+l +oc2{\ - p2qn-l)pn-u
with
~ , > 0, n = 0,l,2,....
We have taken \q\ < 1. If q = 1, then the value of un in F.11.4) is defined
by the limit as q -» 1. This case also gives rise to orthogonal polynomials. For
example, if p = qk in F.11.5) and F.11.6), then An = (n + l)/(n + A), Cn =
(n + 2X — l)/(n + A.) (with a = 1), and the recurrence relation for ultraspherical
polynomials is obtained. Clearly, there are other cases where division by zero may
be involved in F.11.5) and F.11.6). These do not lead to orthogonal polynomials
of all degrees unless q is a root of unity. Consider the situation where this problem
does not arise and let us see what polynomials we get.
We need an expression for an. We have
an = i = a-p)a-Pq) •••(!-fig"-1)
GO An.iAn-2---A0 a (l-q)(l-q2)--(\-qn)
Therefore,
fire )=«oL
which suggests that we take uq = a = 1. In this case, the polynomial is
COS(n — 2k) в.
This expression may appear a little strange at this point. As pointed out before,
taking p = qx and letting q —*¦ 1 gives the ultraspherical polynomials. One should
keep this procedure in mind. In Chapter 10, we give an introduction to objects of

338
6 Special Orthogonal Polynomials
this kind. By the methods developed there, it can be shown that the generating
function is given by
n=0
- re-wqn)
This may appear like a much more complicated expression than the generating
function for the ultraspherical polynomials that it is supposed to extend. But there
is one sense in which it is simpler. Recall that the singularities of the generating
function can be used to get information about the asymptotic behavior of the poly-
polynomials and the weight function (see Theorem 6.6.2). The generating function for
the ultraspherical polynomials has algebraic singularities, whereas the singulari-
singularities here are simple poles. These are easier to deal with. The poles closest to the
origin are at r = e'e and r = e~'e. Near r = e'e, the generating function behaves
like
e2Wq")(l - qn+l) A - re~m)'
So,
„_o
Write the infinite product as Re'*. Then
a n\
n=0
П
Ln=o
- Pe2Wqn)(l -
-e2Wq")(l -e-2
e2Wq")(l
as n —*¦ oo.
By Theorem 6.6.2, we expect the weight function to be
шр{соьв) =
„_o
-emq")(\ -e~2Wq")
(l-2cos2eq"+q2n)
- 2P cos
n —> oo.
1/2
These infinite products are well known in the theory of elliptic functions. In Chap-
Chapter 10, we hope to convince the reader that they are quite natural and tractable,
unwieldy though they may appear now.

Exercises 339
Exercises
1. Evaluate the integral
POO POO
I = / e-x2e2ix'dx = 2 e~x2 cos2xtdx
J-oo J0
(a) by contour integration, (b) by expanding cos 2xt in powers of x and integrat-
integrating term by term, and (c) by showing that the integral satisfies the differential
equation
dl
— = -2tl.
dt
2. Prove that F{x, r) := e2xr~r2 = J2T=o ^f-r" by showing that
= Hn(x).
\r=0
3. Prove that
n/2
lim ( -
> ) (r^
n\
Hint: You can use generating functions, recurrence relations, or Rodrigues's
formula.
4. Prove that un = e~x 1гНп{х) satisfies the equation
< + Bn + 1 - x2)un = 0.
Deduce that
— (u'num - u'mun) + 2(n - m)umun = 0.
Hence prove the orthogonality of Hermite polynomials, that is,
pOO
I umundx=0 form ф n.
J—oo
5. Use the generating function for Hermite polynomials in Exercise 2 to prove
that
Hn(x cosu+y sinu) = и! V
cos* u sinn->c u
6. Let n be a nonnegative integer. Show that
and
r2n _ Bn)! ^- H2k(x)
X =
! А
22«+i Z_^ Bk + l)\(n -k)\
k=0

340 6 Special Orthogonal Polynomials
7. Define
sgnx = {-i, x<a
Show that
у (-1)" H
n=0
8. Use the generating function F.2.4) for Laguerre polynomials to prove that
00 jj f.
-r" = A + 4r2)/2(l + 2xr + 4r2
9. Obtain the generating function F.2.4) from the integral representation F.2.15)
of Laguerre polynomials.
10. Let ф„(х) = е~х2/2Н„(х)/^2пп\, n = 0, 1, 2,.... Denote the relative max-
maxima of \фп(х)\, as x decreases from +oo to 0, by До,п, Ai,n, A2,n, • • ¦ • Prove
that
Ar,n > Mr,n+b П >Г >0.
Deduce that |</>„(д;)| < max \<fo(x)\ = 1. (See Szasz [1951].)
11. Show that the Fourier transform of un(x) = e~*2/2Hn (x) is г"м„ (х) by filling
in and completing the following steps:
Г un(x)eixydx = Г e-x'—eixy+x2'2dx
J-oo J-oo dxn
= цуе^2— Г
dyn J-o
12. Let yjfn{x) = т^рщжщHn(^/2nx)e~7tx . Suppose / is square integrable on
(—oo, oo) and g is its Fourier transform. Let
¦2_^anfn{x), g{x)'
n=0 n=0
oo oo
xf(x) ~ 2_]сп^п(х), xg(x) ~ у dn\/fn(x).
n=0 n=0
(a) Show that an = i"bn.
(b) Use the recurrence relation for Hermite polynomials to obtain
fn(x) = Vn + 1 tyn+i(x) + Jn fn-\{x).

Exercises 341
(с) Use (a) and (b) to show that
cn = Vn + \an + \fnan-\,
n = i~n~l[y/n + \an - *Jnan-\].
(d) Deduce that
roo poo i poo
/ x2\f(x)\2dx + x2\g(x)\2dx > — / \f(x)\2dx
J— oo -/—oo -^^ J—oo
with equality only if /(x) is almost everywhere equal to a constant mul-
multiple of exp(—jrx2).
(e) Rescale to show that (d) implies that (for p > 0)
/¦OO pOO 1 /-OO
p2 / x2\f(x)\2dx + p-2 x2\g(x)\4x > — / |/(*)|2Jx.
J—OO J—OO -^^ J—OO
13.
14.
IS.
(f) Show that (e) implies
Show
Show
j a /„-
Prove
[y°°x2|/(x)|2Jx]
that
that
(-1)пГ(п+а
л/тгГ(а + 1/2)
that
Heisenberg'
1/2 г ,оо
/ xl
\.J-oo
I (\
s inequality:
-,1/2
\g(x)\2dx\
Г(п+а + 1)
n\
2 „-1/2
(See de
E " 1)"
sc+n+l
j—
Э0
oo
Bruin [1967].)
(-1)" ^ Hlk{x)Hln_lk{y)
2
16. Prove that
17. For Re(a + 1, /S) > 0, prove that
and

342 6 Special Orthogonal Polynomials
18. Prove
(а) УA)т!г?;Уя~к = (У + Ъя—
(b) У
19. Prove the identity
\+xy
20. Show that for a > -1, r > 0, and x > 0,
\al1 —r \—^ ^*
21. Prove that for Legendre polynomials Р„(х),
00
Ern\X) „ xr - . Г. 7
r = e Jn(v 1 — x2r).
n=U
n!
More generally,
Л > -1/2.
22. Suppose Pn (x) is the Legendre polynomial of degree n. Then Turan's inequal-
inequality states that
[Pn(x)]2-Pn-i(x)Pn+i(x)>0, п>1, -1<дс<1.
This exercise sketches a proof of Turan's inequality. See Szego [1948] on
which this is based.
(a) Show that if the polynomial
has all real roots, then
U2n_x - UnUn-2 > 0.
(b) The following is a result from entire function theory: Suppose
00

Exercises 343
is an entire function with the factorization
n=0
where a > 0, /S and fin are real, and J2™=o Pn *s convergent. Then Sn (y)
has all real roots.
To obtain Turan's inequality, use Exercise 21 and D.14.3).
23. (a) Use Exercise 19 to show that Sn(y) in Exercise 22(a) with
uk = Pk(x)
has all real roots and thus obtain another proof of Turan's inequality,
(b) Extend Turan's inequality to the polynomials Hn(x), L"(x), and C%(x).
Prove these inequalities by two different methods.
24. For the Legendre polynomial Р„ (x), prove the following results:
(a) / Р„(х)е "xdx=i "\ — Jn+y2(t).
7-i
(b) %^(
J-°o 10 if x > 1 orjc < — 1.
(c) For a nonnegative integer k,
25. Suppose a, /3 > — 1. Show that
, d2y dy
A - хг)-\ + [p-a-(fit + P + 2)x]-f- +Xy = 0
dxl dx
has a nontrivial polynomial solution if and only if к has the form n{n +a +
/3 + 1), where n is a nonnegative integer. This solution is CP^a^\x), where
С is a constant.
26. Prove the following results for ultraspherical polynomials:
(a)
(b)
where
00 ,.
n=0 ^
lim
1/2)» яд
^-"Свх(дс) =
X-l/
(А)„
n'
/г = (i -2xr
(с)
*=o

344 6 Special Orthogonal Polynomials
27. Use Rodrigues's formula to prove
when x ф ±1, and when the contour of integration is a simple closed
curve, around t = x in the positive direction, that does not contain t = ± 1.
J(l-yf(
(b) 2nJ(l-yf(l+
28. Prove that
(а)
-m,m+X
Л+ 1/2 ;
(b)
-2m+V
A+l/2 ;1
The following problems define some important hypergeometric orthogonal
polynomials. For a given nonnegative integer N appearing in the definition of
a discrete orthogonal polynomial, we use the notation
jfc!
29. The Racah polynomials are defined by
n,n + a + P+l,x,x + y+8+l
a + l,/J + «+l,y + l
for n = 0, 1,2,..., N, and where
Л(х)=х(х+у +5+1)
and one of the bottom parameters is —TV. Show that the orthogonality relation
is given by
" (y + S + l)x((y + 5
^ x\((y + 5 + l)/2),(y + S - a + l),(y - /8 + 1),(S
¦Rm(X(x))Rn(k(x))
= M(n + <* + P+ 1Ш + !)»(" ~ Д + 1)и(" + <g - У
(a + /8 + 2Jn(« + l)B(j8 + 5 + 1)„(У + 1)„
\

Exercises
345
where
M =
2)N(-p)
ifa
= -N,
2)N(8-a)N
if у + 1 = -N.
Show that the recurrence relation is given by
X(x)Rn(X(x)) = AnRn+i(X(x)) - (An + Cn)Rn(X(x)) + CnRn
where
(n + a + P + \)(n + a + l)(n + P + 5 + l)(n + у + 1)
Л„ =
and
С„ =
{In + a + /в + 1)Bи + а + /в + 2)
и (и + /в) (и + a + /в - у)(и + a - 8)
Show that the difference equation satisfied by y(x) = Rn(X(x)) is
n{n + a + l)y(x) = B(x)y(x + 1) — [B(x) + D(x)]y(x) + D(x)y(x — 1),
with
(x + a + l)(x + p + 8 + l)(x + у + l)(x + у + 5 + 1)
B(x) =
Bx + у + 8 + l)Bx +y+8 + 2)
and
Bx + у + 5)Bx + у + 5 + 1)
30. The Hahn polynomials are limits of Racah polynomials defined by
lim Rn(X(x); a, p, -N -1,5)= Qn(x; a, p, N)
or
lim Rn(X(x); a, p,y,-p-N-l) = Qn(x; a, p, N).
y->OO
Show that
a + l,-N
' ;1 , n = 0,l,...,N.
(b)
x=0
x\(N -x)\
(-l)nn\(p +!)„(«
NlBn + a + p
„(x;a,p,N)

346 6 Special Orthogonal Polynomials
(c) -xQn(x) = AnQn+i(x) - {An + Cn)Qn(x) + CnQn-i(x), where
_ (n + a + p + l)(n + a + l)(N - n)
"~ Bn + a+p + l)Bn + a + p + 2) '
and
c = n(n+p)(n + a+p + N + l)
" Bn+a + P)Bn+a+P + l)'
(d) Qn(x) satisfies the difference equation
n{n + a + p + l)y(x) = B(x)y(x + 1)
- [B(x) + D(x)]y(x) + D(x)y(x - 1),
where
В(х) = (х-Л0(х+а+1),
D(x) = x(x - P - N - 1).
(e) Use (b) to show that Qn(x; a, p, N) = CnQn(N - x; p, a, N), where
Cn = (-l)"(a + l)n/(P + 1)„. Deduce Corollary 3.3.2.
31. Define the dual Hahn polynomials by
Rn(\(x); y, S, N) := lim^ RnDx); -N - 1, p, y, 8)
and deduce properties corresponding to (a) through (d) in the previous prob-
problem. Observe that dual Hahn is obtained from the Hahn by interchanging n
andx.
32. Show that, for the Hahn polynomials Qn(x) defined in Exercise 31, the fol-
following limit formula holds:
xQn(;a,p,)
33. The Meixner polynomials can be defined by
- c) \
lim Qn[x;b-l, '-^ -, N =: Mn(x; b, c).
N->oc \ С J
Show that
(a) Мя(х; b, c) = 2Fl(-n, -x; b; 1 - 1/c).
^ (b)x c~"n\
(b) > —-cxMm(x;b,c)Mn(x;b, c) = —— -Smn,
f^0 x! (b)n(\ -c)b
b > 0 and 0 < с < 1.
(с) (c-
(Note that an application of the Pfaff transformation (Theorem 2.2.5)
shows that Meixner polynomials satisfy a three-term recurrence, which,

Exercises 347
by Favard's theorem, implies that they are also orthogonal with respect
to a positive measure when с > 1. The reader is encouraged to find this
orthogonality relation, which is obtainable from (b).)
(d) n(c - 1)М„(х) = c(x + b)Mn(x + 1)
-[* + (*+ b)c]Mn(x) + xMn(x - 1).
Observe the duality in n and x exhibited by relations (c) and (d).
34. A way of defining Krawtchouk polynomials is given by
Kn(x; p, N) := lim Qn(x; pt, A - p)t, N).
r->oo
Prove the following relations:
(a) Kn{x;p,N)=2Fx{-n,-x;-N;l/p), n = 0,l,...,N.
^ Л ( x )PX^ - p)N-xKm{x; p, N)Kn(x; p, N)
(-irn\f\-p\"s
(-N)n V P
(c) -xKn(x) = p(N - n)Kn+l(x) - [p(N -n)+ n{\ - p)]Kn(x)
+ n(l-p)Kn-l(x).
(d) -nKn(x) = p(N - x)Kn(x + l)-[p(N -x)+x(] - p)]Kn(x)
+ x(l - p)Kn(x - 1).
Note the relationship of (c) and (d) as in the case of the Meixner polynomials
given in the previous exercise. In fact
Kn(x; p, N) = Mn{x; -N, p/(p - 1)).
35. Define the Charlier polynomials by
С„(х; a) := lim М„(х; b, a/{a + b))
b—юо
or
С„(х; a) = lim Kn(x\ a/N, N).
Deduce that
00 x
(b) ^2 -^cm(x; a)Cn(x; a) = n\a~neaSmn, a > 0.
(c) -хС„(х) = aCn+l(x) - (n +fl)CB(jc) + лС„_1(дс).

348 6 Special Orthogonal Polynomials
(d) -nCn(x) = aCn(x + 1) - (x + a)Cn(x) + xCn(x - 1).
Compare (c) and (d) as in the previous two exercises.
36. Prove that
Urn Ba)n/2Cn(V2ax +a\ a) = (-1)"Я„(х).
a—>oo
37. The hypergeometric representation of the Meixner-Pollaczek polynomials is
These polynomials can be obtained as limits of continuous dual Hahn (or
continuous Hahn) polynomials. Show that
, Sn{{x
(а) Р„(х;ф) = hm
n\(t/sin<p)n
/•00
(b) / Bф^
-лч2^ ! 3m"' Л > 0' 0<ф <7T.
(c) (и + 1)Р„л+1 (х) - 2[х sin ф + (и + X) cos 0]Р„*(х)
+ (и + 2Х-1)Р„л_1(х)=0.
(d) в1'*(А. - гх)Р„А(х + 0 + 2/[xcos0 - (и + A.) si
38. This problem gives generating functions for some orthogonal polynomials,
(a) Wilson polynomials:
fa + ix, b + ix \ / с — ix,d — ix
and
((a+b+c+d-l)/2, (a+b+c+d)/2,a+ix,a-ix -At
4 3\ a+b,a + c,a+d ' A - tJ

Exercises 349
A simple corollary is that
dk fa+x,b + x \ /c-x,d-x
for —oo < x < oo when a,b,c,d > 0 and 0 < t < 1.
(b) Continuous dual Hahn:
and
, „ (a + ix, a-ix \ ^ Sn(x2; a,b,c)t"
(c) Continuous Hahn:
a + ix \ fd-ix \ ^ pn(x;a,b,c,d)tn
г 1^*1 ' гМ = У
> г' 11^11
\a + c J[ \b
and
(a + b + c + d- l)/2, (a + b + с + d)/2, a + гх -4Г
a + c,a+d ' A -О2
(d) Meixner-Pollaczek:
00
A _ егф1)-к+1х{\ - е-'ф1Гх-'х = E Pn& Ф)*"
n=0
and
'X + ix ,.-,. .
^ BV"ein*
(e) Meixner:
t
and
, _ (-x (l-c)t\ ^ Mn{x;b,c)

350 6 Special Orthogonal Polynomials
(f) Charlier:
и
л=0
For more examples of orthogonal polynomials and their properties, see
Chihara [1978, Chapter 6].
39. Prove the inequalities in F.4.19) for a > —1/2. Prove the corresponding
result for —1/2 > a > — 1.
40. Suppose a, fi > — 1. Show that
[ \P?"-P)(xi)\ ~ VVn when max(a, /8) < -1/2.
Here x\ is one of the two maximum points nearest (/в — a)/(a 4-/6 + 1) =
(Compare with Exercise 4.18. Take
n(n + a + p
and show that f'(x) can change sign only at x0.)
41. Show that
f&k ifX>0,
max \c^(x)\ = { "¦
-l^x-1 { \C^(xi)\ if Л < ОД nonintegral.
Here xi is one of the two maximum points nearest 0 if n is odd; x\ = 0 if n is
even.
42. Show that for a fixed с and и —»• oo,
|-0-«-1/2О(и-1/2) if c/n < ^ < я/2,
Pn(a'«(cos#)= <^ - -
[ О(иа) ifO <#<c/n.
[Use the result on the asymptotic behavior of Pn(a^\x) in Section 6.6. Also
apply Exercise 4.18, with
y(x) =
and
1/4 — a2 1/4 — B
+
(и
4sin2(x/2) 4cos2(x/2)
43. Show that the sequence formed by the relative maxima of \L" (x)\ and \L" @) |
is decreasing for x < a + 1/2 and increasing for x > a + 1/2. (Consider the
function n{L«(x)}2 ^2

Exercises 351
44. Prove that the successive relative maxima of \Hn(x)\ is a decreasing or in-
increasing sequence according as x < 0 or x > 0.
45. Use Theorem 6.7.2(c) and Gegenbauer's integral in Theorem 6.7.4 to obtain
Koomwinder's Laplace-type integral for Jacobi polynomials:
P^*'P){x) _ 2Г(«+ 1)
2 r 1"
у1 Г Г1+х-A-х)м2 . г 1
/ / ; HV1-*«COS0
Jo Jo L 2 J
a > /в > -1/2.
46. Note that v(x) = e~x2/2Hn(x) is a solution of
y" + {In + l)y = x2y.
(a) Hence verify that for s = \Jln + 1,
v(x) = An[cos(sx - ИЛ-/2) + /fn(x)],
where
Rn{x) = ~ f t2y(t)sin(s(x-t))dt,
sAn Jo
and
(n\/k\, n = 2k,
А„ = <
l(n!/Jk!)B/s), n = 2*+l.
(b) Use Schwartz's inequality to prove that
\Rn(x)\ <C|x|5/2n-1/4,
where С is a constant.
(c) Deduce that
Hn{x) ~ 2<"+1>/V/V/V2/2cosГ« - ^ as n ^ oo.
47. Show that y(x) = e~x/2L"(x) satisfies the equation
(a) Deduce that, for N - (a + 2n + l)/2,
У(х) = j yi(x)
+ ^N I ^г)а+1У@[У1@У2(х) - yi(x)y2(t)]dt,
where
У\(х) = .

352 6 Special Orthogonal Polynomials
(b) As in Exercise 46, but after much more work, it can be shown that the
integral divided by Г(n + a+I)/n\ tends to zero as n -» oo. Thus prove
that
+a + l)
as n -> oo.
(c) Use D.8.5) to conclude that
as n —»• oo.
48. Let rk denote the number of ways in which к rooks can be placed in different
rows and columns on an m x n chessboard. Let Rmn = Y17=o rkXk ¦ Prove
that, when a is an integer, Rn,n+a — n\x"L^(—l/x).
49. Use the methods of Section 6.9 to evaluate the integral
-== / Hea(x)Heb(x)Hec(x)Hed(x)e-x2/2dx.
Then evaluate it by a different method.
50. Suppose f(x) is expandable in terms of Jacobi polynomials. Let
л=0
where hn is defined in F.2.5). Let g(x) be an average of f(x) denned by
g(x) = A - х)-а-]A - лгГ" / /@A - t)a(l + tfdt.
Jx
Suppose for simplicity that a@) = 0. If b{n) and a{n) are the Jacobi
coefficients of g and / respectively, show that
b(n) =
+
а+^ + 1 Г(л + 1)
^ a(k)[2k + a + fi + 1] Г(*+1)
\ht\ k[k+ a + fi + l] Г(* + а + 1)'
for n = 0, 1,....
A sufficient condition for the result to hold is the convergence of the series
Ti\a(n)\nl+a. Similar results hold forLaguerre and Hermite polynomials.
51. This exercise gives a proof of the following theorem of Hardy [1933]: If /
and its Fourier transform g are both 0(\x\me~x /2) for large x and some m,
then each is a finite linear combination of functions of the form e~x2/2Hn (x),
where Hn{x) is the Hermite polynomial of degree n.

Exercises 353
Note that it is enough to prove the theorem for self-reciprocal (/ = g) and
skew-reciprocal functions (/ = —g). Take / to be self-reciprocal and show
that:
(a) For Re s = a > — 1,
X(s) = Г e-s*2'2f{x)dx
Jo
satisfies
X(s) = s~
(b) When ix(s) = y/(s + l)k(s),
fi(s) =
The function ix(s) may have a singularity at s = — 1 but is analytic at all
other points including infinity. Hence,
л=0
(c) X(s) = 0(\s + \\~p) forsome p, near.? = —1. This can be proven by the
argument below. Let а +1 = т. On the unit circle \s\ = 1, т = \s +112/2.
For |s| < 1
X(s)= o( Пe-Tx2'2xmdx) = Ods + ll"™).
(d) / e-«2V'/2u-M-'--' 1Ч" ' ^-^
2
/•00
/•00
(e) / е-/2(/-Ф)Л = 0,
where Ф is an appropriate linear combination of functions of the form
52. Hardy's theorem given in the previous exercise extends to the following result
ofRoosenraad[1969]:
Set
and define generalized Laguerre functions by
Ca2n(x)=caJx\a+l/2e-*2?2Lan(x2),
Ca2n+X(x) = can+x\x\a+ll2e~xll2xLa+\x2), л = 0, 1, 2,....

354 6 Special Orthogonal Polynomials
For a function / defined for all real numbers, set
If00 (
= -J f(x)U
(xt)
i^
Note that 1Za is the sum of an even and an odd Hankel transform. Check that
?%, are eigenfunctions of TZa, that is,
naCam=im?am, m =0,1,2,....
Theorem Iff and ga = Haf are both O(xm+a+l/2e~x2/2) for large x and some
m > 0, then each is a finite linear combination of the functions C"(x).
As before, it is sufficient to consider the cases where lZaf = ±f.TakelZaf = f.
Show that, for Re s > — 1,
/•00
= / xa+l/2e-sx2/2f(x)dx
Jo
satisfies
So, if n(s) = A +i)a+1A.(i), then (x(s) = fx(l/s). Now complete the proof as in
the previous exercise.

Topics in Orthogonal Polynomials
As we have seen before, we can gain insight into Jacobi polynomials by using
the fact that they are hypergeometric functions. In this chapter, we reverse our
procedure and see that Jacobi polynomials can shed light on some aspects of hyper-
hypergeometric function theory. Thus, we discuss the connection coefficient problem
for Jacobi polynomials. We also discuss the positivity of sums of Jacobi polyno-
polynomials. We mention several methods but here, too, there are situations in which the
hypergeometric function plays an important role. Finally, for its intrinsic interest,
we present Beukers's use of Legendre polynomials to prove the irrationality of
f C), a result first proved by Apery.
It is evident that the Jacobi polynomial P^Y'S\x) can be expressed as a sum:
YH=o cnkPk (*)¦ The significant point is that the connection coefficient cnk is
expressible as a 3 Fj hypergeometric function. This 3 F2 can be evaluated in terms of
shifted factorials under conditions on the parameters a, /3, y, and <5. Surprisingly,
this leads to an illuminating proof of Whipple's 7 F6 transformation. We have seen
that, with the exception of Gauss's 2^1, most summable hypergeometric series
are either balanced or well poised. A puzzling fact is that at the 5 F4 and higher
levels, the series are very well poised. The above-mentioned proof of Whipple's
transformation sheds light on this fact by showing that very well poisedness arises
from the orthogonality relation for Jacobi polynomials.
Fejer used the positivity of the series YH=o sin (fc + \)в to prove his famous
theorem on the Cesaro summability of Fourier series. The positivity of some other
trigonometric series have also been important in mathematics. It turns out that
these inequalities are generalizable to inequalities for sums of Jacobi polynomi-
polynomials. The soundness of this generalization is illustrated by the usefulness of the
inequalities. A dramatic example is an inequality proved by Gasper that played an
unexpected but significant role in de Branges's proof of the Bieberbach conjecture.
For this interesting story, see Baernstein et al. [1986]. We also state and prove a
trigonometric inequality due to Vietoris.
355

356 7 Topics in Orthogonal Polynomials
7.1 Connection Coefficients
Suppose V is the vector space of all polynomials over the real or complex numbers
and Vm is the subspace of polynomials of degree < m. Suppose po(x), p\(x),
рг(х),... is a sequence of polynomials such that pn(x) is of exact degree n; let
qo(x), q\ (x), qj(x),... be another such sequence. Clearly, these sequences form
a basis for V. It is also evident that po(x),..., pm (x) and qo(x),... ,qm(x) give
two bases of Vm. It is often necessary in working with finite-dimensional vector
spaces to find the matrix that transforms a basis of a given space to another basis.
This means that one is interested in the coefficients cnk that satisfy
^ G.1.1)
k=0
The choice of pn or qn depends on the situation. For example, suppose
pn(x) =xn, qn(x) = x(x- l)---(x -n + 1).
Then the coefficients cnk are Stirling numbers of the first kind. If the roles of
these pn and qn are interchanged, then we get Stirling numbers of the second kind.
These numbers are useful in some combinatorial problems and were defined by
Stirling [1730].
Usually, little can be said about these connection coefficients. However, there
are some cases where simple formulas can be obtained. For example,
7%^y cos(n - 2к)в G.1.2)
gives an expansion of Ри(а'а)(х) in terms of Pk~ ' ¦ ' \x). This formula was
derived in the previous chapter from the generating function for C*(x). See
F.4.11). Another example is
= t~^T^). GЛ.З)
This can be obtained from the generating function for L%(x). We have
= A -r)-^1exp(-xr/(l -r))
n=0
-ЕЕ

7.1 Connection Coefficients 357
Notice that in both these cases the polynomials are similar, in that they are orthog-
orthogonal on the same interval and their weight functions are closely related.
The next lemma is a basic result of this section and gives the connection coeffi-
coefficient cnk when qn{x) = P?'*>{x) and pk{x) = Рк(аф)(х).
Lemma 7.1.1 Suppose P^-S) (x) = ?Lo cnkР1"ф) (х). Then
(n + у + S + l)k(lc + y + l)n-kBk + a + p
_
Cnk~
(п-к)\ГBк+а + Р +
n+k,n + k + y + 8 + l,k + a + l\
k + y + l,2k + a + p + 2 ' )'
Proof. From the orthogonality of Jacobi polynomials,
Cnk = ank/hk,
where
[
+Р+1 а + 1)Г(к
Bk + a + P + 1)Г(к +а + Р + 1)Г(к
and
апк= / P^&\x)Pla'P)(x)(\-x)a(\+xfdx
7-1
( ivt Г1 dk
= Чг- / Pi^GO—r[(i-*)B+*O+*)/'+t]<fr
2*Л! У_! dxk
.-x)a+k(l+x)p+kdx.
We have seen earlier that
G.1.4)
dx-n v»/- 2 '«-i (*)• F-3.8)
Therefore,
n(y ,8) /.^.4 ^ * D vK~i~^i^~r^) / \
Г ' I у I — r i ( у I

358 7 Topics in Orthogonal Polynomials
Use this in the integral for ank to get
ank =
22kk\
y+8+ l)k(y + k + !)„_* y^ (-и + k)j(n
22kk\(n-k)\
• Г (l-x)a+k+i(l+x)p+kdx
(п + у + S + \)к{к + у + l)B-tr(fc + а + 1)Г<*г
This is equivalent to the claim in Lemma 7.1.1. ¦
In general, the 3 F2 in the lemma cannot be summed. If we take у = a, then the
3 F2 reduces to a terminating 2 F\, which can be evaluated by the Chu-Vandermonde
formula (Corollary 2.2.3). The 3F2 can again be summed if S = p. For in this
case we get a balanced 3F2 whose value is given by the Pfaff-Saalschiitz identity
(Theorem 2.2.6). Finally, the 3F2 can be summed by Watson's identity if а = p
and у — S (Theorem 3.5.5). It is, however, sufficient to do the а = у case in
Lemma 7.1.1 as the other two cases are consequences of this one.
Theorem 7.1.2
П
ft V / (fy l_
2)„ 2^
jfc=0
Proof. Take у = a in Lemma 7.1.1. The 3 F2 reduces to
(a + p + 2k + 2)n_k
Use this in Lemma 7.1.1 and simplify to get the result. ¦
The case S — fi is a corollary.
Theorem 7.1.3
(a + p+2)n
¦^ (y — а)„_^(а + P + l)k(a + P + 2k + 1)(/? + у + n
jfc=0 '" " « и

7.1 Connection Coefficients 359
Proof. Use Theorem 7.1.2 and the fact that P^\x) = (-1)" P^'a)(-x). Ш
Theorem 7.1.4
p(.Y.Y)(x\ - ^ + ^
rm *• л> ~
B у + l)
? B«
3/2)m_afc(y - a). (a,g)
m
y
?j (a + l)m_2ifc(« + 3/2)m_t(a
Proof. Replace x with 2x2 - 1 and ft = ± 1 /2 in Theorem 7.1.3. Then
(« + 3/2)я
(К - a)n-k(a + l/2)tBfc + « + 1/2)(и + у + 1/2). (,/)
k (
G.1.5)
(a,-i/2) 2
(n-Jk)!(l/2)t(o+l/2)(n+o + 3/2)t k (X
and
2)Bx2 l) C/2)B
(« + 5/2)„
уЧ (у - а)„-П« + 3/2)tBfe + а + 3/2)(и + у + 3/2). (аЛ/1) 2 _
G.1.6)
Now use the quadratic transformation formula from Chapter 3 (see C.1.1)) to
obtain
p(a,a)( = (« p
2й (о+1)„Bл)! "
and
(aa) («+lJn+iw! (l/2J _ ,ч r? , ox
Take m = In and combine G.1.5) and G.1.7) and simplify to get
p(Y,Y)(x\ = (y + ^m
By + l)m
(У ~ <*)f-*B« + 1)ц(« + 3/2)a(y + l/2)f +k (tt.g)
2*
The formula in the theorem for even m follows from this by reversing the order of
summation, that is, by changing к to у — к. The odd case is obtained similarly
from G.1.6) and G.1.8). ¦

360 7 Topics in Orthogonal Polynomials
We can also write the previous theorem in terms of ultraspherical polynomials:
Theorem 7.1.4'
Remark 7.1.1 The relation in G.1.2) is obtained from the above formula by
letting ц, -» 0. Surprisingly, it is easy to obtain Theorem 7.1.4' directly from
G.1.2).
Proof of Theorem 7.1.4' Note that
Differentiate G.1.2) with respect to 9 and divide by — sin в to get
a+i, m V^ W»-*W*(* 2fc) sin(n - 2к)в
„ Ucosd) = > . G.1.10)
"~1V f^ (nfc)!fc! sin0
f (n
к—0
If /t is replaced by n — к, the expression in the summation does not change. So
for n odd the terms of the sum can be paired and for n even the same can be done,
because the term when n = к is zero. Thus, after changing n — 1 to n and X to
A.- 1 in G.1.10), we get
Here we used the fact that
, sin(n + 1N
Repeat this process, that is, differentiate with respect to x and change n — lion
and X to A. — 1 to get
Щ(*2)я + 22*
t=0 v-zn-ic-v
By induction it follows that
for jx = 1, 2, 3, It is evident from G.1.2) that C^_2jt(xIS a polynomial in д.
Hence the right side of G.1.11) is a rational function of д. Since G.1.11) is true

7.1 Connection Coefficients 361
for infinitely many values of ц, it is identically true. This completes another proof
of Theorem 7.1.4'. ¦
It is possible to obtain Dougall's sum of a very well poised 7F6 from Theorem
7.1.3. This evaluation of Dougall's identity gives an insight different from those
suggested by earlier evaluations. To start, take x = 1 in Theorem 7.1.3 to get
(y + l)n(a + p+2)n = (y-a)n
P
Note that
^+0+1 = (|g+^+t)/2) • GЛЛ2)
So
/a + 0 + 1, (a + P + 3)/2, и + у + p + 1, a + 1, -n
Set a + p + 1 = a, a + 1 = ?, and n + y + /J+l=cto rewrite this formula as
яо + р+1 = a, a -t- i =o, ana n + y + p-t-i=:cio rewnie mis rormi
/ a, a/2+1, ft, с, -л; \ _ (а + 1)„(а + 1-ft -
5 4 Va/2,a + l -ft, a + 1 - c, a+ 1+n' У (a + 1 - b)n(a + 1 -
hi« iHpntitv oivpc fhp cum rvf я tprminatino upn/ \x/p11 nni«pH * Т?л Tt i« intp.rp
с)„'
This identity gives the sum of a terminating very well poised 5F4. It is interesting
that after Kummer's sum of the well-poised 2^1 at x = —1 and Dixon's sum of
the well-poised 3F2 at x = 1, most well-poised series that can be summed have
the additional feature of a numerator and denominator parameter differing by one.
This makes the series very well poised. In the above series, this followed from
G.1.12), which in turn came from the orthogonality relation for Jacobi polynomials
G.1.4). This partially explains why the summable well-poised series after 3F2 are
very well poised.
The above 5F4 is only a particular case of Dougall's 7F6. To find the general
case, note that evaluation of a function at a point is an example of a linear operator.
This was the operator applied to the identity in Theorem 7.1.3 to get the 5F4.
A more general operator is an integral with respect to a measure. To obtain
attractive formulas, one chooses the measure suitably. The Jacobi polynomials
can be written as 2^1 hypergeometric series and we know that the integration
of a hypergeometric function with respect to a beta distribution introduces two
new independent parameters into the series. This shows how we may obtain the

362 7 Topics in Orthogonal Polynomials
generalization of the 5F4 formula. Write Theorem 7.1.3 as
(Y + 1)п р f-n, y+p+n+1
^ ;
i211 ,
n\ V У +
а + ? + 2)й^ (л-*)!
(а + j8 + 2A: + l)(y + p + n + l)k(a + \)k
;ut\. G.1.13)
Integrate this with respect to two independent beta distributions. The result is
'-n, у +Р +n + I,a, b \ _ (P + l)n
+ 1)„п\ Л y + hc,d
(у - a)n-k(a + p + l)t(o + P + 2k + l)(y +p + n + l)k(a + \)k
k=Q (n - k)\(P + l)t(o + P + 1)(а + Р + П + 2)kk\
,-k,a + P + k+l,a,b у
\ a + l,c,d )
The 4F3S here cannot be summed without restriction on the parameters. Observe
that the 4F3 on the left-hand side is balanced ifa + b + p + 1 = с + d. This
is also the condition for the 4F3S on the right to be balanced. So assume that the
parameters are chosen to balance the 4F3S. If we further let b = a + 1, then the
4F3 on the right side is reduced to a balanced 3F2, which can be summed by the
Pfaff-Saalschiitz formula. The result is the formula
+P + n + l,a+ I, a +0-c + 2,c-a
a-y-n + l ( + p+ l)/2 + p + n+2p + l +a + p + 2'
when a+P+a+2 = c + d. This is just Whipple's transformation formula from
which Dougall's identity can be obtained, as we have seen earlier.
Remark 7.1.2 The coefficients in Theorem 7.1.4 are nonnegative when у > a >
— 1. This fact is useful in the proof of the positivity of a certain 3 F2 function. This
played a significant role in the first proof of the Bieberbach conjecture. We prove
the inequality in a later section. The nonnegativity of the coefficients in Theorem
7.1.3 holds under the same condition, that is, у > a > — 1. In Theorem 7.1.2,
nonnegativity occurs when S — p = — 1, —2,..., 8 > — 1. For general a, p, y, S

7.2 Rational Functions with Positive Power Series Coefficients 363
the problem of the nonnegativity of the coefficients reduces by Lemma 7.1.1 to
that of a certain 3F2. One way to deal with this is to use three-term contiguous
relations for 3F2S. For details the reader may look to Askey and Gasper [1971].
7.2 Rational Functions with Positive Power Series Coefficients
We start by showing that
m - к + a \ (n - к + a \ fk - a - 2\
m-k ) \ n-k ) \ к ) ~
for a > 0. Lorentz and Zeller [1964] used this to obtain a new proof of a theorem
of Hardy and Bohr. The above inequality is not directly related to orthogonal
polynomials but its proof gives a nice introduction to the method of generating
functions. This method will be used again in this section to prove an inequality
involving Laguerre polynomials. Moreover, G.2.1) is really an inequality for a
3F2, a topic we discussed in the previous section. Note that G.2.1) can be written
as
(« + 1>«(« +DVa (-».-». "«"I. Л, О G.2.2)
m\n\ \— m — a, — n — a
for a > 0.
We prove a more general result, which is the content of the next theorem, due
to Askey, Gasper, and Ismail [1975].
Theorem 7.2.1 If 0 < a < min(?, y), then
min(m,n) i , o\ / ;i\/; <->\
El m — к + p\ /n — k + y\ / к — a — 2\
ы ( -* )( .-* )( * J0'
m,n =0,1,2,.... G.2.3)
Proof. Observe that
k=0
xk.
Thus G.2.3) must be the coefficient of some term in an expansion obtained as the
product of three binomial expansions. In fact, G.2.3) is the coefficient of rmsn in
the product

364 7 Topics in Orthogonal Polynomials
Verify that we can rewrite this product as
-5> S V rn-k J\ n-k
A - rs)a+1
-r)Y+l
= A -г)<*^A -s)"-y
The two factors A — r)a~P and A — s)a~y have nonnegative power series coeffi-
coefficients when ft > a and у > a. This shows that it is sufficient to prove the case
where a = fi = у. Note that
1 - rs 1 1
(l-r)(l-s) 1-r 1
n=\
Thus the expansion of A — rs)/[(l — r)(l — s)] has positive coefficients and it
follows that any positive integer power of this rational function also has positive
power series coefficients. Now write
1-rs r+1 Г 1-rs ]M+1r 1-rs
(l-r)(l-s)J
Since 0 <a — [a] < 1, we need only consider the case a = f} = у andO < a < 1.
Observe that the 3F2 in G.2.2) when written out is
mn(a + 1) mirn — l)n(n — l)(a + \)a
(m+a)(n + a)V. (m+a)(m+a- l)(n + a)(n + a - 1J!
m(m - \){m - 2) л (л - 1)(л - 2) (a + l)a(a - 1)
(m + a)(m + a - \){m + a - 2)(л + a)(n +a- l)(n + a - 2K!
+ •¦
There are both positive and negative terms in this series, so the sign of the sum is
not immediately evident. To show that the series is positive, we transform it into
another series, all of whose terms are positive. For this purpose apply to G.2.2)
Thomae's formula (Corollary 3.3.4),
F (а'Ь'с.Л r(rf)r(g)r(s) fd-a,e-a,s
3 2\ d,e ' ) r(a)r(s + b)T(s+cK 2\ s + b,s + c '
where s — d + e — a — b — с The result, after a little simplification, is
a(a + 1) / 1— m, 1— n, 1— a
^ ;
(m + or) (л +a) \ 1 — m — a, 1 — n — a

7.2 Rational Functions with Positive Power Series Coefficients 365
It is clear that every term of this 3F2 is positive when 0 < a < 1. This proves the
theorem. ¦
Remark Theorem 7.2.1 is equivalent to the statement that
3F2f~IB'"n'"a;l>) >0, m, л = 0,1,2,... G.2.4)
\-m-p,-n-y )
when 0 < a < min(/), y). The condition 0 < a < min(yS, y) is necessary. Take
m = 1 in G.2.4) to get
Let и —>• oo to see that a < fi. By symmetry a < у.
The next problem is to show the positivity of the coefficients A(k, m,n) in the
power series expansion of the rational function
G.2.5)
The A(k, m,n) satisfy a finite-difference equation that approximates a two-
dimensional wave equation. Friedrichs and Lewy wished to use the positivity of
A(k, m, n) to prove the convergence of solutions of finite-difference approxima-
approximations to solutions of the wave equation. Szego [1933] gave a proof using Bessel
functions. He also translated this problem into an equivalent problem about the
positivity of integrals of products of Laguerre polynomials. We follow this direc-
direction here.
Rewrite the left-hand side of G.2.5) as
1 1
Jo
e-xn\-s) e-x/a-t)
dx-
\-r \-s \~t
Recall the generating function for the Laguerre polynomials L"(x),
n=0
Then

366 7 Topics in Orthogonal Polynomials
Thus G.2.6) is equal to
V" / Lk{x)Lm(x)Ln{x)e-^dxrksmtn
Jo
E
k,m,n
and
/•oo
,m,n)= I Lk(x)Lm(x)Ln(x)e~3xdx.
Jo
A more general situation can be treated in a similar way. Note that if f(x)
(x - r)(x -s)(x- t) then the left-hand side of G.2.5) is 1//'A). Write
k,m,n=0
so that
1 f°°
Aa(k,m,n)= / Lak(x)Lam(x)L°(x)xae-3xdx. G.2.7)
l (a + i) Jo
Theorem 7.2.2 Fora > -1/2, Aa(/t, m, n) > 0. For a > 0, йе inequality is
strict, that is, Aa{k, m,n) > 0.
Proo/ In Chapter 6, we computed integrals of the products of three Hermite
or three ultraspherical polynomials. This also gave their nonnegativity. These
integrals were obtained from corresponding linearization formulas. That method
does not work here. But recall that
Thus it is reasonable to consider the positivity of the integral
a+j)(x)(l -x)a(l +х)а+ъЫх. G.2.8)
/•l
/
We already know that
/ Pt'a)(x)P^a)(x)P^a)(x)(\-x2)adx >0 fora >-1/2. G.2.9)
The question is: Can one increase the second parameter f$ in P[, \x) and still
retain positivity in G.2.8)? In fact, we have the formula
) 2{n +1) p{a^(x^ i 2(n + ^ +1} p(a-e)()
G.2.10)
Verify this by noting that the right side vanishes when x — — 1 and that Jacobi
polynomials are orthogonal. Since the coefficients in G.2.10) are positive, we get

7.2 Rational Functions with Positive Power Series Coefficients 367
the nonnegativity of G.2.8) from G.2.9) and G.2.10). This in turn implies the
nonnegativity of Aa(k, m, n) fora > —1/2.
The strict positivity for a > 0 comes from
l Г l
This implies
^oo
[Г((а + l)/2)]2 / Lak(x)Lam(x)Lan(x)xae-3*dx
Jo
к т п
= Г (a + 1) 53 53 YlI(k ~ a' m ~ b> n ~ c)/(a' fo' c)' G-2л!)
a=0 6=0 c=0
where
(i, j, k) = / Lf-1)/2(x)Lf -
Jo
When a > 0, all terms in G.2.11) are nonnegative. So it is enough to find one
strictly positive term. The positivity of the term a=k,b = m,c = 0 follows
from the next lemma, which proves the theorem. ¦
Lemma 7.2.3 For a > — 1, e > 0, we have
(¦OO
e-"Lan{x)Lam{x)xae-xdx > 0. G.2.12)
Proof. Consider the generating function
Jo
= Jo
From the last expression it is clear that if 0 <e < 1, then the coefficient of r"sm is
positive. The result may be extended to larger values of e by iteration as follows.
Since e~(XL"(x) is smooth and integrable, we can expand it in terms of Laguerre
polynomials. Let
k=0

368 7 Topics in Orthogonal Polynomials
For 0 < € < 1, Ct(e) > 0 by our previous remarks. Now
k=0
So e 2exL"(x) can be written as a sum with positive coefficients. Iteration of
this process completes the proof of the lemma. Another proof is in Exercises 6
and 7. ¦
A consequence of Lemma 7.2.3 is the next result. The proof is left to the reader.
Corollary 7.2.4 Let a > — 1, and suppose that f(x) is represented by its
Laguerre—Fourier expansion. Suppose also that the coefficients of the expansion
are positive, that is,
poo
an= f(x)Lan(x)xae-xdx>0, п = 0,1,....
Jo
Then
/•oo
an(€)= f(x)e-"Lan(x)xae~xdx>0, n = 0, 1, 2,..., e > 0,
Jo
unless f(x) = 0, x > 0.
Theorem 7.2.2 is surprising when considered from a different point of view, as
we shall see below. Consider the following result of Sarmanov [1968].
Theorem 7.2.5 If
oo
f(x, y) = J2anK(x)Lan(y)/Lan@) > 0, 0 < x, у < oo, G.2.13)
n=0
then
an= f rn
Jo
where dfi(r) is a positive measure.
In fact, the positivity of G.2.13) for an = r", 0 < r < 1, is a consequence
of F.2.28). Now Laguerre polynomials satisfy a differential equation in x and a
difference equation in и. It frequently happens that a dual result can be obtained
by interchanging n and x. For example, Lemma 7.2.3 is the dual of the positivity
of G.2.13) when an = rn. A dual of Szego's positive integral G.2.7) would be
z) > 0, 0 < *, j, z < oc,
n=0

7.2 Rational Functions with Positive Power Series Coefficients
for some sequence а„. However, Theorem 7.2.5 shows that
369
anLan(z)Lan(O) = / r"dn(r,z),
Jo
with dfi(r, z) > 0 for all z, 0 < г < oo. This is possible only when dfi(r, z) is a
point mass at r = 0, and when ao = с > 0, а„ = 0, и = 1, 2,..., since ?"(г) is
negative for some z > 0.
Sarmanov's paper contains aproof of Theorem 7.2.5. Another proof that makes
more explicit use of special functions is in Askey [1970].
There are some extensions of Theorem 7.2.2 for some a.
Theorem 7.2.6 If a > a0 = (-5 + ч/Г7)/2, then
3
xaLak(x)Lam(x)Lan(x)e-2xdx > 0, a > a0,
/
Jo
k,m,n = 0, 1,.... The only case of equality occurs when к = m = n — 1
a = ao.
For a proof, see Askey and Gasper [1977]. The case a = 0, 1,... is outlined
in Exercise 10. The fact that Theorem 7.2.6 implies Theorem 7.2.2 when a > ao
follows from Lemma 7.2.3.
Theorem 7.2.7 IfO < a < 1, a + b = 1, and a > 0, then
xaLak{ax)Lam{bx)Lan(x)e-xdx > 0,
Г
Jo
k,m,n = 0, 1,....
For a proof, see Koornwinder [1978]. We sketch a proof of Koornwinder's
inequality in Theorem 7.2.7 for nonnegative integer values of a after a discussion
of MacMahon's Master Theorem.
The theorem of MacMahon [1917-1918, pp. 93-98], known as the Master
Theorem, makes it possible to give combinatorial interpretations of coefficients of
series expansions of rational functions in several variables. MacMahon's Master
Theorem can be stated as follows: Suppose
an - \jx\ an
Я21 a22 - l/x2
a\n
а„„ - l/xn

370
7 Topics in Orthogonal Polynomials
Then the coefficient of xx
coefficient of the same term in
k2
x2
xk" in the expansion of 1 / Vn is the same as the
(aux\-\ 1- alnxn)kl • ¦ ¦ (я„1*Н h annxn)k".
As an application of this theorem, consider the following example. An easy
calculation shows that
1
s+t) + -rst = -rst
1/2-1/r -1/2 -1/2
-1/2 1/2 - 1/s -1/2
-1/2 -1/2 1/2 -l/t
The Master Theorem implies that the coefficient of rksmt" in the series expan-
expansion [1 - (r + s + 0/2 + rst/2\~l is the same as the coefficient of rksmt" in
(r-s-t)k(-r+ s-t)m(-r -s + t)n/2k+m+n. The combinatorial interpretation
of this result is as follows. Take three boxes with k, m, and и distinguishable
objects in them. Rearrange these objects among the boxes so that the num-
number of objects in each box remains the same. Then the coefficient of rksmt"
in (r — s — t)k(—r+s — t)m (—r — s + t)" represents the number of rearrangements
where an even number of objects has been moved from one box to a different box
minus the number of rearrangements where an odd number of objects has been
moved to a different box. By Exercise 10 (where this coefficient has been obtained
as an integral of a product of three Laguerre polynomials), this coefficient must
be positive. Thus, we see that in the above combinatorial situation, the number of
"even" rearrangements exceed the number of "odd" rearrangements.
As another application of the Master Theorem, we outline Ismail and
Tamhankar's [1979] proof of Koorwinder's result in Theorem 7.2.7 for a =
0, 1,2,.... Let
poo
Ba(k, m,n)= / Lak{x)Lam((\ - k)x)Lan{kx)xae-*dx.
Jo
A simple calculation, using the generating function for Laguerre polynomials,
shows that
Ba(k m n)rksmt" -
kmn
- О " *)Г - ^ - krt - A - k)St + rst]«+l ¦
Since a is a nonnegative integer, B°(k, m, n) > 0 implies that Ba(k, m, n) > 0.
So we take a = 0. To apply the Master Theorem observe that
1 - A - k)r - ks - krt - A - k)st + rst
= -rst

7.3 Positive Polynomial Sums from Quadrature and Vietoris's Inequality 371
By the Master Theorem, B°(k, m, n) is the coefficient of rksmtn in
[A — k)r — \/k(l — k)s — Vkt]k[— -\AA — k)r
+ ks - Vl -kt]m[-Vkr - Vl -ks]n.
By applications of the binomial theorem, Ismail and Tamhankar show that
В (к, w, ft) = к A — A)
-I 2
> 0. G.2.14)
This proves Koomwinder's theorem for nonnegative integer values of a.
Note that Koomwinder's inequality implies that for 0 < к < 1, a > 0,
m+n
Lam{kx)Lan({\ - k)x) =
with ак,т,„ > 0. This relation can be iterated to give
G.2.15)
G.2.16)
k=0
with ak > 0 when a > 0, J]/=i A.,- = 1, and A, > 0, i = 1, 2,..., j.
Several proofs of the Master Theorem now exist. Perhaps the proof that best
explains its combinatorial significance is due to Foata [1965]. This proof was
later simplified by Cartier and Foata [1969]. A readily accessible treatment of this
argument is given by Brualdi and Ryser [1991]. A short proof using a multiple
complex variables integral was given by Good [1962].
7.3 Positive Polynomial Sums from Quadrature and Vietoris's Inequality
Fejer used the inequality
^ 1 - cos(n + 1N» sin2((n + 1N»/2)
1/2H= n .\Z = ¦ „ 1 ' > 0,
2 sin в /2
sin6»/2
to prove that the Fourier series of a continuous function is (C, 1) summable to the
function. This inequality can be expressed as
~ - -

372 7 Topics in Orthogonal Polynomials
In Section 6.7, we saw that a similar inequality holds for Legendre polynomials,
that is,
k=0 k=0
Fejer used this to study the summability of a series of spherical functions. He also
conjectured that
G.3.3)
These sums are partial sums of the Fourier series
Л -
which was studied because it illustrates the Gibbs phenomenon. It is possible that
the graphs of the partial sums suggested the conjecture to Fejer. We can write
G.3.3) as
E pA/2.,/2)m >°> 0<в<я. G.3.4)
The earliest proofs of G.3.3) are due to Jackson [1911] and Gronwall [1912].
Recall that G.3.2) was obtained from G.3.1) by using Mehler's integral in Section
6.7. There are other integrals in that section that give extensions to sums involving
It is possible to obtain positive sums with terms of either the form
\\) or P^\x)/P^'a\\). Without some applications in mind,
it is difficult to determine which extension is going to be useful. The inequality
in G.3.1) suggests that sums of P^a-P)(x)/P^-a)(I) may be important. Here we
consider a problem in quadrature that provides some confirmation of this.
Let {Pn(x)} be a sequence of polynomials orthonormal with respect to the
distribution da(x) on (a, b). As in Gauss quadrature discussed in Chapter 5,
interpolation is done at the zeros of the polynomials Р„{х), but now the integration
may be with respect to a different distribution. Let xv,v = 1,..., n, denote
the zeros of Pn{x). Let f{x) be a continuous function and let the interpolation
polynomial be given by
E, . , ., . V"^ Pn(x)f(xv)
tv{x)f(xv) := ^ • G-3.5)
v=l v=l r"(Xv)(X Xv)

7.3 Positive Polynomial Sums from Quadrature and Vietoris's Inequality 373
Then we have the approximate formula
where
I
^ = jb Pn(xW(x)
Ja P^(XV)(X-XV
= ["
Ja
v=l
G.3.6)
G.3.7)
Pn(x)Pn+i(xv) - Pn(xv)Pn+1(x)
Pn+l(xv)P^(xv)(x-xv)
dp(x)
К
±i
к Р'
1 rb "
- I / J2
'n(xv) Pn+\(XV) Ja k=Q
f2Pk(xv)Pk(x)dp(x).
Here kn is the coefficient of x" in Pn{x) and the last equation follows from the
Christoffel-Darboux formula (Theorem 5.2.4). Now write
k=0
Pk(x)
Then
K(xv)
К P;(xv)Pn+l(xv)
G.3.8)
G.3.9)
If Ли is positive, then it can be shown that the sum on the right-hand side of G.3.6)
converges to the integral as n —> oo. The proof given for Gaussian quadrature in
Chapter 5 works here as well.
Take the case in which Pn(x) = Р^ю(х), dfi(x) = dx, and (a, b) = (-1, 1).
Then
K(x) =
k=0
РГ'р'(х) / Pp
Ни
where h"'p is given by G.1.4). Write PJ;a'p\t) in hypergeometric form and inte-
integrate term by term to get
-k- l,k+a+p

374 7 Topics in Orthogonal Polynomials
This 2F\ can be evaluated by the Pfaff-Saalschiitz identity (Theorem 2.2.6). The
result is
A)Г(* + ?+1I (M
Г(р)Г(к + 2) \ к
This sum is intractable when written as
E^k
k p(aj
k=0 Гк
It can, however, be written in the form
[
P
rk
G.3.10)
Assume kn > 0 in G.3.9). By Corollary 5.2.6, it follows that
Thus to show that Xv > 0 in G.3.9), it is sufficient to prove that
ВД>0, -1<дс<1.
It is easy to check that
Bk + a + p + 1)Г(к + a + p)
Ck = (кП)\
satisfies 0 < ck+i < Ck when 0 < a + p < 1. So the nonnegativity of Kn(x) in
G.3.10) will follow for a, P>0,a+p<l upon summation by parts, provided
that
^>0, -1<,<1. G.3.11)
k=0 Гк У1)
Observe that D^'p)(—l) = 0 when n is odd. Thus G.3.11) is sharp in the sense
that equality holds at some point in [—1, 1] for infinitely many values of и. Proof
of the inequality G.3.11) for some values of (a, f}) is given in the next section.
Now suppose that dp(x) = A - x)a-y(l + x)P~sdx. We shall look at kv for
some specific a, fi, y, 8. When a = 1/2, fi = —1/2, у = 1, S — 0, the positivity

7.3 Positive Polynomial Sums from Quadrature and Vietoris's Inequality 375
of Xv, using the expression G.3.9), reduces to the positivity of the sum
1/2H.
The two cases
or = 0 =-1/2, у = 1/4, S = -1/4
and
a = p = 1/2, у = 3/4, S = 1/4
lead to the respective sums
cos kx, \c* sin fct, G.3.12)
k=0 k=l
where
^^, * = 0,1,2,.... G.3.13)
Vietoris [1958] proved the strict positivity of these sums for 0 < x < n. A proof
of these inequalities is given below. The inequality
> 0, 0<x<7t, G.3.14)
4=1
extends the inequality G.3.4) of Jackson and Gronwall. To see this let 9 -*¦ ж in
G.3.4). The result is
which vanishes when n is even. So it might appear that the inequality cannot be
improved. Now suppose that all we assume about the series in G.3.14) is that
1 = c\ > c2 > съ > • • •.
Divide the series G.3.14) by sinx and let x -* n. We obtain
1 - 2c2 + 3c3 - 4c4 + • • • + (-1Г+1исй. G.3.15)
For nonnegativity of this series for all n, we require that c2 < 1/2. Take the largest
value of c2 = 1/2. Then сз < c2 = 1/2 and the largest value of сз = 1/2. With
these values of c\,c2, and c3, we have 4c4 < 3/2 or c4 < 3/8. So take c4 = 3/8.
If we continue in this manner, we get the sequence q as defined in G.3.13).
As a first step in the proof of Vietoris's inequality, we show that the two sums
in G.3.12) are the partial sums of a Fourier series just as G.3.3) is.

376 7 Topics in Orthogonal Polynomials
Proposition 7.3.1 Ifck is the sequence defined by G.3.13), then
oo oo , j ч 1/2
ck ik V^ k I t(/2)
oo , j ч 1/2
= V^ ck coskx = I - cot(;t/2) 1 for 0 < x < 7Г. G.3.16)
4=1
Proof. For \z\ < 1, z ф 0, we have
Jt=O
It follows that
_ckz, |z|<l,z#±l.
k=0
Setz — elx,0<x < n, and take the real and imaginary parts to get the result of
the proposition. ¦
Proposition 7.3.2 For m>\,
G.3.17)
\ m 1 ¦Jinn
Proof. Set
'2m \
2^\m)-
It is easily seen that am < am+i for m > 1. Now observe that
lim am = lim
m—>oo /n—>oo
1 1
The proof of the next proposition is from Brown and Hewitt [1984].
Proposition 7.3.3 For ck defined by G.3.13), we have
> Vsin0-2cn+b G.3.18)
4=0
n
2 sin@/2) Y^ ck sinkO > Vsin0 - 2с„+ь G.3.19)
4=1

7.3 Positive Polynomial Sums from Quadrature and Vietoris's Inequality 377
Proof. Observe that for m > n
m m
2sin@/2) Y ckcosk0= Y c*[sin(fc + 1/2H - sin(fc - 1/2H]
k=n+l k=n+\
m-\
= —cn+i sin(n + 1/2H + VJ (ck — Ck+i) sin(k + 1/2H
k=n+l
+ cm sin(m + 1/2H
< cn+i(l - sin(« + 1/2H) - cm(l - sin(m + 1/2H)
< 2cn+i.
By Proposition 7.3.1,
00
Vsin0 = 2 sin @/2) Y ck COS ke
k=Q
= 2sm@/2) > acosA:0+ > ctcoskO \
\ k=0 k=n+l /
< 2sin@/2J_,ckcoskO + 2cn+i-
k=0
This proves G.3.18), and the proof of G.3.19) is similar. The proposition is
proved. ¦
We are now in a position to prove Vietoris's inequalities, which are explicitly
stated in the next theorem.
Theorem 7.3.4 If
/2k
C2k = C2k+l = Тп
then
and
rn(x) = Nc* cos fct > 0, 0 < x < n. G.3.21)
k=Q
Proof Consider G.3.20) first. The result is clearly true for n = 1. So let n > 2.
We need a separate argument for each of the three intervals: 0 < x < n/n,n/n <
x < 7Г — n/n, and 7Г — n/n < x < ж.
an{x) = 2_,ck sin/be > 0, 0 < x < jz, G.3.20)
k=l

378 7 Topics in Orthogonal Polynomials
The positivity of an (x) for 0 < x < л/п is obvious since each term in the sum
is nonnegative and the first term is strictly positive. When л — л/п < х < л, set
x = л — у so that 0 < у < л/п. Suppose n is even, say n = 2m. Then
2m m
Gn(x) — /_](— l)*~'cjt sin^y = y_4c2,t_i sinB/c — l)y — сг* sin2/cy]
/t=l k=l
[sinBk — l)y sin2ky~\
.
*-i -I
The last term in square brackets is positive because sin /// is decreasing in @, л]
and 2ky < 2my = ny < л. So <т„(х) > 0. When n is odd there is an extra term
in the sum, cn = sinwy, which is positive for 0 < у < л/п. Thus an(x) > 0
whether n is even or odd.
Now note that
sin и > и — м3/6.
For the interval л/п < x < л — л/п, which is nontrivial for n > 3, we then have
sinx > sinGr/«) > (л/п)(\ — л2/6п2).
By G.3.19)
2sin(x/2)<Tn(x) > [Gг/и)A - тг2/6и2)]1/2 - 2сй+1. G.3.22)
An easy calculation shows that the term in square brackets decreases for n > л /а/2.
So for n > 3, and by the definition of cn, the right-hand side of G.3.22) is positive
for n = 2m — 1, if it is positive for n = 2m. For the latter value of n, G.3.17)
implies that the right-hand side of G.3.22) is at least equal to
-тг2/24/и2I/2-2л/2].
A simple computation shows that this expression is positive for m > 2. This
proves the inequality in G.3.20).
The inequality G.3.21) is clearly true for n = 0 and 1. Moreover,
rj(x) = - cos 2л: + cosx + 1
7 1 ( П2 1
= COS X + COS X + - = I COS X + - I + - > 0.
Assume that n > 3. For 0 < x < л/п,
и n
У kck sin kx < 0, 0 < x < л/п.
dx ti

7.3 Positive Polynomial Sums from Quadrature and Vietoris's Inequality 379
So rn(x) is decreasing in 0 < * < n/n, and its value at n/n is positive. Note that
[n/2]
rn(n/n) =
- cn-k)cos(kn/n) > 01.
Thus rn(x) > 0 for 0 < x < n/n. Now let ж — n/(n + 1) < x < ж and set
у = ж — x so that
c2k[cos2ky - cosB?
en,
/t=0
where en = 0 if и = 2m — 1 and en = съп cos2my if и = 2ra. The expression in
the sum is positive because cos* is decreasing in 0 < x < n. This implies that
for n = 2m - 1, rn(x) > 0 for 0 < у < ж /п. When n — 2m, we have
rn(x) >
cosy + cos2y — cos3y + • • ¦ + cos2my)
cos 2* + • • •
= c2m Re
iBm+l)x _
- e~ix'2
sin(w + 1/2)* cos mx
~C2m sin(*/2)
cos(m + l/2)ycoswy
~ C2m cos(y/2)
It follows that rn(x) > OforO < (m+l/2)y < n/2, that is, for 0 < у < п/(п + \).
The rest of the argument can be completed as before. Suppose that n > 3 and
n/(n+ 1) < * < ж —n/(n + l). As in the case of an(x) опж/п < * < ж —n/n,
it is sufficient to show that
1/2
1 - ,. " ... ) - 2cn
Ж
n + l
6(n + IJ
+i
0.
Again it suffices to consider even values of n, say n = 2m. The inequality can
be directly checked for m = 2 and 3. For m > 4, apply G.3.17) to see that the
following inequality is stronger:
ж
2m + 1
6Bm + IJ
/nm
This is true for m = 4, and when the left side is multiplied by фп, it is an
increasing function of m. Thus the inequality holds form > 4, and the theorem is
proved. ¦

380 7 Topics in Orthogonal Polynomials
The next theorem, which is apparently a generalization of Theorem 7.3.4, is in
fact equivalent to it. It is also due to Vietoris.
Theorem 7.3.5 Ifa0 > fli > • • • > an > 0 and 2ka2k < Bk - 1)а2цД > 1,
then
sn(x) = ]Pfl* sinkx > 0, 0 < x < л, G.3.23)
k=\
and
n
tn(x) = Y^ak coskx > 0, 0 < x < л. G.3.24)
Proof. For ck as defined in Theorem 7.3.4, let a* = ckdk. Then d0 > d\ > d2 >
¦ ¦ ¦ > dn > 0 and summation by parts gives
sn(x) ~
k=l
n-\
= ^2(dk - dk+i)ak(x) + dnan(x) > 0, 0 < x < л,
by G.3.20). This proves the theorem since G.3.24) can be done in a similar
way. ¦
The Jackson-Gronwall inequality is a consequence of G.3.23). Just take
ak = 1/k.
There is a nice application of these inequalities of Vietoris to the problem of
finding sufficient conditions on the coefficients of trigonometric polynomials to
force all the zeros to be real, and then also yields information about the distribution
of these zeros. Szego [1936] proved the following theorem.
Theorem 7.3.6 If Xo > Xi > X2 > ¦ ¦ ¦ > Xn > 0, andsk and tk denote the zeros
of
р(в) =
k=Q
and
n-l
k=l

7.4 Positive Polynomial Sums and the Bieberbach Conjecture 381
respectively, with their order such that they are increasing in size on (О, л), then
= 1,...,и G.3.25)
±J I \ z/ V <V / \ z/
[п + -], к =
кп Пп+-\ <tk <(k+l)nU + -\, k=l,...,n-l. G.3.26)
If the Xk are not only increasing but satisfy the following convexity-type condi-
condition:
2Xq — X{ > X\ — X2 > X2 — ^.з^'-'^ Xn— i — Xn > Лл > О, G.3.27)
then the right-hand sides of G.3.25) and G.3.26) can be replaced by kn/n and
(k + \/2)n/n respectively.
The Vietoris inequalities can be used to obtain two other trigonometric inequal-
inequalities. See Exercise 17. These two inequalities along with the conditions
Bk - 1)Л*_1 > 2kXk > 0, ?=1,2,..., G.3.28)
lead to the following different improvements in G.3.25) and G.3.26):
k = l,...,n, G.3.29)
kn^n + -)<tk<(k+-jn/\[n+-), k=l,...n-L G.3.30)
For a proof of these inequalities see Askey and Steinig [1974].
7.4 Positive Polynomial Sums and the Bieberbach Conjecture
In the previous section, we saw the significance of showing the positivity of the
sums
The positivity of some of these sums has turned out to be important. We illustrate
this for some specific a and p, though much more is known. For more information,
see Askey [1975].
The strict positivity of G.4.1) for a = fi = 0, — 1 < x < 1 was proved in
Chapter 6, Section 6.7. This implies, after summation by parts, that for Legendre

382 7 Topics in Orthogonal Polynomials
polynomials Pk(x),
G.4.2)
when ak > пк+\ > 0, uq > 0,k = 0, 1,..., n — 1. The next result is due to
Feldheim [1963] and gives the positivity of G.4.1) when a = fi > 0.
Theorem 7.4.1 For 0 < в < n and v > 1/2,
Proof. By the Feldheim-Vilenkin integral (Corollary 6.7.3), we have for v > 1 /2,
^ 2Г(у + 1/2) Г/2 . 2v_2
Take ak = [1 - sin2 0 cos2 0]*/2. Then ak > a^+i > 0 and aQ = 1. So by G.4.2)
the integral is positive and the theorem is proved. ¦
The Jackson-Gronwall inequality
E
sinkO
> 0, 0 < в < тг, G.3.3)
is a corollary of Theorem 7.4.1 when v = 1.
The positivity of G.4.1) for fi = 0 and a = 0, 1, 2,... was needed in the first
proof of the Bieberbach conjecture on univalent functions. See de Branges [1985].
More generally, take a > — 1.
Since Pn(Oa)(l) = 1, the sum we are interested in is
k{afi\x), a>-\. G.4.4)
The first step in the proof, due to Gasper, of the positivity of G.4.4) is to express

7.4 Positive Polynomial Sums and the Bieberbach Conjecture 383
it as a hypergeometric series. Thus
(-\У((\-х)/2у
j^. (k-j)l
(a + lJj((x - l)/2)J yl (a + 2y + 1),
Since the inner sum is
(a + 2j + 2)n_j
(n — j)\
we have
+ 2)„ л,„+о + 2,(а+1)/2 1д:
^ ;G-4-5)
n\ *\ (a + 3)/2,a+l ' 2
There is a formula of Clausen that gives the square of a iF\ as a 3 F2 (see Exercise
3.17). The formula is
ЗГ21 - ¦ '- ¦ 1/2,2a
This 3F2 is nonnegative because it is a square. The 3F2 in G.4.5) is fairly close to
this but different in one numerator and one denominator parameter. We have seen
before that by fractional integration of a pFq, it is possible to get a p+\ Fq+\ with
the necessary extra parameters. So use formula B.2.2) to write the 3F2 in G.4.5)
as
/ —n, n + a + 2, (a +
iF2\ (a + 3)/2,a+l
; st
G.4.7)
for a > — 1. The 2F1 in the integral is really the ultraspherical polynomial
CaJ2(l-2st)/CaJ2(l).
This has zeros in @, 1) and so we do not have a positive integrand in G.4.7). To
get an idea about what should be done, write the 3F2 in Clausen's formula with

384 7 Topics in Orthogonal Polynomials
2a = -k, 2b = к + a + 1. Then
Г (а + 1)
G.4.8)
The proof would be complete if it is possible to write C2 (x) in terms of C(" ~1)/2 (x)
using a positive coefficient. However, this is obtainable from the connection co-
coefficient formula G.1.9). Thus we have proved the next theorem.
Theorem 7.4.2 Fora > -1,
P^0)(x) > 0, -1 < x < 1. G.4.9)
*=o
The integral in Theorem 6.7.2(b) can be applied to G.4.9) to give the positivity
of G.4.1) for -1 < x < 1 when 0 > 0, a + 0 > -1. For a > -0, -1/2 <
0 < 0, Gasper [1977] has shown the positivity for —1 < x < 1, except when
a = —0 = —1/2, in which case this sum reduces to G.3.1), when there are cases
of equality as well as nonnegativity.
7.5 A Theorem of 1\iran
In the last section we proved the Jackson-Gronwall inequality. There is a theorem
of Turan [1952] that shows another way of doing this.
Theorem 7.5.1 IfY^jLo \aj I converges and
a,sin(y + 1/2H >0, 0<ф<п, G.5.1)
> 0, 0 < в < it, G.5.2)
then
^ sing + pa
7=0 J
unless aj = 0, j = 0, 1, 2,
Proof. In the integral formula given by Theorem 6.7.2(d), take a = 1/2, 0 =
— 1/2, and (i=l. This gives
sin(n + 1N» [*/г sinBn
2(и + l)(sin@/2)J"+2 "

7.5 A Theorem of Turan 385
Multiply by 2an(sin@/2)Jn+2 and sum to obtain
sin(n + \)в fn/1 y> /sin@/2)\2n+2 an sin(n + \/2){2ф)<1ф
Write the integrand as
sin Ф „=o
sin2
The strict positivity of this sum for \в < ф < \ж follows from the fact that
n=0
2 Z171
= - /
G.5.4)
This formula is obtained by using the orthogonality of the sine function. Now note
that the closed form of the Poisson kernel,
rnsin(n + - Usin(/i + -W
n=0 \ / \ /
A — r)sin i^sin |V [0 ~ rJ + 4r(l — cos \{ф + if)cos \{ф — if))]
[l — 2r cos i(<? + VO + r2] [l — 2r cos |@ — if) + r2]
G.5.5)
gives its strict positivity for 0 < r < 1. This and G.5.1) make the integrand in
G.5.4) nonnegative, and the theorem is proved. ¦
There is a generalization of G.5.5) to Jacobi polynomials due to Watson. The
result states that
n
+ 2)/2)m+n((a + fi + 3)/2)m+n
(a + l)m(/3+ l
Dr sin2 ф sin2 в)тDг cos2 ф cos2 0)"
A+rJm+2n •

386 7 Topics in Orthogonal Polynomials
where h^ is given by G.1.4). We base our proof on a result of Bailey. For the
result, see Bailey [1935, p. 81] and for the reference to Watson see Bailey, p. 102,
Example 19. Bailey's formula is the following:
2Fi(a,/3;]/;*JFi(«,/?;« +/3 + 1 -y\y)
= y- (a)m+n(fi)m+n[x(l - y)T[y(\ -x)T
tt (y)m(a + p + I - y)nmln\
We sketch a proof of G.5.7) leaving some details to the reader. Start with the
double series
h
G.5.8)
Expand in powers of s and t and show that the coefficient of smt" is
- /)mO + I ~ У)п(У ~ l)»-n
When y + )/' = a + j6 + l, the last factor in the numerator cancels the last factor
in the denominator and so G.5.8) is equal to
i* rn\n\(y)m(y')n
= 2Fi(a, Y-P;r, sJFi(P, y' - a; y'\ t)
, P; y; -s/(\ - j)JFi(a, /3; /'; -
The last step follows from Pfaff's transformation given in Theorem 2.2.5. This
proves G.5.7).
The left side of G.5.6) can be written as
rk. G.5.9)
First consider the simpler series
Replace the Jacobi polynomials by their hypergeometric representations and apply

7.5 A Theorem of Turan 387
G.5.7) to get
к
-k,k+a+P + l 2
p+i ;cos
+ p + i). (-k)m^,(k +a + fi + l)m+n
к[ tT ( + DO + D
(sin 0 sin 0Jm (cos 0 cos 0)
OS20)" + y, (q + ^ + l)^+2m+2n
m+n y, (q + ^ + l)^+2m+2n .
(r sin2 в sin2 0)m (r cos2 в cos2 0)"
¦ (a + ^ + lJm+2n(l + r)-2"-2"—"-1.
Now multiply G.5.10) and the last expression by r(a+^+D/2 and take the derivative.
This introduces the factor Bk + a + fi + 1) needed on the left side of G.5.9).
The right side of Watson's formula follows after an easy calculation. This proves
G.5.6).
Theorem 7.5.2 If p > a > -1 and
Yak \ ; ' >0, -\<x<\,
unless cik = 0, & = 0, 1,..., n.
Proof. As in the proof of Theorem 7.4.1, use the Feldheim-Vilenkin formula to
get
•л/2
2Г(/} + ° Г si
-a)r(a + l)J0
з(",а)/ /,,1 -2 п „„„2 J,\ — 1
¦ 2O 2 ,lk/2Pk (COS в (I -Sin2 в COS2
sin20cos2^2^ ——
k=o rk

388 7 Topics in Orthogonal Polynomials
Let r = A - sin2 в cos2 ф)^2 and и = cos6A - sin2 в cos2 0)~1/2. Then for
0 < 0 < л and 0 < ф < л/2, we have 0 < r < 1 and |w| < 1. We can conclude
from G.5.6) that the sum inside the integral is strictly positive unless f(x) = 0.
We know this because
ёг^л
hf
Now observe that orthogonality of the Jacobi polynomials implies that f(x) = 0
for -1 < x < 1 if and only if a* = 0, /fc = 0, 1,2,..., и. This proves the
theorem. ¦
7.6 Positive Summability of Ultraspherical Polynomials
The (C, 1) means of the formal series 1+2 J2b=i cosk6 are
v^ / к \ 1 fsini(n+lH)
» =1 + 2? 1 cos?<9 = I 2 . } > 0. G.6.1)
Fejer used this positivity to prove the (C, 1) summability of the Fourier series of
a continuous function. The generating function for the sequence <7n' in G.6.1) is
A — r)~2( l + 2Vcosn<9 rn]=—— r. G.6.2)
v \ ^ ) \-r 1-2rcos0+r2
The last equality follows from E.1.16). Now observe that
, n > 0,
1, и = О,
and the generating function for {^^C^cos в)} is
^ ^—Cnv (cos 0)r" = — Yl^I- G.6.3)
This follows from the generating function for ultraspherical polynomials given in
Chapter 6, namely,
v СЦсо%в)гп = -— 1 G.6.4)
' A -2rcos# + Г2)"
n=0
Kogbetliantz [1924] proved the following generalization of G.6.1). For a discus-
discussion of Cesaro summability of infinite series, see Appendix B.

7.6 Positive Summability of Ultraspherical Polynomials 389
Theorem 7.6.1 The (C, 2v + 1) means of the formal series
-Cv(x) -1 < x < 1 v > О
are positive. That is,
, (k -I- d)
-C(jc)>0, -1<jc<1,v>0. G.6.5)
The proof depends on the following lemma.
Definition 7.6.2 A function is called absolutely monotonic if its power series has
nonnegative coefficients.
Lemma 7.6.3 Thefunction 1/[A —rJv(l —2xr+r2)v] is absolutely monotonic
for-I <x < 1.
Proof. Denote the function by [g(r)]v and let x = cos в. Then
A(r) :=log*(r) = -21og(l -r) -log(l -rew) - log(l - re"'6*)
2cosn#) n
n=l
Thus A (r) and
00
vn[h(r)]n
„=o и!
are absolutely monotonic. ¦
Proof of Theorem 7.6.1 The generating function for G.6.5) is
1-r2
1-r2 1
A - rJ(l - Ixr + r2) A - rJv(l -2xr+ r2)v'
The first factor is absolutely monotonic by G.6.1) and G.6.2). The second factor
is absolutely monotonic by Lemma 7.6.3. This proves the inequality G.6.5) and
the theorem. ¦
The lemma has the following corollary.
Corollary 7.6.4 For v > 0, -1 < x < 1,

390 7 Topics in Orthogonal Polynomials
Proof. This follows from G.6.4) and Lemma 7.6.3. ¦
Similar results have been obtained for Jacobi series. The results are very im-
important, but the proofs involve quite complicated formulas. See Gasper [1977] for
many of these inequalities. For example, Gasper proved the following extension
of the inequality in Exercise 14:
f(--y _l<x<
0<A.<a + /6 and «>?>() or « + ?>() and ?> -1/2. Proofs of this
inequality use many of the formulas given here for hypergeometric series, as well
as some others. Note the particular case A. = 0:
G.6.8)
for a + fi > 0, fi > —1/2. Inequality is strict in -1 < x < 1, when a = 1/2,
fi = —1/2 are excluded.
These inequalities have implications for Bessel functions. Using Theorem
5.11.6, it is easy to see that for a > fi - 1
ton - X] * (/,) = 2"г^ + !> / *~PJ«(t)dt. G.6.9)
f0
/
The condition a > ft — 1 is needed for convergence of the integral at t = 0, but it
is not needed for the sum. In an appendix to the paper of Feldheim [1963], Szego
considered the limit when a = /3, and also the resulting integral. He showed that
/ raJa(t)dt>0, x>0 G.6.10)
Jo
when a > a and when a is the solution of
ГЛ..2
Г" Ja(t)dt = 0 G.6.11)
with ja<2 the second zero of Ja(t). A similar result holds for
rpJa(t)dt, -1<а<1/2.
Jo
/0
From G.6.8), we have
Г
t~fi Ja(t)dt > 0, x>0 G.6.12)
/o
when a + ? > 0, -1/2 < /3 < 0. In fact, G.6.8) also holds when a + /? > -1,
/3 > 0. Thus G.6.12) holds when a > /S - 1 and /3 > 0. Gasper [1975] has

7.7 The Irrationality of t, C) 391
shown that the inequality G.6.12) is strict for these a, ft except when a = 1/2,
/5 = -1/2.
Gasper [1975] derived the following interesting identity when /J = —1/2 and
a > -3/2 in G.6.12):
1/2j (t)dl _ Lr(f + 4)] 2a+1x
G.6.13)
This series is clearly positive when a > 1/2, nonnegative when a = 1/2, and
negative when x is a zero of JBa+i)/4(x/2), where —3/2 < a < 1/2. Gasper also
has a similar result for
f (x- t)a'l/2taja(t)dt, a > -1/2.
Jo
See Exercise 25. This and G.6.13) are derived by using the identity
Y2(v + lJ2v+1 ^ (n + v)r(n +2v)
x = > /„.„(*)¦ G.6.14)
See Watson [1944, §5.1] for G.6.14). An approach to G.6.12) via differential
equations can be found in Makai [1974]. For more references, see Askey [1975]
or Gasper [1975].
7.7 The Irrationality of <C)
The topic of this section is unrelated to the earlier parts of this chapter. However,
it involves an interesting application of Legendre polynomials and so has been
included here.
In Chapter 1, we gave Euler's formula for ?Bn) when n is an integer >1. This
showed that ?B«) is irrational. In spite of repeated attempts, Euler was unable to
evaluate ?B« + 1). Later mathematicians have not had better luck. It was only
as recently as 1978 that R. Apery proved that ?C) is irrational. A simpler proof
using Legendre polynomials was given by Beukers [1979]. We follow Beukers's
exposition.
The basic lemma is the following:
Lemma 7.7.1 There exist two sequences of integers {An} and {Bn} such that
0 < \An + BBf C)| < 3(9/10)". G.7.1)
This immediately implies the theorem.

392 7 Topics in Orthogonal Polynomials
Theorem 7.7.2 ?C) is irrational.
Proof. If f C) = p/q, where p and q are integers, then the sequence of nonzero
rational numbers \An + В„?C)| > \/q. The second inequality in G.7.1), how-
however, implies that this sequence becomes arbitrarily small as n increases. This
contradiction implies that ?C) is irrational. ¦
The proof of Lemma 7.7.1 depends on the following lemmas.
Lemma 7.7.3 For nonnegative integers r ands,
/7/^(ш)Ер) r=s
o Jo ^ху { frtk )
= rational number whose denominator divides df,
where dr = lcm(l, 2,..., r), when r > s.
Proof. Note that for о > 0 and r > s
к к l-xy У f^{k + r+o + \){k
-s \k+s + a + 1 k + r+a +
G.7.2)
r—s{s + l+a r + а
Differentiate with respect to a and then set a = 0 to get
1*1 /*i \nn v,, л ( \ j
The second equality of the lemma follows from this. Now take r = s in the first
equation of G.7.2) and again differentiate with respect to a and set a = 0. The
result is
ri /.l
f1 f'logxy^
/ / x у dxdy =
к Jo i-xy ¦" ¦"
This is the first equation of the lemma, which is now completely proved.
For the next lemma, let
Рп{х)=-.~{ХП(\-Х)п},
n\ dxn
which is essentially the Legendre polynomial on @,1).

7.7 The Irrationality of t, C) 393
Lemma 7.7.4 There exist integers An and Bn such that
Jo Jo 1 -
xy
Here dn = lcm(l, 2,..., и).
Proof. Since pn (x) is a polynomial with integer coefficients, the equality follows
from Lemma 7.7.3. Now observe that
~ i = / ~i 7i z~dz.
L-Xy Jo 1-A-.
Then the integral in the lemma can be written as
/ / / 1 7, —,dxdydz-
Jo Jo Jo l ~
Integrate by parts n times with respect to x to see that this integral is equal to
dxdydz.
fff
Jo Jo Jo
A - A - xy)z)n+l
Set
1-z
со =
and rewrite the last integral as
/ f f (l-x)n(l-co)n- f"(y) dxdyd
Jo Jo Jo 1 - A - xy)co
A A A ^(! -x)y(l -y)co{\ -CO)}"
-111
Jo Jo Jo
dxdydco. G.7.3)
-A -xy)co}"+1 У У
The last equality uses integration by parts n times. It is clear that the final integral
is nonzero and so the lemma is proved. ¦
Lemma 7.7.5 For An and Bn as in Lemma 7.7.4,
0 < \An + Bni;{3)\d~3 < 2(V2 - 1L4C).
Proof. The integral in Lemma 7.7.4 is equal to integral G.7.3). We find the
maximum value of the integrand in G.7.3). Let
x(l - x)y(\ - y)co(\ - со)
f(x, у, со) = .
1 — A — xy)co
By solving the equations
V = V = 9/ =0
дх ду дсо

394 7 Topics in Orthogonal Polynomials
it is easy to see that at the maximum, x = уапйш = l/(l+x). Thus / is bounded
byx2(l -xJ/(l +xJ, which is maximized at x = л/2 — 1. Thus integral G.7.3)
is bounded above by
1 Г1 1
L
Jo Jo 1 - xy
= 2(v/2-lLnfC).
This proves the lemma. ¦
We need the following result from elementary number theory due to Chebyshev.
Let n{n) denote the number of primes less than n. Then
тг(л) < 1.06n/logи. G.7.4)
See Ingham [1932, p. 15].
Proof of Lemma 7.7.1. Observe that
dn = IT n[lo8"/logp] < TT pl°g"/l°gp — пя<")_
p<n p<n
p=prime
By G.7.4), it follows that
dn <ni-°6"/iog« =ei06n < 3«
An upper bound for ? C) is given by
dx 3
Therefore, Lemma 7.7.5 gives
(V2+1L
Apery's proof used some sequences satisfying three-term recurrence relations.
For a lively account of Apery's proof, see van derPoorten [1979]. These sequences
were analyzed in terms of contiguous relations by Askey and Wilson [1984]. See
Exercises 28 and 29.

Exercises 395
Exercises
1. Let
n
x(x — 1) • • • (x — n + 1) = yj.s(n, r)xr
r=0
and
n
x" = ^2 S(n, r)x(x - 1) • • • (x - r + 1).
r=0
The integers s{n, r) and S(n, r) are called Stirling numbers of the first and
second kind respectively. Show that
(n, r)s{r, m) = Snm.
Use this to prove that
an = \ s(n, r)br, n > 1,
r
if and only if
bn = 2 S{n, r)ar, n > 1.
r
2. Derive formulas G.1.14) and G.1.15) as indicated in the text.
3. Verify formula G.2.7).
4. Prove that
2 (
Deduce G.2.10).
5. Supposew(x) isapositive integrable function on (—1,1); {pn(x)} a sequence
of polynomials orthogonal with respect to w(x); and {^„(x)} orthogonal with
respect to A + x)w(x). Let /?„A) > 0 and qn(l) > 0. Prove that
(l+x)qn(x) = Anpn+i(x) + Bnpn(x),
where An and Bn are positive. Determine An and Bn in the case illustrated
in G.2.10).
6. Let {pm{x)} be a sequence of polynomials orthogonal with respect to a dis-
distribution da(x) in @, oo). Let ?i, ?2, •••.?* be zeros of pm(x) and let
fk(t)= e-xt№-x)---($k-x)}-l{pm(x)}2da(x), t > 0.
Jo
(a) Show that
f
Jo
f
Jo
(b) Deduce that fk(t) > 0 for t > 0.

396 7 Topics in Orthogonal Polynomials
7. With {pm(x)} as in the previous problem, assume that pm@) > 0. Prove that
e-xtpm(x)pn(x)da(x) > 0, t > 0.
For Exercises 6 and 7, see Karlin and McGregor [1957, pp. 507-509].
8. Deduce Lemma 7.2.3 from the previous problem.
9. Prove Corollary 7.2.4.
10. (a) Prove that
Г LUx)Lam(x)Lan(x)x"e-2"dx
E
rksmt"
Г (a + 1)
k,m,n=0
= [2 - (r + s + t)
(b) Use the multinomial theorem to prove that, except for a constant factor,
the right side of (a) also equals
^- k\m\n\
k,m,n>0
( -k,-m,-n \
'3 2 \ (-a - ifc - m - л)/2, A-а-к-т-п)/2' )'
(c) Let A: < min(w, n). Reverse the order of summation in the above 3F2 to
get
{-\fm\n\2lkT(m + n + a + 1 - к)
(m - k)\(n - k)\T{m +n+k + a + l)
/ -k, (a + I + m + n - k)/2, (a + 2 + m + n - k)/2 \
'32V m+l-k,n + l-k ' )'
(d) Apply Kummer's transformation (Corollary 3.3.3) to the 3F2 in (c) to
show that the expression in (c) is positive for a — 0, 1, 2, This
proves G.2.13) for these values of a.
11. Prove that
(a) Vy ? ein(it
A
12. (a) Deduce from G.3.1) that
n
1H >0
(b) Use the results in the previous problem to prove that
n
'Yjri + 1 - k) sin(k + 1H > 0 forO<0<7r.
k=0

Exercises 397
13. LetSn(x) = Еь=1 п^- Use induction and the fact that the extrema of Sn(x)
lie at Ikn/n, where к is an integer, to prove that Snix) > 0 forO < x < it.
14. Use Theorem 7.5.2 and Corollary 7.6.4 to prove that
J^-л Bv)n-kC2v)k Clix)
> -. > 0, — 1<х<1, A>v>0.
Z-/ (n — jHli-l Cx(l} ~ _ _ > _
/ n V IV ^ 1IV 1 \s К \ I }
Deduce that
> 0, 0 < a < 2, 0 < в < л.
k=o ^ -
Consider the cases a = 1, 2.
15. Define the difference operator ДА by (ДА/)@ = (f(t+h)-f(t))/h. Show
that (-/0*4ДА/)@ = Е?=о(~1)?(*)/(*+?А)> where Akh is the Ath iterate
of Ah.
Suppose / is continuous on [0, 1] and /A) = 1. Prove that the following
statements are equivalent:
(a) / is absolutely monotonic in [0,1], that is, fit) = Eo° a«f"' й„ > 0, f e
[0, 1];
(b) / G C°°@, 1) and /№)@ > 0 for к = 0,1, 2,... and t e @, 1);
(c) (Д^/п/)@) > 0 for n = 1,2,... and 0 < m < n.
One way to show that (c) =»¦ (a) is to use the Bernstein polynomi-
/
which uniformly approximate / in [0,1]. To prove the last equality, use
the result at the beginning of this problem.
16. Let ao > a\ > • • • > am > 0 and 1 < n < m. Show that
m n — \
^atcos^sin(e/2)>^atcosA:esin(e/2) - (an/2)(l + sin(n -
k=0 ft=0
and
m n—\
^atsinJfc0sin@/2)>^aifcsinJfc0sin@/2) - (an/2)(l - cos(n -
k=0 k=0
(Vietoris)
17. Let ck = 2~2kBk). Show that for 0 < x < 2n
t ck sinik + l/4)x > 0 and 2^ c*С08^ + г/4)х > 0.
k=0 k=0
Deduce that the inequalities hold if ck is replaced by ak, satisfying
Bk - l)ak_i > 2kctk > Oforifc > 1.
18. Let ak satisfy the conditions given in the previous problem. Prove that if
0 < v < 1/4 and 0 < x < 2л- or if -1/4 < v < 1/4 and 0 < x < n, then
ak cos(fc + v)x > 0.
k-0

398 7 Topics in Orthogonal Polynomials
19. With ak as in Exercise 17, show that if 1/4 < v < 1/2 and 0 < x < 2n or if
1/4 < v < 3/4 and 0 < x < n, then
У^ v)x > 0.
/fc=0
20. Show that
J^ cos fcx
l+> —-—> 0, 0<х<л\
21. With c;t as in Vietoris's inequalities, show that
for v > 0 and 0 < x < 1.
22. Show that if
*:=0
then
Observe that this implies the terminating form of Theorem 7.5.1.
23. Prove that if
k=0
then
n p(a-li,P+li)( ч
24. (a) Show that [о^(с; x)]2 = iF2(c - 1/2; 2c - 1, c; Ax).
(b) Prove the formula of Bailey that
/2x рл/2
ha(t)dt = 2x / [Ja(x sin<p)f sin<pd<t), a > -1/2
Jo
(c) Show that /* Ja(t)dt > 0, a > -1.
25. (a) Use F.14) to show that
Г >¦ м
Jo
Г (A. + 1)Г(а + ii + 1)Г2(у + 1)л4у_а
^ Г -n,n + 2v, v+ l,(a + fi+ l)/2, (a + д + 2)/2 1
^5^4 [v + 1/2, а + 1,(а + А. + д + 2)/2, (a + A. + ц + 3)/2' J
Bv + 1)„ 2n+2v 2
n\ n + 2v n+v\2J'

Exercises 399
when a + д > —1, A. > —1, and 2v Ф —1, —2,... and the factor
Bл + 2v)/(n + 2v) is replaced by 1 at л = 0.
(b) Take д = Л + 1/2 and set v = (a + A. + 1/2)/2 so that the 5F4 reduces
to a balanced 4 F3.
(c) Take A. = 0 to get F.13).
(d) Take A. = a - 1/2 to get
f\x-t)a-xl2taJa{t)dt
Г(а + 1/2)ГBа
з,,
Г(За + 3/2)
l)/4)n(Ba - l)/4)n Ba
+ 1)„ 2и + 2a 2 /xn
n! n + 2a n+2a\2J'
fora > —1/2. (Gasper)
26. Suppose that {/?n(x)} and {qn(x)} are orthonormal polynomials associated
with the weights co(x) and ш\ (х) respectively. Prove that if
n
qn(x) = У^уск,прк(х),
ifc=0
then
00
ш(х)рк(х) = У^ск,пд„(х)йI(х).
n=k
27. Use Exercise 26 and Theorem 7.1.4' to show that for —1 < x < 1 and
H > (A. - l)/2,
where
Ck,n
2k + A.)(w + 2^)!Г(п + 2д)Г> + fc + Х)Г(к + A - д)
~ Г(А.- д)Г(у)л!А:!Г(л + А: + д+ 1)Г(л + 2A: + 2A.)
Note that c^ > 0 for (A. — l)/2 < д < A.. Deduce the special case
00
CM sin(" + 2k + Vе
k=0
when д > 0, д ^ 1, 2,..., and
n + *)!Г(л + 2ц)Г(к + 1 - д)
28. Show that

400 7 Topics in Orthogonal Polynomials
satisfies the three-term recurrence relation
пъЪп + (л - 1K2>„_2 = C4n3 - 5In2 + 27n - 5J>„_,.
(Apery)
29.
A *+* A
satisfies a three-term recurrence relation. Find it.
30. Complete the proof of Bailey's formula G.5.7).
31. Use G.5.7) to prove Brafman's [1951] generating-function formula for Jacobi
polynomials,
^ (y)n(tt +1 - у + l^f
= 2F1 (y, a + P - у + 1; a + 1; A - r - R)/2)
¦ 2Fdy,<x + P - у + 1; $ + 1; A + r - R)/2),
where
R = A -2xr + r2I/2.
The case у = a is interesting.
32. Complete the proof of G.2.14).
33. Show that, if a > -1Д, w, n = 0, 1,..., then
/¦00
-Vx > 0.
/¦0
/
34. If a > 0, j,k, m, n — 0, 1, 2,..., then prove that
Laj(x)Lak(x)Lam(x)Lan(x)xae-2xdx > 0.
/
Jo

8
The Selberg Integral and Its Applications
Dirichlet's straightforward though useful multidimensional generalization of the
beta integral was presented in Chapter 1. In the 1940s, more than 100 years after
Dirichlet's work, Selberg found a more interesting generalized beta integral in
which the integrand contains a power of the discriminant of the n variables of
integration. Recently, Aomoto evaluated a yet slightly more general integral. An
important feature of this evaluation is that it provides a simpler proof of Selberg's
formula, reminiscent of Euler's evaluation of the beta integral by means of a
functional equation. The depth of Selberg's integral formula may be seen in the
fact that in two dimensions it implies Dixon's identity for a well-poised 3F2.
Bressoud observed that Aomoto's extension implies identities for nearly poised
3-F2-
After presenting Aomoto's proof, we give another proof of Selberg's formula
due to Anderson. This proof is similar to Jacobi's or Poisson's evaluation of Euler's
beta integral in that it depends on the computation of a multidimensional integral in
two different ways. The basis for Anderson's proof is Dirichlet's multidimensional
integral mentioned above. A very significant aspect of Anderson's method is that
it applies to the finite-field analog of Selberg's integral as well. We give a brief
treatment of this analog at the end of the chapter.
Stieltjes posed and solved an electrostatic problem that is equivalent to obtaining
the maximum of an n variable function very closely related to the integrand in
Selberg's formula. Stieltjes's remarkable solution showed that the maximum is
attained when the n variables are zeros of a certain Jacobi polynomial of degree
n. We devote a section of this chapter to Stieltjes's work and show how his result
can be combined with Selberg's formula to derive the discriminants of Jacobi,
Laguerre, and Hermite polynomials.
Siegel used the discriminant of the Laguerre polynomial to extend the arithmetic
and geometric mean inequality. Siegel's result, which we include, contains an
interesting inequality of Schur relating the arithmetic mean and the discriminant.
401

402 8 The Selberg Integral and Its Applications
8.1 Selberg's and Aomoto's Integrals
The theorem given below contains Selberg's [1944] extension of the beta integral.
Theorem 8.1.1 Ifn is apositive integer and a, /3, у are complex numbers such that
Rea > 0, Re? > 0, andRty > -min{l/n, (Rea)/(n - 1), (Rtp)/(n - 1)},
then
/ f
/о Jo
п
Sn(a,p,y)= f
JO
Г (a + U ~ 1)У)Г@ + U ~ 1)У)ГA + jy)
7 = 1
where
А(х)=
The conditions on a, fi, у are needed for the convergence of the integral. For a
discussion of this, see Selberg [1944]. Note, however, that the condition on у is
related to the first occurrence of a pole of the function on the right-hand side of the
integral formula. Selberg's proof of this formula appeared in 1944, but for more
than three decades it was not well known. More recently, Aomoto [1987] found a
simpler proof which depends on a recurrence relation satisfied by a slightly more
general integral.
Theorem 8.1.2 With the same conditions on the parameter a, ft, у and with
к < n,
-Д- (a + (n- j)y)
Anderson [1990] proved a finite-field analog of Selberg's formula and then
noted (Anderson [1991]) that the idea could be carried over to the continuous
case.
8.2 Aomoto's Proof of Selberg's Formula
To motivate this proof, recall the two basic steps of the proof of the formula

8.2 Aomoto's Proof of Selberg's Formula
403
Step 1. Obtain the functional equation fi(a, B) = ^fi(a, P + 1). Though we
did this differently in Chapter 1, it can be done as follows: For Re a > 0, Re p > 0,
/1 g
— [xa(l-xf]dx
dx
r\ 1-Х
= a\ xa~\\ -xfdx-B / xa(\-xf-ldx
Jo Jo
a-\l-xfdx-p xa-\\-xf~xdx
Jo
Step 2. Iterate the first step n times to get
B(a,p+n).
Then apply a change of variables in the integral for B(a, /3 + n) and let n -> oo
to obtain the necessary result.
We apply the same basic procedure to prove a generalization of Selberg's for-
formula. Let
w(x) =
xif-1 J]
and
h =
(8.2.1)
(8.2.2)
1=1
where Cn is the n-dimensional cube and dx = dx\dx2 ¦ ¦ ¦ dxn. By symmetry the
product nf=i xi таУ be replaced by the product of any к distinct variables without
changing the value of the integral. Let Iq denote the integral without the factor
To obtain the functional equation start with
к
dx
С
i=2
= «/A-
Jcn
+ 2y±[ A-
j=2 JC»
к
w(x)dx
Xl)^
- X,
(8.2.3)

404 8 The Selberg Integral and Its Applications
The third term in (8.2.3) is derived from the fact that
\\c \rlC
if jc
\x\ c\xrsgnx
ax x
The next lemma shows that the third integral can be written in terms of h and h-\-
Lemma 8.2.1
L 0 if2<j<k,
L
L
с„ m-xj K.\h-\ ifk<j<n.
\h if 2 < j < k,
lCn x\ - xj у ik if к < j <n.
Proof.
(a) In the first case, where 2 < j < к, the transposition x\ ¦«->¦ xj changes the
sign of the integrand, so the integral vanishes. In the second case the same
transposition leads to
Xl Xj X\
X\ — Xj Xj — X\ X\ — Xj
so
Г UkiXiW(x)dx f Л
/ -1 =/ \\xiw(x)dx = Ik-i.
JCn XX - Xj JCn ^
(b) For 2 < j < к, the transposition x\ *> Xj gives
Л
xxx)
= X\Xj
X\ — Xj Xj — X\ X\ — Xj
which proves the first part of (b). For the second part observe that
x\ _ XXXj
— x\ +
X\ - Xj X\ - Xj
and the last term changes sign in the transposition x\ *> Xj. So its presence
in the integral makes that part zero. The other part coming from x\ gives the
necessary result. Thus the lemma is proved. ¦
Use this lemma to rewrite (8.2.3) as
0 = or/t_, - (« + 0)Ik + Y{n - k)h-x - YBn -k
This gives
a + fi + {In - к - \)y

8.2 Aomoto's Proof of Selberg's Formula 405
Iterate this functional relation to arrive at
к
/
П Aln
Г=1 + ^ + ( n ~ l ~~
The last integral is Selberg's integral, which we write as Sn(a, fi, y). The problem
now is to evaluate this integral. Fortunately, it is possible to apply the functional
equation (8.2.4) to Selberg's integral itself. Note that
Sn(a + 1, в, у) = I I Sn(a, в, у). (8.2.5)
nv .^>/y 11 a _(_ ^ _|_ Bn — у — l)y 'f''-" v -»
Symmetry in a and ft and iteration give
Sn(a, Д y) = -— ljL^5n(«, уб + к, у)
пК ,Р,П 11 (P + (n-j)y)k 'P 'У>
х + Р + Bп — j — \)у)к
08 + (и - У
п
к
к"
Let /: -> oo and use the limit definition of the gamma function to get
/¦00 /-00
/
/¦00 /-00 "
•/ ¦••/ П^е П \xi-*i\lYdx. (8.2.6)
Denote the integral in (8.2.6) by Gn (a,y). Then by symmetry in a and ft, relation
(8.2.6) implies that
П^г(« + <1.-яу) П-=1
Thus we can write
'(a + (n - j)y)F(P + (n - j)y)
Г(а+^ + Bя_^_1)у) W (8.2-7)

406 8 The Selberg Integral and Its Applications
To compute Dn (y), first note that by the symmetry in the variables х\,хг, ...,х„
f /"' /"' Г1
/ w(x;a,p,y)dx=n\ / •••/ w(x;a, fi,y)dxx- ¦ -dxn. (8.2.8)
JCn J0 Jxn Jx2
Since
/•i
lim a I ta~xf(t)dt = /@)
«^o+ Jo
for a continuous function / (for a proof see Exercise 1 at the end of this chapter),
multiply (8.2.8) by a, let a -> 0+, and apply (8.2.7) to get
JCn-\ , = 1 \<i<j<n-
r(e + Bnjl)y)
7=1
Again by (8.2.7), the last integral also equals
Mj-ну)
ГBу + уб + (n + ] - 3)у)
This gives the functional relation
This implies
Thus, Selberg's formula and Aomoto's extension are both proved.
Applying the change of variables and limiting procedure for obtaining (8.2.6)
to Aomoto's integral formula, we get:
Corollary 8.2.2 With the conditions on the parameters a and у as in Theorem
8.1.1,
roc roo к п
/00 /-00 * "
•••/„ Ibll*"'-* П ^--
JV ;=1 i=l \<i<j<n
к п
— 1 |(« + (П — j)y) Y\

8.3 Extensions of Aomoto's Integral Formula 407
To derive another corollary from Selberg's formula, take a = ft and jc, =
A + f,/V2a)/2, let a ->• oo, and apply Stirling's formula.
Corollary 8.2.3 For Re у > -1 /n,
П I*-*
Remark 8.2.1 One can use Carlson's theorem to prove (8.2.7) without having to
let /? go to infinity. It follows from (8.2.5) that (8.2.7) is trae when a and ft are
positive integers. Moreover, both sides of (8.2.7) are analytic functions of a and
p for Rea > 0 and Re? > 0 and they are bounded for Rea > 1, Re/3 > 1.
8.3 Extensions of Aomoto's Integral Formula
Aomoto's formula involves the introduction of the extra factors [}*=1 xt, к < п.
One may ask whether extra factors of the type ПA — xj) can be inserted in the
integrand. The simplest integral of this type occurs when there is no common
variable among the two different kinds of factors. Let
-. j J+k
B(j,k):= JJxt Y[(l-Xi)w(x)dx,
where j + к <п and
The formula is
А Г (a + (и - i)y)VC + (n- i)y)r(iy + 1)
Ul П Г(« + ^ + Bи-/-1)у)Г(у + 1)
n/=iC» + (и - i)Y] ULlfi + (и - i)y]
This is easy to verify. Denote the right side of (8.3.1) by C(j, k). Observe that
Aomoto's integral implies
B(j, 0) = CO", 0) and fi@, к) = C@, k); (8.3.2)
moreover, both В and С satisfy the same recurrence relation
B(j - 1Д - 1) - B(j, k-l) = B(j - 1, k). (8.3.3)

408 8 The Selberg Integral and Its Applications
This proves (8.3.1). Now let
i,k,t)= I T\xi T] A-х,
so that j represents the number of extra x, factors, к the number of extra A —л:,-)
factors, and I the number of variables that overlap among the extra factors. Here
I < j,k <n and j + к - ? < п.
Theorem 8.3.1
Bn - i -
ПЙi [« + B+Bn-i-
where Sn(a, fi,y) is the value of the Selberg integral.
Proof. First consider the case where к = ?. This integral, after renumbering
variables, can be written as
i)w(x)dx.
, j к
j, k,k) — I TT Xj TTA — Xj
Jcn ~t ~t
It satisfies the functional relation that generalizes (8.2.4):
(a+P+Bn-j-k- l)y)B(j + 1 ,k - 1 ,k - 1)
= (a + (n-j - \)y)B(j, к - 1 ,k - 1). (8.3.4)
To prove (8.3.4), start with
r Q j k
0= / ~-T[xir\(\-xi)w(x)dx
j к j к
~fi Y[xiY[(l -Xi)w(x)dx
•'C- i=l 1=2
» i=2 i=l •'C- i=l 1=2
The functional equation (8.3.4) now follows from the following lemma.

8.3 Extensions of Aomoto's Integral Formula 409
Lemma 8.3.2
к т-г; тк
(a) V / ^i=lXi^i=l(l~Xi)w(x)dx = 0.
~~~o JCn x\ ~~ xm
(c) ]T / 'x x —w{x)dx
¦n—j-\-\ n
= (n - j) — ~—! B(j, k-l,k-
m=j
The proof of this lemma is left to the reader.
A direct consequence of (8.3.4) is that
R..,,. W [a+fi + Bn-i-k-l)y]Of
B(j,k,k)= \\ :—— — B(n,k,k)
>Д, [« + (n - i)y]
k (P + (n
fj^ (a + y6 + Bn - г
The second equation follows from (8.3.1) and the fact that fi(n ,k, k) is the same as
B@, k, k) except that a is replaced by a +1. Equation (8.3.5) is, in fact, equivalent
to
* ( + p + in-i- l)y)
+ 1 + Bn - i -
(n - i)y) nLi(
Sn(a,P,y). (8.3.6)
This proves Theorem 8.3.1 for the case к = 1. By writing x\ as 1 — A —x\) in the
integral B{j,k,t) one verifies that
ВЦ + \,кЛ) = В(],кЛ)-ВЦ,к + \,1) forj>?. (8.3.7)
The right side of Theorem 8.3.1 also satisfies this recurrence relation. Thus The-
Theorem 8.3.1 is proved. The above proof is due to Shaun Cooper. ¦
We end this section with the statement of another beta integral of Selberg.

410 8 The Selberg Integral and Its Applications
Theorem 8.3.3 Let Dn = {(*,,..., xn) \ x,¦¦ > 0, ?"=i xt < 1}. Then
•'D»,-=l 1=1
A proof is outlined in Exercise 5.
Historical Remark Hardy and P61ya independently proved the following theo-
theorem on entire functions:
Let f(z) be an entire function of exponential type less than log 2. If f(n) is an
integer for n = 0, 1, 2,..., then f(z) is a polynomial.
This theorem can be restated as follows: 2Z is the smallest transcendental entire
function taking integral values at the positive integers.
Gelfond generalized this theorem as follows:
Iff(z) is an entire function such that
are all integers and f is of exponential type less than p log(l + e(l~p)/p), then f
is a polynomial. The case p = 1 is the result of Hardy and Polya.
Selberg discovered his integral formula when he generalized Gelfond's theorem.
He proved thatp log(l+eA~/')//') can be replaced by log {min [|f=1 A+y,)}. where
yi > 0, У1У2 •¦¦yP = el-p, and | Ili</<,-<,(? ~ ^)l = L To see that this is a
generalization, note that since the yj are distinct,
\ + yd > {\ + 1/у,У2- ¦ ¦ yP)p =
1=1
For references, see Boas [1954].
Corollary 8.2.3 was conjectured by Mehta and Dyson in the mid-1960s. They
considered a gas of N point charges at Xi, хг,. ¦., xN, which are free to move on
the infinite straight line — oo < x < oo. The potential energy of this gas is given by
2
1 = 1 l<i<j<n
where the first term represents a harmonic potential that attracts each charge inde-
independently toward the point x = 0 and the second term represents an electrostatic
repulsion between each pair of charges. An important role in the thermodynamical
study of this system is played by the partition function
ЫР)
= I ¦••/ e-Pwdxx-.-dxn.
J — 00 J — OO

8.4 Anderson's Proof of Selberg's Formula
411
It was the value of this integral that was conjectured by Dyson and Mehta. See
Mehta [1991]. Mehta's book also contains Selberg's original proof of Selberg's
integral formula.
8.4 Anderson's Proof of Selberg's Formula
In Chapter 1, we gave two essentially different proofs of the formula В (a, fi) =
Г{а)Гф)/ Г(а+уб). One was done by constructing a functional equation and the
other by evaluating a suitable double integral in two different ways. The second
method applied to the finite field analogs of the beta and gamma integrals, that
is, to the Jacobi and Gauss sums. Anderson [1991] found a proof of Selberg's
formula which involves the computation of a (In — l)-dimensional integral in two
ways. This proof carries over to a formula for the finite field analog of the Selberg
integral. In fact, Anderson [1990] obtained a proof of the latter result first.
Anderson's proof depends on Dirichlet's generalization of the beta integral given
in Chapter 1: For Re a, > 0,
¦dpn-i =
(8.4.1)
where У is the set p, > 0, YTi=o A = 1 • The formula is used after a change of
variables. To see this, first consider Selberg's integral, which may be written as
nlAn(a,P,y):=n\ [ f " ¦ ¦ ¦ Г \F@)\a-'\F(l)^-l\AF\*dxi ¦ ¦ -dxn, •
Jo Jo Jo
where 0 < X\ < X2 < • • • < xn < 1,
F{t) = (t-xi)(t -x2)---(t-xn) = tn - Fn^t"-1 + ¦¦¦ + (-l)nF0)
and AF is the discriminant of F, so that
We now change the variables from x\, xi, ¦ ¦ ¦, xn to Fo, F\,,.., Fn_\, which are
the elementary symmetric functions of the x, s.
Lemma 8.4.1
where the integration is over all points (Fq, F\, ..., Fn-\) in which the F, are
elementary symmetric functions ofx\,..., xn with 0 < x\ < • • • < xn.

412 8 The Selberg Integral and Its Applications
Proof. It is sufficient to prove that the Jacobian
= |Af |1/2. (8.4.2)
Observe that two columns of the Jacobian are equal when jc, = Xj. Thus
Ylt<j(Xi — Xj) is a factor of the determinant. Moreover, the Jacobian and
П,<;(*( — Xj) are homogeneous and of the same degree. This proves (8.4.2)
and the lemma. ¦
We make a similar change of variables in (8.4.1). To accomplish this, set
and let
V={(t- xx){t - x2) ¦ ¦ ¦ (t - xn) | fc_i < x,¦< fc; i = 1,..., n). (8.4.3)
Lemma 8.4.2 For all F(t) = t" - Fn^tn~l + h (-1)" Fo € V, the map
/¦7-17-1 j-i \ / V5O/ VbW/ \ / . тлп-1-l
(Fo, F\,..., Fn_!) h-> I — ,..., — I = (po, px,..., pn) e R + ,
where Z'(t) denotes the derivative of Z(t), is a bijection and pj > 0 with Ym=o
Proof. Observe that
since the numerator and denominator have exactly n — j negative factors.
Now let
z (t) - Z@
By Lagrange's interpolation formula
F(t) = j^PjZjit) = ^ ~\Р%). (8.4.4)
One can directly verify this by checking that both sides of the equation are poly-
polynomials of degree n and are equal at n + 1 points t — fo, ?ь • • ¦, t,n ¦ Equate the
coefficients of tn on both sides to get

8.4 Anderson's Proof of Selberg's Formula
413
Now for a given point (po, Pi, ¦ ¦ ¦, Pn) with Hpj = 1 and pj > 0, j — 0,..., n,
define F(t) by (8.4.4). The expressions
@ - &-i)(& - O+i) •
and
show that F(f;) and F(f,+1) have different signs and F vanishes at some point
xi+i between f,- and ff+i. Thus F e 2? by (8.4.3). This proves the bijection. ¦
We can now restate Dinchlet's formula (8.4.1).
Lemma 8.4.3 With the notation of Lemma 8.4.2 and Re a, > 0,
*[[\F(b)\ai-ldF0---dFn_l = ^
Proof. We need to verify that the Jacobian
3(po, ••• ,Pn-\)
9(F0,...,Fn_,)
i=0
Since
the Jacobian is
dpi
IteOl
The numerator is a Vandermonde determinant and the result follows. ¦
The final step is to obtain the Bn — 1)-dimensional integral. Let F(t) and G(t)
be two polynomials such that
and
G(t) = (t - yi)(t ~ y2) ¦ ¦ ¦ (t - yn),
0 < Ух < Xi < y2 < • • ¦ < *n-i < Уп < 1-
(8.4.5)

414 8 The Selberg Integral and Its Applications
The resultant of F and G, denoted R(F, G), is given by
\R(F,G)\ =
п
1 = 1 И—1
п
n-\
(Xi)
(8.4.6)
The absolute value of the discriminant of F can be written as \R(F, F')\. The
(In — l)-dimensional integral is
J(F,G)
;G)\r-1dFo---dFn-2dGo---dGn-i
dFo ¦ ¦ ¦ dFn-2(
/
(F,G)
¦ ¦ ¦ dGn-\.
(8.4.7)
Here the integration is over all F and G defined by (8.4.6).
Lemma 8.4.4 Selberg's integral An{a, fi, y) of Lemma 8.4.1 satisfies the recur-
recurrence relation
„{а, р, у) =
Г(а)Г(Р)Г(пу)
(а + Р + (п- \)у)
, y).
Proof. Integrate the Bn — l)-dimensional integral (8.4.7) with respect to dFo •
dFn-2 and use Lemma 8.4.3 with G(t) instead of Z(t) to get
"-'iGd)!"-
Y-2
dG0 ¦ ¦ dGn-\
Г(пу)
= An(a,p,y)
Г(пуУ
To compute (8.4.7) in another way, set F(t) = t(t — Xi)(t — x2) ¦ • ¦ (t — х„),
«о = a, an = P, cij = у for j = 1,..., n — 1, x0 = 0, and xn = 1 so that (8.4.7)
is equal to
/
n — 1
Y-\
dG0 • ¦ ¦ dGn-idF0 ¦ ¦ ¦ dFn-2
(f.G) j=l
f[\G(.xi)\ai-idGo---dGn-1dFo---dFn-2.
(8.4.8)

8.5 A Problem of Stieltjes and the Discriminant of a Jacobi Polynomial 415
Now integrate (8.4.8) with respect to dGo ¦ ¦ ¦ dGn-\ and use F(t) instead of Z(t)
in Lemma 8.4.3 to obtain
K-l/2
n-1
\F'@)\a~i'2\F'(l)f~1'2dF0---dFn-2
Since
and
n в —1 в —1
the last integral can be written as
Equate the two different evaluations of the Bn — 1)-dimensional integral to obtain
the result in Lemma 8.4.4. ¦
Selberg's formula is obtained by iterating Lemma 8.4.4 n — 1 times.
R. J. Evans has shown that Aomoto's extension (8.3.1) can also be proved by
this method. The idea is sketched in Exercise 21 at the end of this chapter.
8.5 A Problem of Stieltjes and the Discriminant of a Jacobi Polynomial
There is a problem of Stieltjes that connects up Jacobi polynomials, the hyper-
geometric differential equation and Selberg's integral in a very interesting way.
It may be stated as a two-dimensional electrostatics problem. Let p > 0 and
q > 0 be fixed. Suppose there are charges of size p at 0 and q at 1 and unit
charges at x\, xi,..., х„, where 0 < x,- < 1, i = 1,..., n. Assume the potential
is logarithmic to get the energy of the system as
T(xux2, ...,xn) = -p^logXj -g^log(l -xt) - ^2 logl*i-•*/!•
i = l i = l l<i<y<n
(8.5.1)

416 8 The Selberg Integral and Its Applications
The problem is to find the location of the charges so that they are in electrostatic
equilibrium. The latter occurs when the energy is a minimum, so we either minimize
(8.5.1) or maximize
n
H(x1,x2,...,xn):=l[x?(l-xiL Д \Xi-xj\. (8.5.2)
i = l \<i<j<n
Theorem 8.5.1 The maximum of (8.5.2) occurs when X\,xi, ¦ ¦ ¦, xn are the zeros
of the Jacobi polynomial pO-p-iM-\)(\ _ 2x).
Proof Since Я is a continuous function of X\, xi, ¦ ¦ ¦, х„ for 0 < x, < 1,
i = 1,..., n, it has a maximum value at some point. If any of the x,- is 0 or 1, the
value of H is 0. So the minimum of T (or the maximum of H) occurs where
— =0, i = l,2,...,n.
Эх,-
Therefore
Y1= 0' ' = 1> 2. ••-.«• (8-5-3)
+ Y1
Xf i X[ X/ Xj
This is a set of n nonlinear equations in n unknowns. Stieltjes introduces a polyno-
polynomial / whose zeros X;, i = 1,..., n satisfy (8.5.3), and he shows that / satisfies
a specific hypergeometric differential equation. Set
n
f(x) = ]J(x - Xi)
(=1
so that the discriminant A of / is given by
n
A- J] (х,-х,J =
l<i<j<n
Take the logarithmic derivative to obtain
, п.
Write (8.5.3) as
— H = 0, i = l,..., л,
2 f'(Xi) Xi 1-Х;
or
x,(l -Xi)f"(Xi) + 2[p -(p + q)xi]f{xi) = 0, i = 1,...,«.
Now consider the expression
x(l - x)/"(x) + 2[p - (p + q)x]f\x).

8.5 A Problem of Stieltjes and the Discriminant of a Jacobi Polynomial 417
This is a polynomial of degree < n, since / is a polynomial of degree n, and it is
zero at xt, i = 1, 2,..., n. Thus the expression is a constant multiple of f(x) and
we deduce that у = f(x) satisfies the differential equation
x A - x)y" + [2p-2(p + q)x]y' + Xy = 0 (8.5.4)
for some constant k. Compare (8.5.4) with the hypergeometric equation
x(l - x)y" + [c - (a + b + l)x]/ - aby = 0,
which has the two independent solutions
'a,b
;x\ and xl C2FA ;x
с ) V 2-е
We see that these are also the independent solutions of (8.5.4) with с — 2p,
a + b = 2p + 2q — 1, and ab = —A.. We get a polynomial solution of degree
n only when a or й is — n. So if b — —n, then a=n + 2/7 + 2<jr — 1 and
Л. = -n(n +2p +2q - I) and
(-WW + 2;+2^1) (8.5.5)
To find k, note that the coefficient of x" in /(x) is 1. So
k _ (-D"Bp)n
(и + 2p + 29 - 1)„'
Except for a constant factor, the hypergeometric polynomial in (8.5.5) is the Jacobi
polynomial pQ-p-^-V{\ - 2x). This proves the theorem. ¦
It is actually possible to find the maximum value of H from Selberg's integral.
Theorem 8.5.2
„2, , Л <2p
H(xu x2,..., *„) = g
g ^ + ^ + ^ + . _ 2Jp+2q+n+]_2
where H is defined by (8.5.2).
Proof. Recall that if \x is a positive measure on a measure space X and
II/IU — ( / 1/1*^Д I < oo, for some к in 0 < A: < oo,
\J J
then
11/11* -> ll/lloo as^^oo. (8.5.6)
In Selberg's integral, take a — 1 = 2/??, p — \ = 2qk, and у = A: and then apply
(8.5.6) together with Stirling's formula to obtain the theorem. ¦

418 8 The Selberg Integral and Its Applications
Since the expression for Я(хь ..., xn) involves the discriminant we have the
next theorem.
Theorem 8.5.3 The discriminant of the Jacobi polynomial P^a^\x) is
2„(в-1) TT J4a
¦Ц (a +
¦Ц (a + P + n + j)"+J-2
Proof. We have
;=0 1 = 1
Bp)n
(-l)V(O) = (-1)"* =
(n + 2p
and
The Chu-Vandermonde identity (Corollary 2.2.3) was used to sum the 2F\. Com-
Combine these results with Theorem 8.5.2 to see that the discriminant of рО-р-^М-Ъ
A - 2x) is
Bp + 2q+n + j - 2)"+J-2
Theorem 8.5.3 follows after an appropriate change of variables. ¦
Similar theorems can be stated for Laguerre and Hermite polynomials.
Theorem 8.5.4 The maximum of
n
U(xux2,...,xn) = Y[xfe-x> Д |jc,- — JCyl
i=i i<i<;'<«
is obtained when X\,x2, ... ,xn are the zeros of the Laguerre polynomial
L%p-1) Bx). The discriminant of Lan(x) is
Theorem 8.5.5 The maximum of
П \*-*А

8.6 Siegel's Inequality 419
is attained when x\,... ,xn are the zeros of the Hermite polynomial Hn(x). The
discriminant ofHn(x) is
1 "
(n-l) 11-
22л(л-1) J
The proofs of these theorems require Corollaries 8.2.2 and 8.2.3 and are left to
the reader as exercises. Other formulations of Theorems 8.5.4 and 8.5.5 are also
possible. These are given in the exercises at the end of the chapter. For references to
the work of Stieltjes and to other methods for the calculation of the discriminants
of the classical orthogonal polynomials, see Szego [1975, §§ 6.7-6.71].
Remark 8.5.1 Stieltjes's interpretation that the zeros of the Jacobi polynomials
are the equilibrium positions of charges placed in the interval [0,1] is useful for
guessing theorems about these zeros. For example, if the zeros of P^"^\\ — 2x)
are written in increasing order 0 < Xi < X2 < ¦ ¦ ¦ < xn < I, then ^ > 0 and
% < 0. Observe that if a is increased the unit charges in @,1) are pushed up
toward 1, and if p is increased they are pushed down toward 0.
8.6 Siegel's Inequality
Siegel [1945] derived an inequality that refines the arithmetic and geometric mean
inequality. He used it to obtain results about traces of algebraic integers, all of
whose conjugates are real and positive.
The idea of the proof is as follows: Let 5 and p be two positive numbers such
that sn > p. Find the maximum of A = Yli<i<j<n(Xi — XjJ considered as a
function of X\,..., xn and subject to the conditions that X\ + ¦ ¦ ¦ + xn — ns and
X\Xi ¦ ¦ ¦ xn — pn, Xi > 0. It turns out that the maximum lies at the zeros of a
certain Laguerre polynomial. In fact, the method by which this is determined is the
same as that used in the solution of the Stieltjes problem of the previous section.
Siegel's inequality drops out easily after this.
Before stating Siegel's inequality we prove the arithmetic and geometric mean
inequality.
Lemma 8.6.1 LetX\,x-i, ¦. ¦, xn be n positive numbers. Then
X\ + X2 + ¦ ¦ ¦ + Xn
$X\X2 ¦ ¦ • Х„ < ,
п
where equality holds if and only if X\ = хг = ¦ ¦ ¦ = xn.
Proof. Start with the inequality in Exercise 1.6. For a, P, u, and и nonnegative
and a + p = 1 write the inequality as
uavp <au + Pv. (8.6.1)

420 8 The Selberg Integral and Its Applications
Here equality holds if and only if и — v. We prove the lemma by induction. It
holds for n = 1. Assume the result true up to n. Then, by (8.6.1) and the inductive
hypothesis,
(XlX2 ¦ • -*B+lI/(B+1) = (*!/П ¦ ¦¦Х1'П)ПКП+Х)Х^Т1)
n /х\Л \-xn\ xn+x
~n+l\ n ) n+1
n + 1
It is clear from the proof that equality holds if and only if all the x,- are equal. This
proves the lemma. ¦
To state Siegel's extension, set
n-k-l
«-1
k=\
and
A =
where the x; are positive. For n > 2 and A ^ 0, let //, denote the unique positive
root of the algebraic equation
Pn(pL) = (Xl---xn)n-l&-1. (8.6.4)
(The polynomial P (t) has only positive coefficients and P @) — 0, so ц, is uniquely
determined.) Siegel's inequality is contained in the next theorem.
Theorem 8.6.2 Let n > 2, А ф 0, ant/ //, ?>e the positive root of (8.6.4); then
where Q is given by (8.6.3).
Remark 8.6.1 Since Qn(fi) > 1 for positive //,, Theorem 8.6.2 is a refinement
of the arithmetic and geometric mean inequality.

8.6 Siegel's Inequality 421
We first prove the following lemma.
Lemma 8.6.3 Let s and p be two positive numbers such that sn > p. The maxi-
maximum of A = Yl(xi ~ xjJ> subject to the conditions
x\ H \-xn=ns and X\---xn = p",
occurs when X\,... ,xn are the zeros of a certain Laguerre polynomial.
Proof. To show the existence of the maximum, observe that for any positive set
of values x3, хц,..., xn the equations
Xi +X2 = ns - (X3H \-Х„), XYX2 = р(ХЪ ¦ ¦ ¦ ХпУ1
have a unique solution that is positive if and only if
x3 H h xn < ns and {ns - (x3 H h xn)}2 > 4p(x3 ¦ ¦ ¦ xn)~l. (8.6.5)
Moreover, x\ = x2 if and only if equality holds in (8.6.5). The conditions (8.6.5)
define a compact domain D in a (n — 2)-dimensional space whose points have
positive coordinates x^,... ,х„. The boundary of D consists of the surface x\ — x2
so that the maximum of Д is inside D. By Lagrange's method of undetermined
multipliers, at the maximum point
дф
— =0, k = l,...,n,
дхк
where
ф(хи...,хп) = -log A - A.(xi H h xn) + Atlog(xi ¦¦¦xn)
and X and [i are some constants. In fact, ц, will be seen to be the positive root of
(8.6.4) when the x, maximize A.
As in the solution of Stieltjes's problem of the previous section, show that
/(•«) = n7=i(^ - ^) satisfies
——— -X-i =0, k=l,2,...,n.
f'(xk) xk
So xf"(x) — (Xx — n)f'(x) is a polynomial of degree n that vanishes at X\,..., xn
and hence is a constant multiple of /(x). This constant has to be — Xn. Therefore,
/ satisfies the differential equation
xy" - (Xx - ix)y' + Xny = 0.
Set t = Xx to transform this equation to
ty" -{t- ц)у' + ny=0.

422 8 The Selberg Integral and Its Applications
This equation is satisfied by the Laguerre polynomial L^~l)(t). So we get
f{x)=kL^-l\kx) (8.6.6)
for some constant k. This proves the lemma. ¦
We make a few observations necessary to complete the proof of Siegel's in-
inequality. First note that the discriminate of the polynomial in (8.6.6) is, according
to Theorem 8.5.4,
n-l
^-„(„-1) Д{(д + jyu + jy + lj (8 6 ?)
Also observe that since
the constant к in (8.6.6) is (-l)nn!/A.n. So
xi-\ \-xn ii + n - 1
5 = =
n к
and
00»
к"
This implies
s"p~l = П~ =ПA+ + ft_i) =g"(^ (8-6-8)
and (using (8.6.7))
p"-iд-i = - TT (^f) = Pn{ii). (8.6.9)
n\ ±A- \n — к)
Proof of Theorem 8.6.2 Suppose that yt, V2,..., yn are n positive numbers such
that
З'Н + Уп ,
=5 and У\---уп = Р-
n
Let ixo be the solution of equation (8.6.4) with these v in place of the x and let Ao
denote the discriminant using the v. Since Pn(t) is an increasing function and
snp-x = QnUi) > Qnino).
we have ц, < ц,0- However, Qn(t) is a decreasing function and so
snp-
This proves Siegel's inequality.

8.6 Siegel's Inequality 423
The following corollary is due to Schur [1918].
Corollary 8.6.4 For positive numbers X\,..., х„
Proof. Set
R(t) = P{t)Qn-\t) = Д ^ "~_
Use (8.6.4) and the inequality in Theorem 8.6.2 to eliminate p = x\ ¦ ¦ ¦ xn and
obtain
sn
n(n-l) > дЛ(д)_ (8.6.10)
Show that R(t) is an increasing function of t > 0 by proving j-t log R(t) > 0.
Thus R{ix) > Л@) and
sn(n-l)
This is Schur's inequality. ¦
Siegel applied his inequality to the derivation of some results on traces of positive
algebraic integers with positive conjugates. We include statements of two of these
results as they are quite interesting. A proof of one of them is sketched. The details
are left to the reader, since they involve an easy application of the Euler-MacLaurin
summation formula.
Before we state the next theorem, observe that the equation
g(v) :=(l+uJlog(l + (l/t/))+logt;-t;-l =0 (8.6.11)
has exactly one positive root, say, в. This is true since g@) = — 1, g(oo) = oo,
and
g'(v) = 2A + v) log(l + A/w)) - 2 > 0 forv > 0.
Theorem 8.6.5 Suppose the algebraic equation with integer coefficients
xn +a1x"-1 +---+an =0
has all positive roots X\, xi, ¦ ¦ ¦, xn. Let в be the unique positive root of (8.6.11)
andletko = еA + A/в)Ув'; thenforanyk < ко, there exists a number N — N(k)
such that
x\ Л + xn > kn
for all n > N.

424 8 The Selberg Integral and Its Applications
The proof depends on the following lemma.
Lemma 8.6.6 Ift is any positive number satisfying APn(t) > 1, then
sn > Qn(t).
Proof. Since p = х\Хг ¦ ¦ -xn = {—\)nan is a positive integer, Theorem 8.6.2
and Corollary 8.6.4 imply that
We have seen that Q{t) is decreasing and R{t) is increasing for t > 0. It follows
for t > 0 that
1 > mm(Qn-l(t), ДЛ@) = Qn~l(t)mm(l, AP(t)).
This proves the lemma. ¦
Proof of Theorem 8.6.5 Start with the Euler-Maclaurin summation formula (see
Appendix D)
Jo Jo
(Recall that Bx{x) = x - \.) Take f(x) = (n - x) log(x + vn - 1) in (8.6.12)
2
and show that
n-\
k=\ k=2
— -g(v)n2 + O(nlogn).
2
Since
lim{l - ulog(l + A/u))} = logAo,
it follows that s = (xi + ¦ ¦ ¦ + xn)/n > X, for any X < Xo and all n greater than
some suitably large N(k). This proves the theorem. ¦
Siegel computed the value of Xo = 1.7336105 The best possible value of
the constant that could replace Xo in the theorem is certainly < 2. This follows
from the fact that 4 cos2 -, for odd prime p, has trace equal to 2n — 1 < 2n, where
n = (p — l)/2. The following theorem was also proved by Siegel, but because its
proof is longer we omit it.

8.7 The Stieltjes Problem on the Unit Circle 425
Theorem 8.6.7 Suppose f is an algebraic integer ф 1 or iC ± V5). Suppose
all the conjugates x\,X2, ¦ ¦ ¦, xn of% are positive. Then
3
x\ +x2 H \-xn > -n.
8.7 The Stieltjes Problem on the Unit Circle
In Section 8.5 we considered Stieltjes's problem for the unit interval. Dyson and
others have looked at the situation where the freely moving charges lie on a thin
circular conductor of unit radius. See Mehta [1991] for references. We work out
the case in which n unit charges are placed on the unit circle, although the more
general case with a charge of size q at в = 0 and one of size p at в = л can also
be treated similarly.
The potential energy of the system with n unit charges on the unit circle is
log\eiek-ew'\. (8.7.1)
In the equilibrium position, W is a minimum.
Theorem 8.7.1 The minimum value of W is — | log n and is attained when e'0k
for к = 1,..., n are roots of the equation x" ± 1 = 0.
Proof. Write \eWt - ew> \ as follows:
\еЪ - eie' | = 2 sin (9-^-^\ = -(«'* - eie')e-m**>I2 for вк > в,.
\ 2 ) i
This shows that at a minimum
л dW . n-\
° Wkl\
V
As in the Stieltjes problem, define
so that
f"(eWt)
/() eiek _ ei
Thus / satisfies the equation
xy" - (и - 1)/ = 0
or
f(x) = Cx" + D.

426
8 The Selberg Integral and Its Applications
Since the coefficient of x" in / is 1 and the roots lie on the unit circle, С — 1 and
D = ±1. Thus
f(x)=xn±l.
The least value of W is — \ log Д, where Д is the discriminant of f(x). Since
Д= JJ/'(e1'*) = f[nei(n-m
k=\ *=1
this minimum is — | log n. ¦
Remark 8.7.1 The partition function for this charge distribution is given by
П
¦«>.¦
This integral is sometimes called Dyson's integral, but it is a special case of
Selberg's integral, as can be seen after a suitable transformation. Its value is
ГA + (pN/2))/(T(l + P/2))N. Incidentally, one may use the last integral to
evaluate the discriminant of xn ± 1. Forrester and Rogers [1986] have consid-
considered a different distribution of charges on the unit circle, which leads to Jacobi
polynomials. See Exercise 13.
8.8 Constant-Term Identities
Take fi = 2k, where к is a positive integer, in Dyson's integral (8.7.2). Its value is
equal to the constant term in the Laurent expansion of the product
П '-*
Zi
(8.8.1)
To prove this, set Zj = e10', so that
2k
\zj -zA2k = (Zj-
= (l - |
l - Z?
Now observe that any power other than 0 of Zj vanishes on integration. From the
value of Dyson's integral given in our last remark, we have
ст.П (¦-?)'-2^ = ??. (8^)
+к)п (k\)n

8.8 Constant-Term Identities 427
where C.T. stands for "constant term of." More generally, Dyson conjectured the
result contained in the next theorem, which was first independently proved by
Gunson and Wilson. The elegant proof given below is due to Good [1970].
Theorem 8.8.1 Ifa\, аг, ¦ ¦ ¦, an are nonnegative integers, then
ГТ
ze
Proof. Let p(x) = П"=1 (х -Zi). Then
X - Zj)p'(Zj) = '
since the left side is a polynomial of degree < n — 1 that is equal to 1 at the n
points Zi,i = 1,2,... ,n. Rewrite the identity as
=i \<k<n zi Zk
Let x = 0 to get
E П i-J— =
Now put
and multiply (8.8.3) by Fn(a\,..., an) to arrive at the recurrence relation
Fn(au...,an) =
Obviously, C.T. Fn (ai,..., an) must satisfy the same relation:
n
C.T. Fn{ai an) = Y^ C-T- F(au...,aj-i,aj - \,aj+\,... ,ak).
Also C.T. Fn @,0,..., 0) = 1 and if ak = 0, then
C.T. Fn(ai,..., an) = C.T. Fn_i (a,,..., ak-U ak+1,..., an).
The last relation holds because ak = 0 implies that only nonpositive powers of
Zk appear in Fn{a\,...,an). It is easy to check that (a\+a2-\ +an)\/

428 8 The Selberg Integral and Its Applications
{a\! a2\ ¦ ¦ ¦ an!) also satisfies the same recurrence relations and initial conditions.
This proves Theorem 8.8.1 by induction. ¦
Morris [1984] derived the following constant-term identity from Selberg's in-
integral.
Theorem 8.8.2 Suppose p, q, and r are nonnegative integers. Then
г1 т ТТл xp / i * \ TT
~ j}t (P + U - l)r)\(q + (j - l)r)!r!"
The proof of this identity is left as an exercise.
8.9 Nearly Poised 3F2 Identities
In Chapters 2 and 3 we gave several derivations of Dixon's sum for a well-poised
3F2. A terminating form of this identity also follows from Selberg's formula. The
more general result of Aomoto allows us to sum a few nearly poised 3F2. Recall
that a series is nearly poised if all but one of the pairs of upper and lower parameters
have the same sum.
To see that Selberg's formula gives the terminating Dixon sum, take n = 2 and
у = у to be positive integers in Selberg's integral. We get
el A
/ /
o Jo
0 Jo \ *i
Expand A - x2/xiJy by the binomial theorem and integrate term by term with
respect to xi to get
2v
/ У^ x?+r~'(l — X2)
Jo Г(г + 1)ГBу -г + 1) ГBу + а +р -г)
_ Д* ГBу + 1)ГBу +а- г)Г(Р)Г(а + г)Т(Р)
~ f^ ГBг + 1)ГBу -г + \)ГBу +а+р- г)Г(а + Р + г)
Г(а)Г(РJГ(а + 2у) ( -2у, а, -а - р - 2у + 1
Г(а+Р)Г(а + Р + 2у) \ -а-2у+1,а
FJ2y, а,а р 2у + 1 \

8.9 Nearly Poised 3F2 Identities 429
This, together with the value of Selberg's integral for n =2, gives
-2у,а,-а-0-2у+1 \ _ ГA + 2у)Г(а + р)Тф + у)Г(а
2+l+9 ' ) ~
+ 2у)ГA + у)Г(а + 0 + у)'
This is the terminating form of Dixon's identity we mentioned earlier.
The next theorem gives the values of some terminating nearly poised series that
can be derived from Aomoto's formula and its extension. This theorem was also
obtained by Bressoud [1987] in the more general setting of basic hypergeometnc
series.
Theorem 8.9.1
f -2k, a, -a - p - 2k \ _ (a + l)tQ8)t(lJ*
' 3 4 a2ka+p ' )
-a-2k,a+p ' ) (a
г (-2к,а,1-а-р-2к \ _ (а)к(Р + 1)к(\Jк(а + P)
3 2V la2ka + p+l ' )
l-a-2k,a + p+l ' ) (а+Р)к(\)к(аJк(а+р +
l,l-a-p-2k \ (a + l)k(P)k(lJk(a
1 1 - a - 2k, a + P
-2k,a,-a-p-2k \ (a)k(P + l)k(lJk
;1
\-a-2k,a + p ' У (a + p)k(\)k(aJk'
-2k,u + \,\ -a-p-2k
l-a-2k,a + p + 2
(a + l)k(P + 1М1)п(а +
+ P
Proof. Consider the integral
when к is a nonnegative integer. Expand (jci — x2Jk by the binomial theorem and
integrate. The result is
I^ Г(а1+2к)Г(р1)Г(а2)Г(р2) (-2k, 1 - ax - px - 2k, a2
р2K 2{ 1 - a{ - 2k, a2 + p2 '
The value of / from Aomoto's formula can be found in the following cases:
(a) «i = a2 + 1 (= а + 1), p2 = Pi (= P),
(b) «i = a2 (= a), p2 = Pi + 1 (= p + 1),
(c) a2 = a, + 1 (= a + 1), px = p2 (= fi),
(d) «! = a2 (= а), Д = ft + 1 (= P + I)-

430 8 The Selberg Integral and Its Applications
These give the first four cases of the theorem. Theorem 8.3.1, which extends
Aomoto's formula, can be applied when
(e) a2 = «i + 1 (= a + 1), p2 = Pi + 1 (= P + 1).
Thus the theorem is proved. ¦
The reader may also use Theorem 8.3.1 to work out the case
cti=a2 + l(=a + l), Pi = p2 + 1 (= 0 + 1)- (8.9.1)
There are nonterminating extensions of the sums in Theorem 8.3.1. See Exercise
15 for one example. It is possible to sum similar, almost very well poised 5F4
series. See Bressoud [1987] for some terminating cases, again done there for basic
hypergeometric series.
8.10 The Hasse-Davenport Relation
The finite-field analogs of the gamma and beta integrals are the Gauss and Jacobi
sums respectively. It is worthwhile to look into the analog of Selberg's integral.
Anderson discovered his idea for the evaluation of the integrals by studying the
finite-field analog. Evans [ 1981 ] conjectured formulas that are analogs of Selberg's
integral formula, and Anderson [1990] proved a special case of it. Later Evans
[1991] used Anderson's ideas to obtain the complete result. In this section, we
prove the Hasse-Davenport relation, which can be viewed as another finite-field
analog of the multivariable beta integral of Dirichlet, which was used in Anderson's
proof of the Selberg integral. In the next section, we give the statement and proof
of the analog of Selberg's formula due to Anderson. Though this is a particular
case of the more general known result, it contains the essential ideas that help
explain the origin of the concepts used in Section 8.3. In this and the next section
some knowledge of finite fields is assumed.
Every finite field is a finite-dimensional vector space over the field Z/pZ =
Z(p), for some prime p. A finite field F thus has q = pm elements for some integer
m > 0 and a prime p. Denote by Fs the finite field with qs elements. For a e Fs
define the trace and norm of a from Fs to F as
TrFs/F(a) = a + aq +aq2 + ¦¦¦+ aq"'
and
NFs/F(a) = a ¦ ofi ¦ ¦ ¦ -ofi' .
Check that the trace and norm of a belong to F and that
NFs/F(a +p) = NFs/F(a) + NFs/F(P)

8.10 The Hasse-Davenport Relation 431
and
NFs/F{aP) = NFi/F(a)NFs/F(P).
We omit Fs/F in the notation for trace and norm if the context is clear. Fora e F,
aq = a. This implies that Tr(a) = sa and N(a) = as when a is viewed as
an element of Fs. More generally, Tif,/f(oc) = fTrf(ayf (a) and NFl/F(a) =
[NF(a)/F(u)Y/d, where d = [F(a) : F] = the dimension of F(a) over F.
Suppose that a is a root of the monic irreducible polynomial
f(x) =xd - cxxd~x + c2xd~2 ¦¦¦ + {-\)dcd e F[x]. (8.10.1)
Lemma 8.10.1
(a) The trace and norm of a from F(a) to F are given by
Tr(a) = c\ and N(a) = cj.
(b) If a is viewed as an element of Fs 5 F(a), then
Tr (a) = ^ci and N(a) = cjd.
Proof.
(a) In this case, Tr (a) = a + aq + \-aq and N(a) = aaq ¦ ¦ ¦ aq . Since
the automorphism /3 -» /3q of any finite field, which contains the field F with
q elements, fixes F, it follows that 0 = (f(a))q = f(aq). Thus if a is a root
of /, so is aq. This implies (a).
(b) This follows from (a) and the discussion preceding Lemma 8.10.1. ¦
We now extend the idea of a Gauss sum, introduced in Chapter 1 for Ъ{р), to
any finite field. For convenience in writing, denote Z(p) by Fp (not to be confused
with Fs, the finite field of dimension s over F.)
Let \j/ denote the additive character on F defined by
f(a) = ^FIF'(a\ where fp = e2ni/p. (8.10.2)
A multiplicative character on F is a homomorphism x from F — {0} to the complex
numbers, that is,
X(«/J) = X(«)X(/?)- (8.10.3)
By convention x @) = 0. In what follows, let x denote a nontrivial multiplicative
character. Define the Gauss sum g(x) by
Х(а)ф(а). (8.10.4)
aeF

432 8 The Selberg Integral and Its Applications
We are interested in the connection between g(x) and
), (8.10.5)
where
x' = x о NFs/F and x/r' = f о TrFt/f. (8.10.6)
Check that x' and ^' are characters of the appropriate kind on Fs. A relationship
between g(x) and g(x') was given by Davenport and Hasse [1934]. We state and
prove it after a couple of lemmas.
Lemma 8.10.2 Suppose a e Fs is a root of the irreducible polynomial (8.10.1).
Then
Х'Ш'(а) = [x
where x' and \jf' are defined by (8.10.6).
Proof.
X'(a)f'(a) = x(N(a))f(Tr(a))
(by Lemma 8.10.1)
The next lemma is a well-known result on finite fields. ¦
Lemma 8.10.3 The polynomial xq' — x is the product of all monic irreducible
polynomials in F[x] of degrees that divide s. (F has q elements.)
The proof is left to the reader.
The essential idea in the proof of the Hasse-Davenport relation given here is
due to Weil [1949]. Weil's proof is much simpler than the original. We follow
the account given by Ireland and Rosen [1991, §11.4], which contains a further
simplification of Weil's argument by P. Monsky.
The Hasse-Davenport relation is given in the next theorem.
Theorem 8.10.4 -g(x') = (-g(x)Y-
In order to make the proof more manageable, we break it down into two lemmas
followed by the completion of the proof. Suppose / is the monic polynomial
xn — c\xn~l + сгхп~г + ¦ ¦ ¦ + (—l)ncn, where с, е F. Define a complex-valued
function к on the set of monic polynomials with coefficients in F by the equation

8.10 The Hasse-Davenport Relation 433
Lemma 8.10.5 The function X is multiplicative, that is, if f and g are monk
polynomials in F[x], then A.(/g) = X(f)X(g).
Proof. Suppose f{x) = xn - cxxn~x + c2xn~2 + h {-\)ncn and g{x) =
xm - dxxm-1 + ¦¦¦ + {-\)mdm. Then
/(*)*(*) = xm+n - (ci + dl)xm+n-1 +¦¦¦ + (-l)m+ncndm.
By definition
= х(с„Жс1)х(<*«Ж<*1) =
Lemma8.10.6 g(x') — X^/(deg/)A.(/)s/deg/, where the sum is over all monic
irreducible polynomials in F[x] whose degrees divide s.
Proof. Suppose a e Fs satisfies an irreducible polynomial / of degree d. Then
X'(a)xfr'(a) = [x{cn)f(ci)Yld = X(fYld by Lemma 8.10.2. This implies that
conjugates
where the summation is over all the conjugates of a. Since every element of Fs is
the root of a monic irreducible polynomial whose degree divides s and conversely
(in fact, these polynomials are the only irreducible factors of xq" — x), the result
is proved. ¦
Proof of Theorem 8.10.4. Consider the L-function given by the formal power
series L(X,t) = J2/monicM/)?deg/> where the summation is over all monic
polynomials in F[x]. The following identity is easy to verify:
L(x,t)= ^ M/)'deg/ = П (i-M/)'*87),
/ monic / irred.
where the product is over all monic irreducible polynomials. Here it is understood
that X(l) = 1. Write the L-function as
n=0 \deg/=n
The coefficient of t is
(x - a) = Y,X(a)f(a) = g(x),
aeF aeF
and the coefficient of tn for n > 1 is

434 8 The Selberg Integral and Its Applications
The factor qn~2 arises because, for a given pair c\, cn e F, there are qn~2 ways of
writing the other coefficients of a polynomial. We have shown that
L(k, f) = l+ g(X)t =
f irred.
Take the logarithmic derivative and multiply the equation by t. Then
tg(x)
or
_ _ J2 (deg/)A(/)J/de^
s=i
Equate the coefficients of ts on each side and use Lemma 8.10.6 to complete the
proof of the theorem. ¦
Remark 8.10.1 In Exercise 1.50, the Dirichlet L-function ПA - x(p)P~Tl
was introduced. The prime ideals of the ring of integers p are the ideals generated
by prime numbers p. The number of elements in the field Ъ(р) is p. For the ring
of polynomials F[x], the prime ideals are generated by irreducible polynomials. If
F has q elements, then F[x]/(f(x)), where / is irreducible, has qdesf elements.
Thus, when t = q~s,
f irred. / irred.
is called an L-function. Note that Л is multiplicative on F[x] just as x is multi-
multiplicative on Z.
8.11 A Finite-Field Analog of Selberg's Integral
We begin by recalling the definition and some elementary properties of the resultant
of two polynomials. Suppose К is a field and let
f(x) — aox" + a\xn~x H + an,
g(x) = boxm + bix1"-1 +•¦¦+*„
be two polynomials in K[x].

8.11 A Finite-Field Analog of Selberg's Integral 435
The resultant of / and g, R(f, g), is defined by the (m +n) x (m +n) determinant
по
b0
а\
ао
Ъх
bo
а\
Ьх
а„
по
ьт
ап
ът
¦ а„
(8.11.1)
bo h ... bn
It can be shown (an exercise for the reader) that the resultant vanishes if and only
if either / and g have a common nonconstant factor or the leading coefficients of
/ and g are zero.
Suppose that / and g can be factored as
/ = ao(x - xi)(x - x2) ¦ ¦ ¦ (x - xn),
g = bo(x - yi)(x -y2)---(x- ym).
Then
R(f, g) = ¦
* " У & = < П
(8.11.2)
(8.11.3)
It follows that
R(f,gh) = R{f,g)R{f,h).
It is also clear from the definition that if either g or / is a constant, then
R(f,bo) = bno, R(ao, g) = a?. (8.11.4)
Now let F be a field with q = p" elements where p is an odd prime. Let x and
rjr be the multiplicative and additive characters defined in the previous section. For
all positive integers a, define the Gauss sum
(8.11.5)
(8.11.6)
(8.11.7)
x€F
Extend the definition to all integers a by the requirement
g(a+q-l) = g(a), andlet g*(a) = q/g(-a).
For all positive integers a, fi, y, n, the Selberg sum is defined by

436 8 The Selberg Integral and Its Applications
where the sum is over all monic polynomials / of degree n in F[x], A/ is the
discriminant of /, and S — x(9~1)/2-
To evaluate this sum we need a property of the following L-function: For a
monic polynomial V e F[x], let
w
where W ranges over the monic polynomials in F[x]. Let G be the product of all
distinct monic irreducible polynomials that divide V. Denote the multiplicity of
the monic irreducible polynomial / as a factor of V by ord/V. We say that V is
primitive if for every factor / of V, q — 1 does not divide ord/V.
The next lemma plays a role in the evaluation of Selberg's sum similar to the role
of Lemma 8.4.3, which is a consequence of Dirichlet's beta integral formula, in
the evaluation of Selberg's integral. The proof of this lemma employs the Hasse-
Davenport relation (proved in the last section).
Lemma 8.11.1 Suppose V is primitive and of positive degree. Then
degL(t, V) <degG-l
and the coefficient e(V) o/rdegG^ is
e(V) = S(AG)x(R(V, G'V
f\G
Proof. It follows from (8.11.2) that R(V, W) depends only on the value of
W mod G. Suppose m is an integer > deg G. For a given polynomial S of degree
< deg G — 1, there are qm~ deg G polynomials Q of deg m — deg G such that
G<2+Sisofdegm. Write
( X(R(V,W))\tm.
Since V is primitive, x(R(V> $)) Ф 1 for some polynomial 5, and hence
deg W=m S
Thus L(t, V) is a polynomial of degree < degG — 1.
To find e(V), consider the double sum
X(R(V,U)),
\ I r I
и w
where W ranges over monic polynomials of deg G — 1 and U ranges over poly-
polynomials of degree < deg G and ReSoo denotes the residue at infinity. When U is a

8.11 A Finite-Field Analog of Selberg's Integral 437
constant, say a e F, then
„ UW, «•¦*,,• U(l/x)W(l/x) 1
—ReSoo ax = coefficient or 1/x in ¦ —т = а.
G G(l/x) x
(The factor 1/x2 comes in because the transformation x -> 1/x changes dx to
2.) By (8.11.4)
When U is not a constant, then, by an argument similar to the one given before,
the sum over all W vanishes. Thus
^2 (8.11.8)
a€F
To evaluate ц in another way, start with the fact that the sum of the residues of
UW/G at finite points plus the residue at inifinity is 0. The finite poles are at the
zeros of G. This implies
UW ^ UW
-Reso
G ^ G '
Res i)S
Here r\f denotes a root of G that comes from the irreducible factor /. The residue
at r]f is given by U(r) f)W(r) f) /G\t] f), where G' denotes the formal derivative of
G. The sum of residues at all the roots of / is then
(U(.t)f)W(r)f)\
\ G'(r,f) )¦
Use this to write ц, as
U W f\G
Here N denotes the norm, and we used (8.11.2) to obtain the expression involving
N. To simplify /x, observe that when R(V, W) ф 0,
otdfV
= Ф Tr I
ord/V
(8.11.9)
\G'(rif)JJ
It is clear that since V is primitive, the sum over U in ц vanishes when R(V, W) =
0. Thus the terms in the sum indexed by W that have R(V, W) = 0 can be dropped.

438 8 The Selberg Integral and Its Applications
This observation, together with (8.11.9), (8.11.2), and (8.11.3), implies that
ff (8.11.10)
f\G
In the above expression the Gauss sum gf is over the field F(r]f), where [F(r)f) :
F] — degree of /. By the Hasse-Davenport relation,
gf(-ordfV) = (-l
Now observe that Y[f^G(—l)aeef~1 is equal to the sign of the permutation of the
roots of G effected by the qth power automorphism of the algebraic closure of the
field F. Since q is odd, this value is equal to S(AG). Thus
д = €(V)S(AG)x(R(V, G'))Y[g(-ordfV)^f. (8.11.11)
Compare the two expressions for /x, (8.11.11) and (8.11.8), to prove the lemma. ¦
Theorem 8.11.2 Suppose that the numbers a, a + y,..., a + (n — l)y; p, p +
y, ..., p + (n — \)y; y, 2y,... ,ny are not divisible by q — 1. Then the Selberg
sum can be written as
g4
S( я = П g4a + JY)g4p + iY)g4u
i
Proof. The proof is by induction. It is sufficient to prove that
с/ а ч g*(u)g*(P)g4ny)
S(a * Y) Sia + YP +
Sn(a> *' Y) g*(a + P + (n
For this рифове, consider the double sum
)"Q(lfR(P, Q)y),
P Q
where the sum on P ranges over monic polynomials of degree n — 1 and Q is over
monic polynomials of degree n. First note that
x(Q@)aQufR(P, QY) = x(Q@)aQ(ifR(QY, P))
So we can take V = Qy or V = xa(x - lfPY in Lemma 8.11.1, since the hypo-
hypothesis of the theorem implies that V is primitive. The lemma also implies that the
sum over P (respectively Q) is zero if Q (respectively P) is not square-free.
Therefore, summing over P with Q square-free, Lemma 8.11.1 implies that
g*(ny)

Exercises 439
This follows since, if Q = Q\ ¦ ¦ ¦ Qs is a factorization into irreducible polynomi-
polynomials, then deg V = deg Qy = ny and orda V = y. Thus by (8.11.2),
S = X(-ir+n{n-l)y/2Sn(a, p, y)g\y)n/g4ny). (8.11.13)
Similarly, summation over Q with P square-free and V = xa(x — 1)PPr gives
n-l
•*(а + р + (п-1)У)
Set
T = —x(x- 1)P = (x- 1)P +xP +x(x- \)P'
dx
and observe that
x(R(xa(x - \fPY, T)) = x(T@)aT(l)fiR(P, T)y)
= X((-l)"P@)aP(lf R(P, x(x -
and
R(P,x(x - 1)P') = R(P,x)R(P,x- 1)R(P, P')
= P@)P(l)R(P,P').
These relations and the fact that S(Ax^-i)P) = 8(AP) give
"
S = x(-ir+n(n-1)r/2Sn-i(a + У,Р + У, Y) 8(а)?У_?8(У) v (8.П.14)
g*(a + p + (n- \)y)
A comparison of (8.11.13) and (8.11.14) shows that (8.11.12) is true, which proves
the theorem. ¦
Exercises
1. Prove that if / is continuous on [0, 1], then
lim a [ ta~
The case in which / is differentiable is easier and can be done by integration
by parts.
2. Suppose that / is a complex measurable function on a measure space X with
a positive measure ц, and that ||/||oc > 0. Prove that if \\f\\p < oo for some
0 < p < oo, then
II У* I ¦ II f> ||
as p —>• oo.

440 8 The Selberg Integral and Its Applications
3. Work out the details of the proofs of Corollaries 8.2.2 and 8.2.3, that is, prove
the formulas
/00 /-OO * "
¦••/ IixUx"le'X П \*-хА
= П(« + («- j )r) П
and
J-оо ./-oo \ ^,.1 J l<i<jsn
4. Denote the first integral in Exercise 3 as Gn (k; a,y) and employ the method
used in proving Aomoto's formula to show that
Gn(k; a, y) = [a + (n- k)y]Gn(k - 1; a, y).
Obtain another evaluation of Gn(k; a, y) from this recurrence relation.
5. Here is one proof of Selberg's alternative beta integral formula (Theorem
8.3.3). With R+ = [0, oo), set
(ГЫ П iXi-xj^flxf-'e-^dx.
\i=1 / 1 <(<;<« i=l
(a) Show that A.^+*+na+(n-1)n>'-1e-xG(A.) = G{\)X^le-k.
(b) Integrate (a) with respect to Л over [0, oo) and show that
/,,„ Г1 i V^n -iP+k+nu+n(n-\)y \.LXi '
JR+ [L+2^1xi\ i = l
=1(а + (л-у» " г (а + (л - у
(с) Change variables so that
/ \ -i
show that
д(уи...,у„)
and obtain Selberg's formula.
6. Prove Lemma 8.3.2.
7. Verify Equations (8.3.6) and (8.3.7) in the proof of Theorem 8.3.1.

Exercises 441
8. Prove Theorems 8.5.4 and 8.5.5.
The next five problems are similar to Steiltjes's maximum problem and
have similar solutions.
9. Let there be a positive mass p at the fixed point x = 0 and unit masses at the
variable points x\, x2, ¦ ¦ ¦, xn in [0, oo) such that
x\-\ Yxn < nK,
where A" is a given positive number. Show that the maximum of
j=\ l<i<j<n
is attained if and only if the {Xj} are zeros of the Laguerre polynomial L^ (ex),
where a = 2p — 1 and с = (n + a)/K.
10. Suppose there are unit charges at the variable points xi,..., xn in the interval
(—oo, oo) such that
х\л +хгп <nL,
for a given positive number L. Show that the maximum of
V(X\,...,Xn) = Y[ \xi~xj\
l<i<j<n
is attained if and only if the {jc, } are the zeros of the Hermite polynomials
Hn(Cx), С = V(n
11. Suppose there are n unit masses, n > 2, at the points x\, хг, ¦ ¦ ¦, xn in [—1, 1].
Find the positions of these points for which fJi <, <, <„ \x% — xj I is a maximum.
12. Suppose that the n unit masses exist at jci ,..., xn in [0, oo) and satisfy the
condition
X] +x2-{ Yxn < nK,
where К is a given positive number. Find the positions of these points for which
IIi<,-<;-<n I** - XA is a maximum.
13. Suppose charges are distributed on a unit circle. At в = 0, fix a charge +q
and at, в = ж, fix a charge +p. Distribute 2N freely moving unit charges at
6»i,..., 02n so that
O<0j:< ж, j = \,...,N and ж <0j <2tt,j = N + 1,...,2N.
Show that the potential
IN 2N
T = ~qY, logl 1 - eWt | - p Y, logl 1 + e1'* I - XI 1оё1е'е' " e"J I
/k=l k=\ l<k<j<2N
is a miminum when the #,s are zeros of the Jacobi polynomial
рй-1/2,г1/2) (Cos^)> о < в < 27Г. See Forrester and Rogers [1986].

442 8 The Selberg Integral and Its Applications
14. Prove that, with appropriate conditions on the parameters for convergence,
00 /-OO "
/ Г1A+^)~^~2У(Й~1) П \(xi-xj)\24xl...dxn
T
/
/
r Г{а + (у - 1)у)Г(р + U - 1)к)ГA + jy)
and
Bтг)" J-o
\(Xi -Xj)\lYdxx...dxn
1
П Г(а _ 0- _ \)у)Г(р - (у - 1)У)ГA + у)'
For the second integral, begin with Cauchy's beta integral.
15. Show that
3F2( а~ХЛс ;l)
\a+ l-b,a+l -c J
(а- 1)Г(а + 1 - b)T(a + 2 - c)T(a/2 + 1)Г(а/2 + 2-b-c)
~ (b - l)(c - 1)Г(а/2 + 1 - Ь)Г(а/2 + 1 - с) Г (а + 1)Г(а + 1 - * - с)
Г (а + 1 - Ь)Г(а + 1 - с)Г(а/2 + 1/2)Г(а/2 + 3/2 - Ъ - с)
{Ь- \)(с - 1)Г(а/2 + 1/2 - Ъ)Т{а/2 + 1/2 - с)Г(а)Г(а + 1-Ь-с)'
Note that, if а = 2л, the second term vanishes and the first can be evaluated
by setting a = —In — e and letting e ->¦ 0. When a — —In — 1, the first term
vanishes and the second can be evaluated by a similar limit.
16. Evaluate the integral in Remark 8.7.1, that is,
One approach is to set jc, = tan@,/2) in the second integral in Exercise 14.
17. Prove Morris's constant-term identity contained in Theorem 8.8.2.
18. Complete the missing steps in the proof of Theorem 8.6.5.
19. Prove the arithmetic and geometric mean inequality as follows: Let x\ <
xi < хт, ¦ ¦ ¦ < xn. Show that with s = (xi + • ¦ • + xn)/n, s(x\ + xn — s) >
xixn. By the inductive hypothesis the result holds for the n — \ numbers
Jt2,..., xn-\, x\ + xn — s. Now obtain the necessary result.
20. Fill in the details in the proof of Theorem 8.9.1. In particular, work out case
(8.9.1).

Exercises 443
21. Consider Sn(a,p,y;u) = J IFCO)!"-1 IF(l)^1AF\y-2F(u)dF0dFi...
dFn-\, where и is a parameter,
(a) Show that
[
F(t)€T> i=Q
A iz'(fr)r
11
,-=0 x
(This follows from Lemma 8.4.3.)
(b) Let
¦F(u)dF0 ¦ ¦ ¦ dFn_2dG0 ¦ ¦ ¦ dGn-i.
Show that
1 d Г(у)п
In(ct, P, Y\ u) = -—Sn(a, p, у; и) ¦ ——-
n du 1 (ny)
(с) Now prove by induction that
Sn(a,p,y;u)
m-\
I ж—, I П \
= Sn(a,p,y)
To prove this, let Tn(a, p,y;u) denote the sum inside the braces show
that
Tn(a, p, y; u) = nTn-i(a + y,P + y,y; u).
(d) Use (c) to show that the introduction of the factor
x\x2---xm(\ -дгт+1) • • • A -xm+i)\ m + l <n,
inside the Selberg integral, multiplies its value by
(e) Use the Chu-Vandermonde identity to sum the expression in (d).
The following problems on finite fields use the notation and definitions
given in Section 8.10.
22. For a e Fs show that
(a) 2>F,/F(a) € F, NFs/F(a) e F,
(b) Trpjp maps Fs onto F,
(c) There exists a /? e F such that
= 0-

444 8 The Selberg Integral and Its Applications
23. Define ga{x) = J2t€F Х(?Ж«О- Show that
(a) \ga(x)\=q1/\
(b) IfX ^н^,thenga(x)s«(X-1) = X(-l)<^
24. (a) Prove Lemma 8.10.3,
(b) Take F = Ъ/рЪ = Z(p) in (a). Let Nd denote the number of monic
irreducible polynomials of degree d in F[x]. Use (a) to show that
d\s
(с) Prove that
25. Prove that
1
00
= У2 (# of monic polynomials of degree n)tn
n=0
= П а-'57)-1
/ irreducible
where Nd has the same meaning as in the previous problem. Now deduce the
result in Exercise 24(b).
26. Verify formulas (8.11.4).
27. Prove that R(f, g) = 0 if and only if either / and g have a common noncon-
stant factor or the leading coefficients of / and g are 0. (Note that / and g
have a common factor if and only if there exist polynomials f\ and g\ such
thatdeg/i < deg/ - 1, deggi < degg - 1, and fgx = fxg.)
28. Prove that if / = ao(x - x\) ¦ ¦ ¦ (x - xn) and g = bo(y -yi)---(y- ym),
then

Spherical Harmonics
The aim of this chapter is to introduce the basic functions necessary for Fourier
analysis in higher dimensions. One way to view cos пв is as a restriction to the
unit circle of the homogeneous polynomial [(x + iy)" + (x — iy)"]/2, which
is a solution of the two-dimensional Laplace equation. Spherical harmonics are
restrictions to the spherexj* +x| H \-x^ = 1 of homogeneous polynomials that
are solutions of the и-dimensional Laplace equation. These functions are related
to the ultraspherical polynomials studied in Chapter 6.
An important result in this chapter is the addition theorem for ultraspherical poly-
polynomials, which generalizes the addition formula for the cosine. A useful tool in
the proof of this result is a theorem of Funk and Hecke on an integral of a product
of a continuous function and a spherical harmonic. Our presentation owes much
to Miiller [1966]. We also employ the Funk-Hecke formula to obtain the Fourier
transform of a function on R", which is the product of a radial function and a
spherical harmonic.
The final six sections of the chapter show how spaces of spherical harmon-
harmonics of a given degree give irreducible representations of SUB), the group of all
2x2 matrices
a —i
b a
of determinant one. Representation theory provides a very important approach to
the study of special functions. Unfortunately, we do not have space for it here. The
reader may consult Vilenkin [1968] or Miller [1972]. Here we content ourselves
with showing the manner in which Jacobi polynomials appear in representations
of SUB) and deriving an addition theorem.
9.1 Harmonic Polynomials
Solutions of the Laplace equation
Y,^>=0 (9.1.1)
445

446 9 Spherical Harmonics
are called harmonic functions. We are particularly interested in polynomial so-
solutions. Since any polynomial in the variables x\, x2, • • •, xn is a sum of a finite
number of homogeneous polynomials of different degrees, we concentrate on
these.
Definition 9.1.1 A polynomial Hm(x), which is homogeneous of degree m and
satisfies (9.1.1), is called a harmonic polynomial.
Interest in these polynomials arises from the fact that they are useful in finding
harmonic functions with given boundary values. In particular, one would like to
determine such functions in the unit ball YH=ixl < 1- To motivate the study
of these polynomials and functions, consider the case n = 2. In this situation we
usually write и — u(x, y), where x and у are real. It is clear that z = x + iy can
be used to determine harmonic functions. Since
and
=B(Bi)(, + ly)
dyl
it follows that z" = (x + iy)" is a harmonic polynomial. Similarly, z" = (x — iy)"
is also a harmonic polynomial. There is a technical problem in using (x + iy)" and
(x — iy)n to find harmonic functions with a given boundary value on x2 + y2 = 1,
because it is hard to see some of the important properties of these polynomials
when they are given in terms of (x + iy)" and (x — iy)". This problem can be
resolved by using polar coordinates.
Setx = r cos в and у = r sin 6» so that z — re'9 and z" = r"e'"9. Then
un(x, y) = r" cos пв = [{x + iy)n + (x - iy)"]/2
and
vn(x, y) = r" sinne = [(x + iy)" -(x- iy)"]/Bi)
are harmonic polynomials of degree n. The Poisson integral
r2
_ r2
- 2r cosF> - ф) + r2
then solves the problem of obtaining a harmonic function u{x, y) = v(r, в) with
Итг_и- v(r, в) = /(в), 0 < в < 2n.
Note that the Poisson kernel
X_r2 _ _~
1 - 2r cos в + r2
Thus the kernel in the above integral is an infinite sum of harmonic polynomials.

9.2 The Laplace Equation in Three Dimensions 447
The polynomials un(x, y) and vn{x, y) can also be expressed as
к even
and
On the circle x2 + y2 = 1, it is simpler to view these polynomials as cosnO
and sin пв respectively. Note that, for n = 1, 2, 3,..., there are two independent
harmonic polynomials of degree n. For n = 0, there is just one. This is reflected
in the coefficients in the expansion of the Poisson kernel.
9.2 The Laplace Equation in Three Dimensions
We have seen that it is convenient to use polar coordinates to study the harmonic
polynomials on the unit ball in two dimensions. We could have employed polar
coordinates right in the beginning by writing the Laplace equation as
d2v 1 d2v 1 dv
^2+72W2+-r^ = °- (9'2Л)
A set of solutions of the form U(r, в) = R(r)TF) can be obtained by separation
of variables. We have
r2R"T + rR'T + RT" = 0
or
r2R"+rR'_ T" _
R ~ ~~T ~ C'
where с is a constant. So
r2R" +rR' -cR = 0,
which is Euler's equation with a solution of the form R = rl. The exponent A.
satisfies
A. (A. - 1) + к - с = 0
or

448 9 Spherical Harmonics
For polynomial solutions we must have A. = 0, 1, 2,..., and с = n2 for an integer
n. Thus T satisfies the equation
T" + n2T = 0.
Two independent solutions are cos пв and sin пв.
The above calculations can be carried over to three dimensions. We write
Laplace's equation in spherical coordinates as
2dv\ 1 9 / dv\ 1 d2v _
r Y) + ^eYe \sm0Ye) + ^TeW2~
= —c.
The ranges for в and ф are 0 < в < n and 0 < ф < 2л. Separating variables
U(r, ф, в) = R(r)FD>)T(e) gives
(r2R')' 1 (smOT')' 1 F" _
R + sin6> Г + sin2в ~F ~
Therefore,
(r2R'Y
R
and
1 (sineT'Y 1 F"
sin6< Г sin2 в F
The first equation can be rewritten as
r2R" + 2rR' -cR=0.
Again, R = rx gives A. (A. - 1) + 2A -c = 0 or A. (A. + 1) -c = 0. Write с =
n{n + 1), so that X = n and A = —и — 1 are the solutions. Again, for polynomial
solutions take A = n, a nonnegative integer. Then
(sin 6T)' , F"
sine- -+n(n + 1) sin2 6>H =0.
T F
So, F" — dF = 0 for a constant, say, d = —m2. Then T satisfies the equation
1 d ( dT\ { m2 Л
sin6>— ) +<n(n+\) 5— \T = 0.
sinddey de) \ sin2 6> /
Set x = cos в and Т(в) = у(х) so that this equation becomes
4
dx

9.3 Dimension of the Space of Harmonic Polynomials of Degree к 449
This differential equation was mentioned in Chapter 3 as a second-order equation
with regular singularities at — 1, 1, and oo and no other singularities. In Riemann's
notation, its set of solutions was given by
-1 oo 1 \
m/2 n + 1 m/2 x >. C.9.5)
—m/2 —n —m/2 )
Notice that the indices associated with 1 and — 1 are the same. From the discussion
in Chapter 2, and in particular from Equation B.3.6), it follows that the set of
solutions can also be written as
(l-x2)m/2P{ 0 n+m+1 0 x \ s A -x2)m/2v.
By Theorem 2.3.1, v satisfies the equation
A -x2)y" - 2A + m)xy' + (n- m)(n
This equation has polynomial solutions when n — m > 0 is an integer. In fact,
the polynomial solution is the ultraspherical polynomial C™_m2(x). The reader
may verify this by comparing the equation with the differential equation for Jacobi
polynomials in Chapter 6. See Exercise 6.25.
We have shown that the Laplace equation is satisfied by rn A — x2)m^2Cn_^ (x)
cos тф and r"(l — x2)m//2C™_m2(x)sinm0, where x = cos# and 0 < m < n.
These are 2n + 1 independent solutions. We will show that these span harmonic
polynomials of degree n in three dimensions. Observe that only one solution,
namely rnCxJ2{x), does not depend on ф.
9.3 Dimension of the Space of Harmonic Polynomials of Degree к
Let Vjt denote the vector space of homogeneous polynomials of degree к in n
variables. Each polynomial p e Vk has the form
p(x) =
\a\=k
wherea = (ab a2, • • •, an),x = (xux2, ¦¦-, xn),ca =ca,,a2i. ..,„„, xa =x"' •••*"",
and |a| = YH=i ai< with a' nonnegative integers. The dimension of Vk is the num-
number of и-tuples (аь аг, ¦ ¦ ¦, а„) with YH=i a' = ^- Represent a given и-tuple
(ai, аг, - - -, а„) by a sequence of к dots and n — 1 vertical lines as in the example
given below:

450 9 Spherical Harmonics
There are ai dots before the first line, then a2 dots between the first and the
second line, and so on. Clearly, there is a one-to-one correspondence between
such sequences and the и-tuples. The total number of dots and lines is n + к — 1
and so the number of different arrangements of dots and lines is
A second way of seeing this is that dk<n is the coefficient of xk in the expansion of
A — x)~n = A — x)~l • • • A — x)~x, for this coefficient is the number of solutions
of Еаг = к with a,- nonnegative. Yet another argument in a special case is given
by Anno and Mori [1986]. It is evident that not all the homogeneous polynomials
are harmonic. For instance, in the case of two variables, the dimension of the space
of harmonic polynomials of degree к > 0 is 2 but dk,2 = к + I.
To find the number of independent harmonic polynomials of degree к in n
variables, write the homogeneous polynomial
к
P(x) = ^2xJnAk-j(xi,..., xn-\),
j=o
where А*_у- is homogeneous of degree к — j in x\,..., xn-\. Apply the operator
^ 92 Э2
ii илп
to p(x) to get
к
Ap(x) = ^2j(j
j=2
k-2
If p(x) is a harmonic polynomial, then we must have
So once Ak and Ak-\ are given, the remaining A, are determined. Therefore, the
number of linearly independent harmonic polynomials of degree к in n variables
is
Ck,n = <4,«-l + dk-l,n-l
k-l J fc!(n-2)!

9.4 Orthogonality of Harmonic Polynomials 451
Observe that ckt2 = 2 for к > 0 and ск^ = 2к + 1, which reconfirm the statements
at the end of the first two sections.
9.4 Orthogonality of Harmonic Polynomials
The harmonic polynomials of different degrees in two variables are orthogonal
when integration is over the unit circle; the same is true in three variables over
the unit sphere. This is clear by looking at the polynomials given in the earlier
sections that formed a basis for the space of harmonic polynomials in two and three
dimensions. This result continues to hold in higher dimensions. Before proving
this generalization, note that polar coordinates in n dimensions, (r,0\,..., вп-г, ф)
can be defined by the equations
x\ = r cos 0i,
X2 = a-sin 0i cos 02,
x3=A-sin0i sin 02 cos 03,
xn-\ = л sin 0i • • • sin0ra_2cos0,
xn = r sin 0i • • • sin0n_2sin0,
where 0 < 0, < 7Г and 0 < ф < 2т:.
Let Hk(x) and Hj (x) be homogeneous harmonic polynomials in n variables of
degrees к and j respectively with j ф к. By Green's theorem,
0= / [Hj(x)AHk(x) - Hk(x)AHj(x)]dV
J\x\<\
= f
•/№1=1
r=\
r=l
(9.4.2)
where ? = x/|x|, |x| = r, and da>(%) is the invariant measure on the surface of
the sphere. We used the fact that the normal derivative on the sphere is in the radial
direction. The homogeneity of Hk{x) gives
9
9
э7'
r=l
Substitute this in (9.4.2) to arrive at
(* - j) f HjG)HkG)da>G) = 0. (9.4.3)
The functions Hk{%), which are restrictions of homogeneous harmonic polyno-
polynomials to the surface of the sphere in Rn, are called spherical harmonics. Some-
Sometimes they are also called surface spherical harmonics and the Hk(x) are called

452 9 Spherical Harmonics
solid spherical harmonics. A notation sometimes used when expressing spherical
harmonics in polar coordinates is
where в = (въ в2,..., <9„_2)-
Let F be the space of real-valued continuous functions on the sphere |?|2 =
?f + f | H + %l = 1. An inner product on this space can be defined by
{f,g)= I /(?)*(?)<M?), farf.geF. (9.4.4)
J\l-\ = l
The conjugate of g is taken in the integral, if complex-valued functions are used.
The result contained in (9.4.3) is stated in the next theorem.
Theorem 9.4.1 Spherical harmonics of different degrees are orthogonal with
respect to the inner product (9.4.4).
Spherical harmonics of the same degree may or may not be orthogonal. For
example, cos кв and sin кв are two independent spherical harmonics in two di-
dimensions and are orthogonal, but cos кв + sin кв and cos кв are not orthogonal.
Using Gram-Schmidt orthogonalization, it is possible to choose an orthonormal
basis of ckin = Bk + n - 2)(k + n- 3)!/(fc!(n - 2)!) spherical harmonics of
degree к in n variables. Denote the members of this basis by
Theorem 9.4.1 and the definition of Skj(i~) imply
(SkJ, Sk'J'} = Skk'Sjj'-
9.5 Action of an Orthogonal Matrix
Recall the definition of an orthogonal matrix.
Definition 9.5.1 Ann xn matrix О is called orthogonal ifOO' = /, where O'
is the transpose of O.
It is clear that if О О' = I then О' О = /, that is, O' is also orthogonal. Write
a vector x € RN as
/xt\
X =
X2
\Xn/

9.5 Action of an Orthogonal Matrix 453
and write the inner product of two vectors x and у in R" as
(x,y) -x'y =x\y\ +х2у2~\ Vxnyn.
If О is an orthogonal matrix, then
{Ox, Oy) =x'O'Oy=x'y = (x,y). (9.5.1)
So, if we write (x,x) = ||x||2, then ||Ox|| = ||x||. In particular, if ||?|| = l,then
We now show that the Laplace equation remains invariant under the action of
an orthogonal matrix. Write
(f)-(f f)'-
\OX J \OXl OXn J
Then the Laplace operator is given ЪуD-)'D-). For a change of variables x' = Ox,
This proves our claim.
Consider the action of the orthogonal matrix О on a spherical harmonic Hk{%)
defined by the mapping #*(?) -> Hk(O%). This action transforms an orthonor-
mal basis Skj(^), j — 1,2,..., с*,„ to another orthonormal basis Skj(Ol-), j —
1, 2,..., ck n for the space of spherical harmonics of degree к inn variables, since
Moreover,
~ ki,Sk,tG) (9.5.3)
e=i
and the coefficients (Ак}1) form an orthogonal matrix. This follows from the relation
}1
Ct.n
e=i
* = / SkJ(OS)SkJ.(OS)d(oG) = Sjr.

454 9 Spherical Harmonics
9.6 The Addition Theorem
The spherical harmonic of degree к in two variables, cos k6, has the addition
formula
coskO coskOo + sinkd sink90 = cosk@ — во). (9.6.1)
We note two interesting properties of this spherical harmonic. The only orthogo-
orthogonal transformation that fixes the point (cos во, sin#o) on the circle is the reflection
about the line в = во; the unique independent spherical harmonic of degree k,
which is invariant under this transformation is cosk(e — во). Moreover, since
sinke = cos(ke — |), we can view coskd as the basic spherical harmonic in
two variables. This uniqueness continues to hold in higher dimensions but with a
different basic function. The goal of this section is to obtain these generalizations.
There is no nonconstant harmonic polynomial that is invariant under all orthog-
orthogonal transformations. But there is exactly one, modulo a constant factor, invariant
under transformations that leave one point fixed. To prove this, start with the fol-
following preliminary lemma.
Lemma 9.6.1 Up to a constant factor, there exists at most one harmonic polyno-
polynomial of degree к that is invariant under all those orthogonal transformations that
leave one point r\ of the unit sphere fixed.
Proof. Since the inner product (Ox, rj) = (x, 77) for all orthogonal transforma-
transformations О that leave /? fixed, it is sufficient to prove that there is, at most, one (up to
a constant factor) harmonic polynomial Hk(x) that depends only on r and (x, 77).
From the homogeneity of Нк(х), we have
Hk(x) = co(x, r,)k + ar(x, г?)* + c2r2(x, r))k-2 + ¦¦¦ .
It is an easy calculation that
A[rl(x, rj)m] = m(m - l)(rj, r])rl(x, j])m~2 + l(? + 2m+n- 2)гг~2(х, rj)m.
Then AHt = 0 implies that the coefficients ck satisfy the relations
(к - m)(k -m- \)cm + (m + 2)Bk -m-2 + n- 2)cm+2 = 0
for m = 0, 1, 2,... and c\ = 0. This shows that c0 determines Hk(x), and the
lemma is proved. ¦
We now show the existence of the harmonic polynomial of Lemma 9.6.1. Let
Skj($), j = 1,2,..., Ck,n, be the orthonormal basis of the space of harmonic
polynomials of degree к in n variables. The form of (9.6.1) and the remarks after
that suggest the consideration of the function F(l=,rf) defined below. First define

9.6 The Addition Theorem 455
the vector function
and the orthogonality of the matrix С = (Akjt), we have
thensetF(?, 77) = S^'SCt?), where ? and 77 are points on the sphere. From(9.5.3)
kjt)
, Or,) = SiOtf
= S^yCCSir,) = F(?, 77),
when О is an orthogonal transformation. It is clear that F, as a function of ?, is
the restriction to the unit sphere of a harmonic polynomial of degree k. Moreover,
if О is any orthogonal transformation that fixes 77, then F(Ot;, 77) = F(?, 77). This
proves the existence of the function we were looking for. Its uniqueness was proved
in Lemma 9.6.1.
It is clear that the function F(^,r]) depends only on the inner product (?,17).
Denote it bkPk((%, /?)) and normalize by taking Pk((n, rf)) = Pk{\) = 1.
Definition 9.6.2 The function F(?, 77) = bkPk{(%, 77)) is called the zonal har-
harmonic of degree к with pole rj.
To determine bk, take ? = 77 to get
Ct.n
bk = Y^(SkJ(ri)J;
7=1
then integration over the sphere with respect to da>(rj) gives bkcon = ск,п or
bk = ckttl/con, where con = 2(я)"/2 Г (и/2) is the surface area of the unit sphere
x\ + x\ + ¦ • • + x2n — 1. Now multiply the equation
7?)) = ?5t
y=i
by itself and integrate with respect to 77. The result is
"I У|Ч|=1
ri))]2da>(Ti)

456 9 Spherical Harmonics
Therefore,
f
J\r
(? i)jG, r,))dw(r,) = —Skj. (9.6.2)
\r,\=l ck,n
This orthogonality relation for Pk will help us identify the function. Rotate ? to
ej = A,0,..., 0) and take ц = te\ + Vl — t2r]', where |/?'| = 1 and the first
component of rj' is zero. A change to polar coordinates gives
(9.6.3)
The Jacobian is given by
dx\dx2 ¦ ¦ ¦ dxn = rn~x sin"~2 0\ ¦ ¦ ¦ sin2 0„_з sin6n-2drd6\ ¦ ¦ ¦ dQn-2d4>,
which implies that, on the sphere,
dcon = sinn~2 eideidcon-i = A — t2)^~ dtdo)n^\. (9.6.4)
The orthogonality relation (9.6.2) can be written as
Pk(t)Pj(t)(l - t2)^dt = Шп Skj. (9.6.5)
<»n-\Ck,n
Thus Pk(t) = AC(k~2)l2 (t) is an ultraspherical polynomial. Since Pk(\) = 1, the
constant is given by A = 1/cf ~2)/2(l).
We have, therefore, proved the following addition theorem.
Theorem 9.6.3 LetSk ,_/(?)» 7 = 1,2,..., ckn, be an orthonormal set of spherical
harmonics of degree k. Then
(9.6.6)
Remark 9.6.1 This result contains (9.6.1) as a limiting case. Recall that
C(n~2)/2(t)
lim —7—тгт^—=cos^0, where t = cosd.
п^г cf-2)/2(l)
Remark 9.6.2 Since the integral (9.6.5) can be directly evaluated from the prop-
properties of ultraspherical polynomials, the value of ck<n, the dimension of the space
of spherical harmonics of degree k, can be computed from (9.6.5).
To see what the addition formula (9.6.6) looks like for n = 3, consider the set
of independent spherical harmonics in three variables listed at the end of Section
9.2. Rewrite them in terms of the associated Legendre function defined by

9.6 The Addition Theorem
457
It is easily verified that
so that an orthonormal set of spherical harmonics of degree к for n = 3 is given
by
/2Jt + l
V Ал
where
Pk(x), Атсо$тфР™(х), АтиттфР™(х), т = 1,...,
**m —
(k - m)\Bk + 1)
Now take
(к+т)\2л '
= (cos a, sin a cos 0i, sin a sin ф\)
and
ц = (cos/в, sin (i cos 02, sin $ sin fa)
so that (?, 77) = cos a cos /8 + sin a sin /в cos ф when ф = ф\ — фг- Since
ck3 2k+1 л n
— = — and Pk(l) = 1 = q/2(l),
co3 Ал
(9.6.6) gives
Pk (cos a cos /8 + sin a sin /в cos ф)
= Pk(cosa)Pk(cosP) + 2^
m=\
(k-m)\
(k+m)\
Pf(cos a)P™(cos /?) cos тф,
(9.6.8)
where ф = ф\ — фг-
The addition formula (9.6.6) shows that the ultraspherical function Ck
((?, t])) is the basic spherical harmonic in n dimensions, analogous to coskd in
two dimensions. Observe that it is possible to find ckn points щ, rj2,..., r)Ct n on
the sphere x\ + x\ + ¦ • ¦ + x% = 1 such that the matrix
(9.6.9)
Sk,2(r,CkJ
Sk,CkJr,Ctn)
is invertible. Now consider the system of скуП linear equations by choosing
77 = rji, rj2, ¦. ¦, rjCtn in (9.6.6). This system of equations can be solved uniquely
for SkJ (|) in terms of cf ~2)/2((?, щ)). We have proved the following:

458 9 Spherical Harmonics
Theorem 9.6.4 It is possible to choose points щ, т/2, ¦.., r\4n such that every
spherical harmonic can be expressed in the form.
9.7 The Funk-Hecke Formula
In this section we prove the Funk-Hecke formula, which will be useful in finding
a basis for the space of spherical harmonics of degree k. This leads to the addition
formula for ultraspherical polynomials.
In the following material we write the inner product of two vectors a and p
in R" as a ¦ p instead of (a, P). Observe that any continuous function / on the
interval [—1,1] extends to a continuous function of two variables g(a, P) on the
sphere defined by g(a, P) = f(a ¦ fi). If either a or fi is kept fixed, then we have
a function on the sphere. Now consider the integral
F(a, p) = f f(a ¦ r,)Cin'2)/2(p ¦ r,)d(o{r,). (9.7.1)
J\r,\=\
For an orthogonal transformation O,
F(Oa, Op) = / f(Oa ¦ т»С?~2)/2(Ofi ¦ ^dcotq)
J\r,\=l
= I /{а-О*т,)С?-2I2{р-О1т,)й<о{т,).
By the invariance of measure under orthogonal transformations, it follows that
F(Oa, OfJ) = F(a, /?). (9.7.2)
As a function of p, F is a spherical harmonic and depends only on a ¦ p. The
argument of the previous section implies that F(a, P) is a constant multiple of
Cf ~2)/2(a ¦ P). Therefore,
J\n\=l
f(a ¦ г^СГ^'ЧР ¦ r,)dw(r,) = \кСГ"'\и ¦ p). (9.7.3)
To find kk, take a = p = e\ = A,0,..., 0) and set r\ = te\ + Vl — t2r]' where
the first component of r( is zero. A calculation similar to the one used to derive
(9.6.5) gives
= f Г
J\f)'\=l J-l
= (On-y J f(t)C(kn-2)ll(t)(\ - t2)(n-3^2dt, (9.7.4)
-l
where fj' is obtained from 77' by removing the first component.

9.8 The Addition Theorem for Ultraspherical Polynomials 459
The Funk-Hecke formula is contained in the next theorem. It was first published
by Funk [1916] and a little later by Hecke [1918].
Theorem 9.7.1 Let f(t) be continuous on [—1, 1] and Sk(%) be any surface
harmonic of degree k. Then for a unit vector a,
/ f(a ¦ r,)Sk(r,)dco(r]) = XkSk(a), (9.7.5)
J\r,\=l
where kk is given by (9.7.4).
Proof 1. By (9.7.3), the result is true when Sk(rj) is replaced by C(k~2)l2(P ¦ rj).
Theorem 9.6.4 says that any Sk(r]) is a linear combination of C(k~2)/2{fit ¦ /?). Thus
(9.7.5) follows. ¦
Proof 2. An integrated form of the addition formula (9.6.6) is
Ь.№ = (п-2)/2 / СГ2)/2(? ' r,)SkiJ(r,)dco(r,). (9.7.6)
ШпС\ " A) J\t)\=l
Now SkJ (?) can be replaced by any spherical harmonic Sk(t;), because the Skj (?)
form a basis for the space of such functions. Multiply (9.7.3) across by
Sk(P)dco(f3) and integrate with respect to /8. The theorem follows after an ap-
application of (9.7.6). ¦
Remark 9.7.1 Formula (9.7.6) suggests another way of arriving at the zonal
harmonic function. The map ф : SkJ -» Skj(%) is a linear functional on the
finite-dimensional space of spherical harmonics of degree k. So ф is given by an
inner product, that is, there exists a function g% such that
Written out in full, we have
SkJG)= I SkJ(r,)g^r,)dr,. (9.7.7)
One can then prove
9.8 The Addition Theorem for Ultraspherical Polynomials
To motivate the technique used to find a basis for the space of spherical harmon-
harmonics, we start with an integral formula for ultraspherical polynomials obtained in
Theorem 6.7.3. The formula can be written in the form
-,(n-2)/2
/i гм„ •-» i i . .
(9.8.1

460 9 Spherical Harmonics
To obtain a proof different from the one given in Chapter 6, consider the integral
g(x)
У|«»_1|=1
[x
r]n-i]kdo}(fjn-i),
where x = (xb . ..,*„), ei = A,0,..., 0), 17 = te\ + Vl -t2rjn_u t = 77 • eb
гпА?\п_1 is obtained from /;n_i by removing the first entry, which is zero. In what
follows we write цп~\ instead of f]n-\. We first show that g is a harmonic function.
Observe that
|4 = *(* - 1) /
and for j = 2,..., и,
I t(t 1
where ey = @,..., 1,..., 0) with 1 in the y'th position. So
лк-г
7=2
n_l) =0
since
j=2
Thus g(x) is a solid spherical harmonic of degree k. It is clear that g is also invariant
under all orthogonal transformations that fix e\. If we take x — eu then the value
of g(x) is &>„_!, the volume of the unit sphere in n — 1 dimensions.
Set I = x/\x\ and then ? = t€\ + Vl — /2?n-i- From the previous remarks,
is a multiple of CJ,"' ^ (t). The normalization at t = 1 gives
cf-2)/2(o
-=Ы.м
(9.8.2)
Now apply the procedure used in the derivation of (9.6.5) to the above integral to
get (9.8.1).
The next step in the derivation of the addition theorem for ultraspherical poly-
polynomials is to take any spherical harmonic of degree к in n variables and express it
in terms of a spherical harmonic of a different degree in n — 1 variables. For this
purpose consider the integral
(9.8.3)
i»-il=i

9.8 The Addition Theorem for Ultraspherical Polynomials 461
It is immediately obvious from the argument used above that g is a spherical
harmonic of degree к in n variables. Moreover, by the Funk-Hecke formula, the
integral can be written as
«(?) = / [t + iy/l ~t4n-i ¦ ijn-i]kSj(ijn.i)da)(iin.i)
¦/141=1
= S,(?,_,)flfc-2 f(t + iV^2s)kC^)l2(s)(\ -
Rodrigues's formula for ultraspherical polynomials,
A x) Lk(x)- V *)
implies that the term A - s2)(n~4)/2Cf~3)/2(s) in the integrand is a constant times
(l2)J
Integration by parts j times then gives
- t2y'2 J
where AT is a constant. By (9.8.1) the last integral is proportional to CJk*f~2)/2(t).
We have, therefore, proved that 5Д|л_0A - t2)i/2CJk±f~2)/2(t) is a spherical
harmonic of degree к in n variables. Recall that t and ?„_! are related by te\ +
Vl — t2^n^i = ?, a point on the и-dimensional unit sphere. With the notation of
(9.4.5), Sjfi(i-n-\), ? = 1,2,..., Cjttl-\, form a basis for the surface harmonics of
degree j in n — 1 variables. Thus the set of functions
ЗД„_1)A - t2y/2Ciy-2)/2(t), j = 0,... Д, I = 1,..., <:;,„_,, (9.8.4)
forms an orthogonal basis for the vector space of spherical harmonics of degree к
in n variables. The orthogonality is easily verified, and since ck<n = Ylkj=ocj.n-u
there are the correct number of vectors to form a basis.
Observe that, since
j = f [C]ktf-2)l\t)}\\-t2y+(n-3)'2dt
J — 1
-(п-2)/2)Пк-])\(к + (п-2)/2У
the functions in (9.8.4) form an orthonormal basis when multiplied by 1/y/A] =
Bj. If this basis is used in (9.6.6) with
?='<?! +

462 9 Spherical Harmonics
and
T] =
then we have
(n - 2)шп
By (9.6.6) the inner sum is
2/+»-3 (n-3)/2
; ^7^; (?n-i • Щп-
<un_i(n-3) J
This gives the next theorem due to Gegenbauer [1875].
Theorem 9.8.1 For an integer n > 3,
cf-2)/V
j=Q
'<"k-j (*)<¦'j V9n-1 • Лп-l), (У.
w/геге
Г (и - 3J2^(^ - у)![ГG + (и - 2)/2)]2By + и - 3)
u
Remark 9.8.1 The result is true for и = 2 and и = 3 but a limit has to be taken.
The case и = 2 is the addition formula for the cosine function and и = 3 gives the
addition theorem for Legendre polynomials, (9.6.8).
Formula (9.8.5) is often written as
Cf~ )/2 (cos a cos p + sin a sin fi cos ф)
к
= Y, «.(sin «У C^("-2)/2(cos «)(sin pyc?f-2)/2 (cos p)Cf )/2(cos ф).
j=o
(9.8.5')
The addition theorem can be extended to any spherical polynomial C?(x), X > 0
by analytic continuation, since both sides of (9.8.5') are rational functions of

9.9 The Poisson Kernel and Dirichlet Problem 463
к = (n — 2)/2. This identity holds for complex Л as long as no poles occur. It is
also worth noting that (9.8.5') can be obtained from (9.6.8) by differentiating with
respect to ф.
Remark 9.8.2 The addition formulas for trigonometric functions have general-
generalizations to elliptic functions that are different from those given by (9.8.5). Elliptic
functions satisfy an addition formula of the type
f(u + v) = A(f(u), /(«)), (9.8.6)
where A(x, y) is an algebraic function. Weierstrass proved that the only solutions
of (9.8.6) are algebraic functions, algebraic functions of ecm for some constant c, or
algebraic functions of elliptic functions. Apparently, Weierstrass never published
this result though he mentioned it in his lectures. See Copson [1935, p. 363].
9.9 The Poisson Kernel and Dirichlet Problem
The solution of the Dirichlet problem for the unit disk is given by the integral
I r2n
- 2r cosF> -ф)+г2
The Poisson kernel,
\-r2
(9.9.1)
1 - 2r cosF> - ф) + r2
is the sum of all the two-dimensional zonal harmonics with poles at ф. In n dimen-
dimensions, the zonal harmonic of degree к with pole at r\ is, according to (9.6.6), given
by
2k + n-2 (n_2)/2
(n - 2)шп
In Chapter 6, we saw that
00 л, , ^
(I ¦ I?)-
f-? (n - 2)an K
The generalization of (9.9.1) is contained in the next theorem.
Theorem 9.9.1 Suppose f is a continuous function on the n-dimensional sphere.
Let a be a point on the sphere and 0 < r < 1. Then
If \-r2
u(ra) = — / /^Tf
is harmonic inside the sphere and u(a) = f(a) on the sphere.

464 9 Spherical Harmonics
This theorem is proved using the same method as we used in the two-dimensional
case.
9.10 Fourier Transforms
In Chapter 4, we considered the Fourier transform of functions of two variables.
If the point (x, y) is identified with the complex number x + iy = z = re'e', then
we can write the Fourier series of an integrable function f(re'e) as
(9.10.1)
The Fourier transform of (9.10.1) was seen to be expressible in terms of Bessel
functions. This continues to be true in higher dimensions as well. First, we need a
definition.
Definition 9.10.1 A function f : R" -> R is called radial if there is a function
/o(m) on 0 < и < oo such that f(x) = /o(M).
We generalize the functions fk(r)e'ke in (9.10.1) to functions expressible as
the product of a radial function and a harmonic polynomial in higher dimensions.
The main result of this section concerns the Fourier transform of such functions.
The presentation is based on Bochner [1955], though the proofs of the basic results
are different.
Lemma 9.10.2 For any spherical harmonic &(?) of degree к in n variables,
(9.10.2)
If l=i '" ""'
Proof. By the Funk-Hecke formula, integral (9.10.2) equals
/-1
Now apply Gegenbauer's formula:
The result follows. ¦
Let / e L\{Rn) and let Tf be its Fourier transform:
Tf{y) = [ e-2ni{yx)f{x)dx. (9.10.3)
JR"

9.10 Fourier Transforms 465
Theorem 9.10.3 Suppose f e Li(Rn) is of the form f(x) = /0(|*|M*(?). Then
Tf(y) = F0(\y\)Sk(rj),
where
/•OO
F0(t)=2jriktl-n<2 / Ms)Jk^+n/2B7rst)sn^ds (9.10.4)
Jo
and у = \y\r].
Proof. It is easy to see that with x — (x\,..., xn) and Xj = s^j, we have
Tf(y) = Г F0(s)s"-1 ( f e-2
Jo \^l?l=i
The result now follows from Lemma 9.10.2.
An interesting consequence is the next result, which is obtained by combining
Theorem 9.10.3 with the fact that for Re(/t + v) > 0,
lim /q _ ,VK.,. -. rA+(y_
To obtain (9.10.5), apply Pfaff's transformation (Theorem 2.2.5) to the 2F\ in
D.11.4) and then take the limit.
Corollary 9.10.4 For a spherical harmonic
lim
(9.10.6)
Г((к-а)/2)
when \y\ ф 0 and у = \y\r].
Proof. Take F0(t) = (V2~nt)ae^et in Theorem 9.10.3 and then use (9.10.5). ¦
A particular case of (9.10.6) is worthy of note. Take a = —n/2. We have
(9.10.7)
\У\2
This implies that |jc|~"/2^(^) is an eigenfunction of the Fourier transform with
eigenvalue i". Observe that this transform is actually the Abel mean of the Fourier
transform.
One may restate Theorem 9.10.3 for the case where / e L\{R") is of the form
f(x) — fo(\x |)Sjfc(jc) and Sk(x) is a homogeneous harmonic polynomial of degree
к. It is sufficient to remark that Sk(x) — \x\"Sk(%).

466 9 Spherical Harmonics
Theorem 9.10.5 Supposing f is integrable and of the form fo(\x\)Sic(x);
Tfiy) = F0(\y\)Sk(y),
where
poo
F0(t) = 2nikrk^n+l / Ms)Jk+in_lB7rst)si2n+kds.
Jo
It is well known that the Fourier transform of е~ж^ is е~л^ . A similar result
holds when these exponential functions are multiplied by spherical harmonics.
Theorem 9.10.6 For any homogeneous harmonic polynomial Sk{x) of degree k,
f e-2*i(yx)e-*M2Sk(x)dx = ^e-^Skiy).
Thus е"~лМ Sk(x) is an eigenfunction of the Fourier transform with eigenvalue ik.
Proof This can be derived from Theorem 9.10.5 and the following formula of
Sonine:
f°°
/ Л
Jo
(st)e''2tv+ldt = ^-e"s2/4. (9.10.8)
9.11 Finite-Dimensional Representations of Compact Groups
Representation theory provides an important and powerful approach to special
functions. Unfortunately, we can do no more than devote a few sections to this
topic. After giving basic definitions, we show how Jacobi polynomials appear in
representations of 5GB). We also explain how spaces of spherical harmonics give
irreducible representations of 5GB).
Suppose G is a group and V a finite-dimensional vector space. Let GL{V)
denote the group of linear transformations from V onto V.
Definition 9.11.1 A finite-dimensional representation of G in V is a homomor-
phismfrom G to GL(V). If G is a topological group, we assume that the homo-
morphism is continuous.
Thus if
U : G i-> GL(V)
is a representation, then U(gig2) = U{g\)U{g2)- The linear mappings U{g\),
U{g2), and U{g\gi) can be represented by matrices if we choose a basis for V.

9.11 Finite-Dimensional Representations of Compact Groups 467
Suppose dim V = n and {x\, x2, ...,*„} is a basis for У. It is clear that the matrix
entries Ujj(gig2), 1 < i, j < n, satisfy the relation
n
Uijigigi) = Yl Uik{gi)Ukj(g2), 1 < i, j < n. (9.11.1)
k=\
If a different basis [x[, x'2,..., x'n] is chosen, then there exists a matrix P such
that
U(g) = PU'(g)P~l
for all g e G. Here U'(g) is the matrix representation of U corresponding to the
new basis.
Suppose U\ and Ui are two representations of G in the vector spaces V! and
V2. The two representations are called isomorphic, or more briefly, Vi and V2 are
isomorphic, if there is a linear isomorphism
Г : Vi -> V2
such that
T о Ui(g) = U2(g) oT for all g € G.
Now suppose that the vector space У has an inner product (jc, y) defined on
it. Let [xi,X2,...,xn] be an orthonormal basis of V with respect to this inner
product. Then
j ,xi). (9.11.2)
Definition 9.11.2 A representation U : G \->- GL{V) is called unitary if
(U(g)x,U(g)y) = {x,y)
for all x, у е У and g e G.
In this section, we study the representations of the compact group 5GB). For
compact groups, it is always possible to define an inner product on У such that a
representation U of G in У is unitary. This is easy to prove, if the existence of an
invariant measure on G is assumed. In fact, there exists a unique measure dg on
G such that, for a continuous function /,
(a) / f{g)dg = f f(gh)dg,
Jg Jg
where h and g e G, and
(b) I dg=\.
Jg

468 9 Spherical Harmonics
The first condition gives invariance under right translation. It can be shown that the
measure dg is also invariant under left translation. This invariant measure is often
called the Haar measure. See Halmos [1950]. The only group we study in detail
here is SUB). For this group the invariant measure is easy to construct. 5GB) is
the group of all matrices with complex entries of the form
a b
-b a
with determinant one, that is, \a\2 + |b|2 = 1. Suppose
a0 bo'
h= • и -
—bo ao
is also in 5GB). It is easily checked that gh is constructed by the parameters a\
and b\ where
a\ = aoa — bob,
b\ = boa + aob.
Thus
dg = da л da л db л db (9.11.3)
is an invariant measure on 5GB). A simple calculation shows that
dgh = da\ л da\ л db\ л db\
= (Mol2 + \bo\2Jda AdaAdbAdb
= dg. (9.11.4)
Suppose that ( )i is an inner product on the vector space V. For a given repre-
representation U of G in V, define a new inner product by
(x,y)= f (U(g)x,U(g)y)idg. (9.11.5)
Jg
That U is unitary with respect to this inner product follows from property (a) of
the Haar measure. Note that
(U(h)x, U(h)y) = f (U(g)U(h)x, U{g)U{h)y)xdg
Jg
= f (U(g)x,U(g)y)idg
Jg

9.12 The Group 51/B) 469
Let U: G -> GL(V) be a representation of G, and let W be a subspace of V.
We say that W is invariant under the action of G if G(g) maps W onto itself.
This gives another representation of G, namely Uw'-G —> GL(W). Note that
?V(g) = ^(g)lve- Uw is called a subrepresentation of G. If f/ has no nontrivial
subrepresentations, then U is called irreducible.
Suppose G is a compact group; then there is an inner product on V with respect
to which U is unitary. If W is an invariant subspace of V, then the orthogonal
complement of W is also an invariant subspace. This is easily verified. In this
situation, the matrix for U(g) has the form
fUw(g) 0
" V 0 Uw±(
By continuing this process, we see that
v = Wi e w2e---e wk, (9.11.6)
where W\,... ,Wk are irreducible. It can be shown that this decomposition is
unique up to isomorphism. Thus, if
v = w[ e w!, e • • • e wt,
then к = I and, after renumbering if necessary, W, ~ W[.
9.12 The Group SUB)
Recall that 5GB) is defined as the group of matrices of the form
\ h), |fl|2 + |fc|2=l. (9.12.1)
-b a)
Thus, the group is defined by three parameters, which one may choose to be \a \, arg
a,andargb. Whenab ф 0, one can uniquely choose another set of three parameters
ф, ty, and в called Euler's angles. These are obtained from the relations
a = е1(ф+ф)/2 cos -в, b = гУ w>~l/')/2 sin -в, (9.12.2)
where 0 < ф < 2тг, 0 < в < л, and -2тг < rj/ < 2л. When ab = 0, the
correspondence between a, b and в, ф, \jr is not one to one. Another way of
writing the relation (9.12.2) in terms of matrices is
a b
-b a
(9.12.3)

470 9 Spherical Harmonics
Set
with й and b as in (9.12.2). Then (9.12.3) is equivalent to
g(A>, 0, f) = 8(ф, 0, Q)g(p, 0, 0)g@, 0, ir). (9.12.4)
It also follows from (9.12.2) that
2|fl|2-l, e» = -??-, and ** = ??[. (9.12.5)
|a||fc| fc \a\
A question that arises here is the following: If g@, 0, ф) = gi@i, 0i, \js\)
8г(Фг, 02, V^), then what is the relation of 0, 0, ty to ф\, 9\, 1^1 and (/>2, O2, ^2?
This general case follows from the consideration of the particular case when ф\ =
ilr1 = ij/2 = 0. Then
cos i^i г sin \в\ \ ( cos \e2ei<k/2
i sin 10i cos \в\ ) \ i sin \вге~1фг11 cos i
So
a = cos 6i cos в2е sin ^0i sin
22 2 2
a = cos ^6>i cos -в2е1фг'2 - sin \в\ sin -в2е~1ф2/2,
(9.12.6)
b = г ( cos -0i sin -в2е1ф2'2 + cos -02 sin -91e~i<h
From (9.12.5) and (9.12.6), we get
cos 0 = cos 0i cos 02 — sin 0i sin 02 cos 02,
1ф sin 0i cos 02 + cos 0i sin 02 cos 02 + i sin 02 sin 02
sin0
... ..,, cos 1в\ cos i02e"*2/'2 — sin i0i sin 1в2е~1ф2^2
COS \d
This gives the formulas for the product g @, в\, 0)g @2, 02, 0). To obtain the general
case the following remarks are sufficient. Observe that g@i, 0,0)g@, 0, V0 =
#@i ~Ь0, 0, V0, #@, &¦> 1^)^@» 0, ф2) = ^@, 0, ^"Ь^2), ^@, 0, O)g(iff, 0,0) ^
g@, 0, ^r), and ^@, 0, Vi)g@2,0,0) = g(fi + 02,0, 0). Apply these relations
to
, 0, 0)?@, 0i, 0)g@,0, ^i)?@2, 0, 0)?@, 02, 0)?@,0,

9.13 Representations of SUB) All
The general case follows immediately.
Remark The invariant measure dg for G — SUB) defined by (9.11.3) when
written in terms of Euler angles is
dg = - sinfi'ded<pd\/f.
Usually the measure is normalized to
dg = г s
16tt2
so that
' dg=l.
L
Observe that dg is half the product of the normalized measure on the sphere,
sin ededcp/4rt, and the normalized measure on the circle, d\js/2n.
9.13 Representations of SVB)
Let VN+\ denote the (N + l)-dimensional vector space consisting of homogeneous
polynomials of degree N in two complex variables with complex coefficients. If
P e VN+i, then
N
P(x1,x2) = '22rkxklx?-k, (9.13.1)
where x\ and x2 are complex variables and ru ,k = 0,..., N, are complex con-
constants. We also write P(x) = P(x\,x2) with
(x\
X =
\X2
A representation of SL2(C) in VN+\ can be defined by
U(g)P(x) = P(g'x), (9.13.2)
where
g=( ,), ad - be — \,a,b,c,d e C.
\c d)
Note that P(g'x) — P(ax\ + cx2, bx\ + dx2). It is easy to check that
U(gig2) = U(gl)U(g2). (9.13.3)
It can be shown that U gives an irreducible representation of SL2(C). In fact, all
the finite-dimensional irreducible representations of SL2(C) are of this form. The
restrictions to the compact subgroup 5GB) of these representations give all the
finite-dimensional irreducible representations of 5GB).

472 9 Spherical Harmonics
It is standard practice to write the polynomial in (9.13.1) in a slightly different
form. Let N = 21, so that ? is an integer multiple of 1/2. Write
i
P(xl,x2)='^2rnxtl~nx%+n. (9.13.4)
n=-l
Here I + n takes integer values from OtoN = 21.
Associate with P a nonhomogeneous polynomial Q given by P(x, 1) = Q(x).
Thus,
i
Q(x) = Y, г"Хг~п, (9.13.5)
n=-t
and the homogeneous polynomial corresponding to Q is obtained from
P(xux2)=xleQ(xl/x2).
Denote the space of all nonhomogeneous polynomials of the form (9.13.5) by Ht
and the representation of 5GB) corresponding to U in the space Ht by Tt. This
implies that
Tt{g)Q{x) = (bx+dJlQ((ax+c)/(bx+d)). (9.13.6)
Since 5GB) is compact, the inner product (9.11.5) shows that it is possible to
choose an inner product in Ht such that Tt is unitary. With respect to this inner
product, the basis {l,x,..., x2t} of Ht is orthogonal. In fact, we have the lemma
given below. Since the above inner product is defined up to a constant factor,
assume that A,1) = B?)!.
Lemma 9.13.1 (хе-т,хе~п) = (? - n)\(l + n)\Smn, -I < m, n < I
Proof. Let
еи'2 О
О е-»/2
From (9.13.6), it follows that Tt(g)xe'k = е~а'хе~к. Since Tt(g) is unitary,
(re~m xe"n) — lTAQ~\xl~m ТЛдЛх1~п\
This implies that (xe~m, xe~") — 0 for m ф п.
The case m = n requires a little more work. Take
/cos^r —sin^A
О r^ I I
\ sin I? cos|f J

9.14 Jacobi Polynomials as Matrix Entries 473
and observe that, by the first part of the theorem,
0 = (xe-n,xe-n+l) - (Tt(g)xe-n, Te(g)xe-n+l}
= (ut+nvt-n,ut+n--ivt-n+l), (9.13.7)
where
f ¦ l \ l л ( l \ ¦ l
и = I sin -t \x + cos -t and v = I cos -t \x — sin -t.
Take the derivative of (9.13.7) with respect to t and set t = 0. The result is
(i + n){xl-n+\xl-n+l) -(l-n + \){xl~n,xl~n) = 0.
The theorem follows when the condition A, 1) = B?)! is used. ¦
9.14 Jacobi Polynomials as Matrix Entries
Choose the functions
1/л„(г) = — —, _?<„<?, (9.14.1)
V(^-n)!(?+«)!' " -
as an orthonormal basis for the space Ht. Then, for g e SL2(C),
{ax+c)l~n{bx+d)l+n J4 „
Td8)fn(X) = V(? -h)!(/ + hI =ЕСШшЫ. (9.14.2)
Now use Taylor's formula to see that the coefficient t^n(g) is given by
m)\ dl~
Set у + 1 = a(bx + d). Since ad — be — 1, we have ax + с = y/b and
(9.14.3)
If g e 5GB), then the use of Euler's angles provides a simpler formula for
n(g). Consider the decomposition
Ъ(8(ф, в, f)) = Tt(g((t>, 0, 0)Oi($@, 0, 0))ГИ«(О, О, f)). (9.14.4)
It follows from (9.12.3) and (9.14.2) that
Tt{gD>, 0, 0М„ = e-'"V», -?<«<?. (9.14.5)

474 9 Spherical Harmonics
This means that Т^(ф, О, О)) is a diagonal matrix given by
/ eW 0 \
The matrix for Te(g@, 0, i/r)) is similar. Write tlmn{g{0, в, О))
Equation (9.14.4) gives
(9.14.6)
); then
Denote tlmn(Q), 0 < в < n, by P^n(co&0), so that
(9.14.7)
Since
,0,0) =
cos % i sin f
2 2
sin ^ cos x
с dr
the quantity fee in (9.14.3) equals — sin2 | = (cos в — l)/2. Thus, replacing у with
(г - 1)/2 in (9.14.3), we obtain
{t-n)\{t + пЩ1-т)Г
Al-m
[A )/-"A )*
(i г)[A г)A + z)]. (9.i4.8)
This shows that Р^„ (г) can be written in terms of a Jacobi polynomial. It is a
constant multiple of
ч(
)
9.15 An Addition Theorem
It follows from (9.12.7) that when
?(<?, 0, V) = S@,6»ь OXgOfe, ^2, 0)
we have
COS0 = COS0] COS02 — Sin0] sin02COS02- (9.15.1)
Use this in (9.11.1), (9.14.4), and (9.14.7) to get an addition formula:
е-{(тф+пф)Р1п(соь9) = ? е-'к**Р*к(са&е1)Р?„(со*в2), (9.15.2)

9.15 An Addition Theorem 475
where в, ф, and \fr are given by (9.12.7). When ф2 = 0 and 0 = 9\ + 62 < n, we
have 0 = ^ = 0. In this case
(9.15.3)
The addition formula (9.15.2) is the analog of Graf's addition formula D.10.6)
forBessel functions. Formula (9.8.5) is an analog of Gegenbauer's addition formula
for Bessel functions. An analog of this for Jacobi polynomials was found when
fi = 0 by Sapiro [1968] and in the general case by Koornwinder [1972, 1975].
Koornwinder used the addition formula to derive the Laplace-type integral for
Jacobi polynomials given in Exercise 6.45 and an integral formula for a product
of Jacobi polynomials. Koornwinder [1974] observed that it is possible to obtain
the product formula from the Laplace-type integral by using a result of Bateman
[1932, pp. 392-393]. Bateman's result is
P^\(l+st)/(s + t))
=> bk,n{s + t) 7^-jr > (9.15.4)
where bkn is denned by (9.15.4) when t = 1,
Э 1
г +(Ba + l)coth7? + B^ + l)tanh7?)— \F = 0, (9.15.6)
Эту2 Эту]
r" (-1-) *=o
Bateman proved (9.15.4) by showing that both sides of the equation are solutions
of the partial differential equation
^ + (Ba + l)cot§ - B0 + Dtanf)^
Э2 Э
+ (B + l)h B^ l)h)
when
Now recall the formula in Exercise 6.45:
, x _ (I _ X\U2 -I"
h i'v 1 - x2 hcos# dma^(u,9),
(9.15.7)
where a > ft > —1/2 and

476 9 Spherical Harmonics
This implies
(x + y)n —
-IT
Jo Jo
2
"I П
dmaj(u,6).
J
When this formula is combined with (9.15.4) and (9.15.5) the result is the product
formula for Jacobi polynomials given by
1 Кф) y) + A - x)(l - y)}/2
-x2)(l - y2)u cos в - l]dma,p(u, в). (9.15.8)
9.16 Relation of SVB) to the Rotation Group SOC)
The description of elements of SUB) in terms of Euler angles suggests a con-
connection of SUB) with the group of rotations in three dimensions. The explicit
relationship is given below. An interesting consequence of this connection is that
the spaces of spherical harmonics in three variables are seen as the irreducible
representation spaces of SUB).
The rotation group SOC) consists of 3 x 3 matrices g with real entries and de-
determinant 1 such that the transpose of g is also its inverse. These are the orientation-
preserving linear mappings g from /?3 to /?3 such that for x € /?3, |x|2 = |gx|2.
To define a homomorphism ф from SUB) to SOC), first identify the points
x = (xi, x2, хз) € /?3 with 2x2 Hermitian matrices of trace 0:
/ —хз Xi - ix7
\X]+IX2 X3
Note that detMx = -|x|2. For g e SUB), define
gMx^ forx€#3.
It is clear that <p{g)x is a Hermitian matrix with trace zero and det[0 (g)x] = —|x|2.
Moreover, <p(g) is an orientation-preserving linear mapping. Thus <p(g) can be
identified with an element of SO C), and we have a homomorphism ф from SUB)
to SOC) whose kernel is easily seen to be {±/}. It can also be shown, though
we do not do so here, that ф is an onto mapping. Thus ф gives an isomorphism

9.16 Relation of SUB) to the Rotation Group SOC) 477
of SUB)/{±I] onto SOC). Recall that the Euler angle ^ in (9.12.2) ranges over
[—2л-, 2л) whereas the range for \jr in a rotation is [0,2л-). This is related to the
fact that ±g € SU{2) are associated with the same rotation in SOC).
It is easy to check that ш\ (t), o>2(t), л>з(?) е SUB), defined by
cos( isin(\ /cost - sint
A)
wi0) = , • • , , , , •
i sin t cos t ) V sin t cos ?
e" 0
О е~"
correspond to rotations (by an angle of 2t) about the x\, x2, and X3 axes respectively.
Thus by (9.12.3) and (9.12.4) we can view any rotation with Euler angles ф,в,ф-
as a product of a rotation by the angle \j/ about хз, a rotation by the angle в about
x\, and a rotation by the angle ф about хз.
The object of the remainder of this section is to show that Hk(x), the space
of harmonic polynomials of degree к in three variables provides an irreducible
representation of SOC) and hence of SUB) as well.
Let ф be the homomorphism from SUB) onto SOC). We have seen that
0(g!) = 0(g2) if and only if g\ = ±g2. It is clear that if U is a (irreducible)
representation of SOC) in a vector space V, then ф ¦ U is a (irreducible) rep-
representation of SU{2) in V. Conversely, if Г is a (irreducible) representation of
SUB) in У such that Г (-identity) = identity, then T gives rise to a (irreducible)
representation of SOC).
Recall that in the notation of Section 9.13, the proof of Lemma 9.13.1 shows
that the functions xl~k e Ht (not to be confused with Hk(x)) for к = —?,...,?
are eigenvectors of Tt(g) corresponding to the eigenvalues e2lk'(k = —?,...,?)
when g = л>з(—t). This fact may be used to prove that if U is a representation of
SUB) in V with dim V = 21 + 1 and e2Ut occurs as an eigenvalue of ?/(w3@),
then (?/, V) is isomorphic to Gi, #^). We have
у = #tl e я*2 е ¦ • ¦ e нкр
for some integers k\,... ,kp. If p = 1 and &i = •?, then we are done. If p ф 1,
then &, < € and the eigenvalues of ?/(л>з(О) are of the form e2""' with \m\ < €.
Thus V ~ #f and the result is proved.
Now define a representation t/ of 5OC) in Hk(x) by
U(g)p(x) = p(g~[x)
for x e /?3, p e Hk(x), and g e ?ОC). This in turn gives a representation t/ • ф
of SUB). Now observe that (jcj + гх2)* е Hk(x) and that w3 (r) maps to a rotation
by an angle 2t about хз. Thus U(ф (л>з(?))) has (xi + гх2)А as an eigenvector with
eigenvalue e2lkt. Thus by the result of the previous paragraph, Hk(x), the space of
harmonic polynomials of degree k, is an irreducible representation of SUB).

478 9 Spherical Harmonics
Exercises
1. Verify that the only polynomial solution of the differential equation
A - x2)y" - 2A + m)xy' + (л - m)(n + m + l)y = 0,
where т,я eZ,n-m>0,is the polynomial ^С™_Ш2 (х).
2. Let Vic denote the vector space of all homogeneous polynomials of degree к
in n variables. For a = (а\,...,а„), let xa = x x ¦ • ¦ x"n and
Da = да>+а2+-+а»/дх^дх22 ¦ ¦ ¦ дх«\
For a polynomial P in n variables, let P(D) denote the differential operator
obtained by replacing xa by Da in P(x). For P, Q e Vk, define (P,Q) =
P(D)Q (or P(D)Q if complex coefficients are used). Prove that ( , ) is an
inner product on Vk-
3. Let Д be the Laplace operator and к > 2. Show that Д : Vjt ->¦ Vjt_2 is
mapping by proving that there is no nonzero vector in Vjt_2 that is orthogonal
(with respect to the inner product in Exercise 2) to the range of A.
4. Let Hk С Vk denote the subspace of harmonic polynomials. Suppose Lk =
{P€Vk\ P(x) = \x\2Q(x), Q e Vk-2}- Prove that
Vk = Hk ® Lk.
5. Use the result of Exercise 4 to prove that if P e Vk, then
P{x) = P0(x) + \x\2Pdx) + ¦¦¦ + \x\2ePe(x),
where Pj is a homogeneous harmonic polynomial of degree к — 2j, j =
0,1, ...,?. Deduce that any polynomial in n variables and restricted to the
unit sphere is a sum of spherical harmonics.
6. Use the results of the previous problems to show that с к ,„ = dimension of the
space of spherical harmonics of degree к in n variables = dim Vj — dim Vk-2
+ k- 1\ _ /n+k-3'
к ) \ Jfc-2
7. Prove that it is possible to choose points щ, ту2, •.., ЦСкп such that the matrix
(9.6.9) is invertible.
8. Show that the function g^(ту) in (9.7.7) is given by ?SkJ (l-)Skj(rf).
9. Derive the addition formula (9.8.5) from (9.6.8) by differentiation.
10. Prove Theorem 9.9.1.
11. Use (9.8.5') to derive:
(a) Gegenbauer's product formula
sin#sin0cos
Jo

Exercises 479
where A. > 0 and
:' = / (^
(b) The integral formula (9.8.1) for ultrasphencal polynomials.
12. Prove that when one lets X -*¦ 0 in formulas (a) and (b) of Exercise 11, one
obtains the well-known formulas
cosn#cosn0 = -[cosn(# + ф) +cosn(# — ф)],
cos пв = -[еш + е-ш].
13. 1л1х,у е R",x = Щ,у = гг), |f| = \ц\ = l,and# > r, and note that
(a) Let J2"=i У] эГ = r(rl' v)- Use Taylor's theorem to prove that
(-D* i. i. ¦,
X —
(b) Deduce Maxwell's formula that
14. Deduce the following formulas from (9.15.2):
1 Г
(a) P^(cos6>i)P/n(cos6>2) = — / е'(кф2~тф'
mk кпУ In J_n
1 /¦"
(b) Pi (cos 0i)Pf (cos0г) = — / Pi(cos6\ cos 02 — sin0i sin02co:
2л- J-n
where ф, г/г, and 0 are as defined in the text. In (b) t is an integer.
15. This problem gives a generating function for spherical harmonics in three
dimensions. Let x = (xi, x2, X3) and un = (—2?, 1 — t2, i + it2). Then define
Hk(x) by
n
(u ¦ x)" = [x2 + гх3 - 2xit - (x2 - ixj,)t2f = t" ^ Hk(x)tk.
k=-n
(a) Show that Я* = (-\)kH~k.

480 9 Spherical Harmonics
(b) Prove that H*(x) is a homogeneous harmonic polynomial by showing
that V2(m • x)" = 0.
(c) Let v — {—2s, 1 — s2, i + is2) and define the polynomial ф(и, v) by
-'111=1
Use the technique of Lemma9.6.2 to show that </>(«> v) is a constant times
(u ¦ v)n. [Note that и ¦ и — 0 and v ¦ v — 0.]
(d) Define S*(f) = |х|-"Я*(х), where % = x/\x\. Show that
1) / 2« ч ,p
Л Г(п+C/2))
(e) Also prove that
16. Show that the associated Legendre functions P™ (x) defined by (9.6.7) satisfy
the recurrence
form = 0, 1,2,....
17. Derive the formulas
Г(к+т + 1) Г i— t
(a) Pk(x) = — / (x + Vx2-\ cos x/rf cosmx/rdx/r,
л l (к + 1) Уо
m+ 1) Bsin6>)"
/
Jo
where Rex > 0, 0 < в < л, and m is a nonnegative integer.
18. Show that
19. Show that
]2
(—IVя f°°
=-f—^— / uke~ucose Jm(u sin 6)du.
(k - my. Jo
[Ptm(cos0)]2
22>G-m)!G+m)!G!J(?-7)!
20. Prove the relation (9.10.5), that is,

10
Introduction to ^-Series
When one is counting, one may use a generating function to keep track of the
number of objects being counted. This is yet another way in which hypergeometric
series arise. For example, the finite binomial theorem is usually written as
(x + у)" =
4=0
The coefficient Q) counts the number of ways n — к х and к у can be arranged.
The usual argument is to observe that the first у can be put in any of the n places,
the second in n - 1 places, and so on, until the
n{n- 1) ••¦(«- k + l)
ways are obtained. However, note that the first у could have been in any of the
к spots, the second in any of the remaining к — 1 spots, through the k\h, so that
k\ of those arrangements are the same. Thus we can represent the number of
combinations as
n(n -!)•••(« -k + 1) n!
k\ k\(n-k)\
The last expression makes the symmetry in к and n — к as clear as it is by counting
the x first rather than the y.
Observe that since x = у = 1 gives
the binomial coefficients provide a refinement to the cruder result that 2" is the
total number of arrangements of x and у in n places.
A further refinement is possible. One nice way to illustrate this is to consider
lattice paths on the first quadrant, starting at @, 0) and moving to (n — к, к) by n
steps, each either one unit to the right or one up. Consider the case of two moves
481

482
10 Introduction to «^-Series
@,0)
@,0)
il.l)
Figure 10.1
shown in Figure 10.1. In the first case there is no area under the path; in the second
the area is one. We will split the ("k) paths according to the area under the curve.
Since
(x + yJ = xx + xy + yx + yy
we can keep track of the area by rewriting each term with the xs first and adding
a unit area whenever yx is changed to xy. We will use a parameter q to do the
counting, requiring that
yx = qxy.
Since we wish to collect the ^s together, also assume that
УЯ = ЧУ,
xq = qx.
As an exercise, work out (x + yL: The coefficient of x2y2 comes from six
pictures, and the generating function for these six pictures collected by areas
under the graph is
l+q+2q2 + q3+q4.
A little reflection shows that this is
demonstrating that there is some structure to this coefficient. It is possible to rewrite
this in a form that suggests a general form for the coefficients, but there is a more
elegant way to derive this formula. Recall Pascal's triangle property for binomial
coefficients:
A0.0.1)
This can be explained by a combinatorial argument, since the (и +1 )-th spot could
contain an x in Q) ways or а у in (k1l) ways.

10 Introduction to «^-Series 483
However A0.0.1) also comes from
We use this method for finding the ^-binomial coefficients \nk\ that are defined
by
П г -i
(x + y)" = Y, III x"~kyk A0-°-2)
*=o \-ки
when
yx = qxy,
xq = qx, yq = qy.
First
(x+y)n+1 =(x+yT(x + y)
gives
4=0 L J9 4=0
Since
ykx = qkxyk
we have
[T] =[t]9*+[*-i] • A0-°-3)
This is a ^-extension of A0.0.1). In the case when q = 1 there is only one Pascal
triangle relation. In the q-case there is a second. Using the same argument,
(x + y)n+l = (x + y)(x + y)n
gives
Relations A0.0.3) and A0.0.4) can be combined to give
Гл1 {\-qn+l-k)\ n
k\q
Iteration leads to
[л 1 _ {\-qn+l-k)---{\-qn)\n~\
UL (l-qk)---(l-q) [o\:

484 10 Introduction to q-Senes
But
[oj,
and so
Гл1 (l-g)---(l-g")(q; q)n
[k\q (\-q)---(\-qk){\-q)---(\-q"-k) (q; q)k{q; ?)„_*'
A0.0.5)
where
]J-qi)- (Ю.0.6)
Another way to write A0.0.5) is
n\a
where
л!, = A + ?) • • • A + q + ¦ ¦ ¦ + qn'X) = (q; q)n{\ - qT". A0.0.8)
The question naturally arises whether there is a commutative extension of the
binomial theorem that uses ^-binomial coefficients. To derive this result, replace
у with xy in A0.0.2). This is possible since y(xy) = (yx)y — q(xy)y. Then
Observe that (xy)k = xyxy ¦ ¦ • xy — xkykqk<-k'1^2. Also,
(x + xy)(x +xy)---(x+ xy)
= x(\ + y) ¦ ¦ ¦ x(l + y)x(\ + y)
= X(l +y)...x(l + y)x2(\ + qy)(\ + y)
Therefore,
A +y)(l+qy)---(l+ qn~ly) = J2 qk(k~l)/2 \ " | /. A0.0.9)
k=0
Replace у with y/x to obtain
П г
(x + y)(x + qy)---(x+ q"~ly) = X] ? ' qm-\)/ixn-kyk_ (Ю.0.10)
k=0 L

10.1 The q -Integral 485
This is a ^-extension of the binomial theorem. The noncommutative binomial the-
theorem is due to Schutzenberger [1953]. The ^-binomial theorem was independently
known to several mathematicians of the nineteenth century. The interpretation of
the g-binomial coefficient in terms of areas under lattice paths is due to Polya
[1984, Vol. 4, p. 444].
The infinite q -binomial theorem can be seen as an analog of the formula for the
beta integral on @, 1) in terms of gamma functions. To show this, we introduce
a ^-integral. This was explicitly done by Thomae and Jackson, but the essential
idea was discovered by Fermat. We also develop the g-extensions of the gamma
and beta functions.
A generalization of the q-binomial theorem is the i i^i formula of Ramanujan.
This can be considered a ^-extension of the beta integral on @, oo). Ramanujan's
formula and one of its consequences, the Jacobi triple product identity, are very
important in number theory. We show how they imply results on representations
of numbers as sums of squares.
The remainder of the chapter is devoted to the study of a few other important
q-btta integrals and to developing the elementary theory of basic hypergeometric
(or q-hypergeometric) series. We also give a very short exposition of the theory
of g-ultraspherical polynomials. We note that some of these infinite series and
products are also modular functions.
10.1 The ^-Integral
Even before the systematic development of calculus by Leibniz and Newton in
the latter half of the seventeenth century, mathematicians from many parts of the
world attempted to evaluate the integral
/ xadx.
Jo
For example, Archimedes computed the case a = 2. He did this in two different
ways, one using the value of I2 +22 + ¦ ¦ • +n2, which was familiar to the Baby-
Babylonians in 1700 B.C., and the other using the sum of a finite geometric series. In
the early seventeenth century this integral was computed for other small values of
a (up to nine according to some accounts). The difficulty experienced by those
mathematicians was the problem of treating the sums 1* + 2k + ¦ ¦ ¦ + nk in a gen-
general way. In the 1650s, Fermat, Pascal, and others found a method for this. Fermat
also gave an easier way of computing the integral, using a geometric series. In
his studies of the Greeks, Fermat must have noted that Archimedes also used a
geometric series in the quadrature of a parabola. For more history and references,
see A. Edwards [1987] or C. Edwards [1979].
Decompose the interval [0, a] into subintervals using a geometric dissection,

486 10 Introduction to ^-Series
that is, subintervals with endpoints {xn}^ where xn = aqn, 0 < q < 1. In this
case the sum approximating the integral is
OO 00
n=0 n=0
oo
= aa+\\-c
A0.1.1)
Fermat considered the case a = l/m, where ? and m are positive integers. Set
t = #1/m and write A0.1.1) as
J fin+n \ -L I -(-••• -)- (ГП+n — l
-» ^ a(e+m)/m asr^l. A0.1.2)
Thus, Fermat evaluated the integral when a is rational.
Thomae [1869] and later Jackson [1910] introduced the ^-integral defined by
™ oo
/ f(x)dqx = 'S2f(aqn)(aqn - aqn+1). A0.1.3)
Jo «=o
We call dqx the Fermat measure. Jackson also defined an integral on @, oo) by
/¦OO °°
/ f(x)dqx = (\-q) У] f(q")qn. A0.1.4)
Jo „ГГсо
Notice that
roc
lim / f(x)dqx - I f(x)dqx.
N—> oo Jq Jq
The idea here is that on A, oo) the division points are at q x,q 2,q 3,...when
0 < q < 1. It is easy to see that, when f(x) is continuous on @, a),
/a ra
f(x)dqx= / f(x)dx. A0.1.5)
Jo
It is clear that we can write the ^-integral for any continuous function, but
it is important to keep in mind that the resulting sum should be interesting and
manageable. After all, Fermat used the #-integral because a geometric series can
be summed.

10.2 The ^-Binomial Theorem 487
Suppose that f(x) = xa~~l A — x)^; we are interested in obtaining an analog
of the beta integral. From A0.1.3), the sum corresponding to this f(x) is
n=0
We are unable to sum this series because of the term A — q")P l. So we look for
a function fq(x) such that
fq{x) ^ xa~\\ - х)р~1 asq^T,
and for which the ^-integral /J fq (x)dqx can be evaluated in an appropriate form.
It seems very likely that xa~x should be retained as it is. Then we might deal with
A — x)$~x by expressing it as a power series in x and then deciding what to do
with the coefficients. By the binomial theorem,
for |
kl
We now need the ^-analogs of &! and more generally of (a)^ and finally an analog
of the binomial theorem itself.
10.2 The ^-Binomial Theorem
To define к \q, the #-analog of к!, note that A0.1.2) indicates that we should replace
an integer m with \ + q-\ (- qm~l = A - qm)/{\ - q). Thus,
_ A-<?)(!-<?2) ¦¦¦(!-<?*)
and we can replace the shifted factorial (a)k with
Now write
(a; q)k = A - a)(l - aq) ¦ ¦ ¦ A - aqk~l). A0.2.1)
We see that the series corresponding to Y^o (a)kXk/'k\ is
^^хк. A0.2.2)
q;q)
This series can be summed and its evaluation in terms of infinite products is given
in the next theorem, called the #-binomial theorem. To see how to sum it, consider

488 10 Introduction to ^-Series
the following proof of the binomial theorem. Let
k=0
First differentiate
*=i v '
To remove ga+i(x), consider
-(« +
2^
k=l
J} X = ~Xga+i(x)
*=i
Eliminate ga+\ (x) from the two equations to get
=
ga(x) 1-х'
which implies
ga(x) = A - xTa.
Application of this idea to the evaluation of A0.2.2) requires the ^-difference
operator. This operator is defined by
fix) — fiqx) fix) — fiqx)
j v j \ч j v j v<f ) (Ю.2.3)
X — Q'X A — ^)X
We now state and prove the q -binomial theorem.
Theorem 10.2.1 For\x\ < \,\q\ < 1,
(n-
(a; q)k k _ iax;
k=0 iq; q)k (*; q)c
where (a; q)x = ЦТ=оA ~ ас1кУ
First Proof. Let
k=0

10.2 The ^-Binomial Theorem 489
Apply the q-difference operator Aq to both sides. Then
a;q)k _ Krt-i
ч *¦ 4 /X
or
Л0О - /a(?Jc) = A - a)xfaq{x).
Now consider
or
fa(x) — A -ax)faq(x).
Eliminate faq(x) from the two equations to get
1 — ax ,
fa(x) = ~ faiqX).
1-х
Iterate this relation n times and let n —> oo to arrive at
(ax; q)n (ax; q)^ (ax; q)x
fa(x) = ~ —fa(q X) = fa@) = ~ " ¦
(x; q)n (x\ q)oo (x\ q)x
This proves the theorem. ¦
Second Proof. The infinite product (ax; q)oo/(x;q)oc is uniformly and abso-
absolutely convergent for fixed a and q in \x\ < 1-е and so represents an analytic
function in \x\ < 1. Consider its Taylor expansion in \x\ < 1,
F(x) = q)°° = ]T Anxn.
Clearly,
— ax)
F(x) =
A -x)

490
This implies
10 Introduction to ^-Series
*хП = о -a
n=0 n=0
Equate the coefficients of xn on both sides. Then
(a; q)n
{q\q)n'
This completes the second proof.
Remark 10.2.1 The infinite product in the g-binomial theorem also arises natu-
naturally when we look for the analog of A — x)~a. To see this, suppose a is a positive
integer n. A possible <?-analog of A — x)~" is
1
(l-q"x)(l-qn+lx)
(l-x)(l-qx)---(\-q"-1x) A-*)A-?*)••• (л;?)»'
The last expression is meaningful even if и is not an integer, so more generally we
consider (ax; q)oo/(x; q)oo-
There are many interesting special cases of Theorem 10.2.1.
Corollary 10.2.2
(a)
(b)
00 n i
Г-^- = -——, M<i,M<i.
±L (q; q)n (x; q)oo
/ l)"/
= (x;q)oo, \q\<l-
«=o
= (*: ?)лг = A -
A -
(Euler)
(Euler)
(Rothe)
OO
where the q -binomial coefficient is
In
= (q;q)n/(q;q)k(q;q)n-k-
Proof.
(a) Set й = 0 in Theorem 10.2.1.
(b) Replace a with I/a, and x with их and then set a = 0.

10.2 The ^-Binomial Theorem 491
(c) Seta =q~N.
(d) Seta =qN. ¦
Remark 10.2.2 The g-binomial theorem was apparently discovered indepen-
independently by several mathematicians including Gauss [1866], Cauchy [1843], and
Heine [1847]. It seems that the first statement of the g-binomial theorem in ap-
approximately the form given in Corollary 10.2.2(c) was published by Rothe [1811].
(He stated this as A0.0.10), but with a misprint.) For the references to Euler in
Corollary 10.2.2, see Andrews [1976, p. 30].
In most cases we do not give detailed proofs to justify limiting processes. It is,
however, interesting and important to know what is involved here. Hence we end
this section with Koomwinder's [1990] proof of the fact that the ordinary binomial
theorem is obtained from the ^-binomial theorem as q ->• 1~.
The proof of the theorem depends on the following lemma; we omit the proof.
Lemma 10.2.3 Suppose ц, к, к are real; 0 < ц — к < к; /л + k> 1; and
„-lit _ e-(k+k)t
°
Then f'(t) <Oift > 0.
Theorem 10.2.4 Suppose к and /i are real. Then
lim
uniformly on {x e С : \x\ < 1}, if ц, > к, ц, + к > 1, and uniformly on compact
subsets of {x, e С : \x\ < \,x ф 1} for other choices of'k and ц.
Proof. First observe that since
(q x; q)oo (q x; q)i (q + x; grHo
= A0.2.4)
(q^x; q)^ (q^x; q)m (gr/x+mx; q)oo
we can choose ? and m appropriately so that fi + m > к + ? and fi + к + I +
m > 1. Moreover, the first quotient on the right-hand side of A0.2.4) tends to
A — x)l~m uniformly on compact subsets of [x eC: \x\ < \,x ф l}as^ ->• 1".
Consequently, we need only consider the case where fi > к and /x + к > 1. By
the ^-binomial theorem, the left side of A0.2.4) is
A0.2.5)

492
10 Introduction to (/-Series
It is easy to check that (q11 — qk+k)/(l — qk+l) increases with q for к + к > \i
and so
< lim
к-ц+к
Let m be the largest integer such that к + m — 1 < \i. Then for \x\ < 1
\-q
l~qm
1 -q"
, + m A. — и + n — 1
\-q
m + 1
n
Thus the series A0.2.5) from the rath term onward is majorized by the convergent
series
sup
0<o<l
\-q
l-q"
(k - fi + т)„_„
(m + 1)„_ш
This implies that we can take the termwise limit in A0.2.5) to get
lim
This proves the theorem.
n=\
n\
A0.2.6)
In a similar way, if we replace x with A — q)x in Corollary 10.2.2(a) and let
q ->• 1", the series converges to ex:
lim
1
(A -q)x\q)c
A0.2.7)
and we have a ^-analog of the exponential function. The sum in Corollary 10.2.2(b)
also gives a series that converges to the exponential series, but the infinite product
form is equivalent to A0.2.7). Similarly,
lim (-A -q)x;q)oo = ex.
A0.2.8)
The functions used in A0.2.7) and A0.2.8) are occasionally given other names:
1
eq(x) :=
(A -q)x;q)oo'
Eq(x) := (-A -q)x;q)oo.
A0.2.9)
A0.2.10)

10.3 The g-Gamma Function 493
10.3 The 4-Gamnia Function
We return to the problem of finding an analog of the beta integral over @,1). By
Theorem 10.2.4 it is reasonable to replace
X \ 1 л j
with
Xa~l(qx;q)oo/(qfix;q)oo.
We write the <?-binomial theorem as
y. («¦+';?)„*¦ = (ах;д)ж(д;д)ж Л)
* J (ft ft ft • ft\ (Y' ft \ (ft' /7 |
Replace x with qa and a with q& in A0.3.1) to get
,_, (qx\ q)c
j>
-dqX = ^—^—' J;^v-"^. (Ю.3.2)
x) (qa; q) oc (qp;q) oc
This is the q -extension of
^"'(l -xf~xdx =
/о
To write A0.3.2) in this form we need the ^-version of the gamma function.
The existence of a useful n \q indicates the possibility of a convenient analog of
the gamma function. We follow Euler's procedure and look for an interpolation
formula by using infinite products. Now
(q;q)n
(\-q)n
The last expression does not require и to be a positive integer, so we set
n\a = = ; , 0 < q < 1.
4 (\ „\n (\ n\n(nn+\- -^ ^
Tq(x):= ^;q)°° A-<?)'"* when \q\ < 1. A0.3.3)
Here we take the principal values of qx and A — q)l"x. Then Tq(x) is a mero-
morphic function with obvious poles at x = — n ± 2nik/ \ogq, where к and n are
nonnegative integers. It is not difficult to see that the residue at x = — n is
A - q)n+l
(q-";q)n\ogq-1'
Because Tq(x) has no zeros, its reciprocal is an entire function. It is left to the
reader to check these facts. These properties of Yq (x) are similar to those of Г (x).
We can now write A0.3.2), which is just another form of the g-binomial theorem,
as follows:

494 10 Introduction to ^-Series
Theorem 10.3.1
(qpx\q)eo q Tq(a+p)
Remark 10.3.1 At first sight it might appear that we could have replaced A —
л:)^1 with (ql~px; q)oo/(x; q)oo, but this is not as useful. The function A -л:/
is positive in the interval of integration @, 1) and 1-х vanishes at 1. In the q-
integral the set of points over which the summation is carried out is q", n =
0, 1, 2,..., that is, a discrete set of points in [0, 1]. The first point to the right of
the interval where we want our function to vanish is q~l. To get such a function
replace xq~P by x in both infinite products. The result is (qx; q)<x/(qPx; q)^.
The function (q1~^x; q)oo/(x, q)oo could have been used, but the ^-integral would
beon[0,</].
Theorem 10.3.1 gives us one reason to accept Tq{x) as the natural (/-analog
of Г (л:). Another reason is the following: The Bohr-Mollerap theorem states that
Г (л:) is the unique function satisfying the functional equation
f(x + l)=xf(x), /A) = 1,
and is also logarithmically convex. It can be shown that Tq (x) is the only function
that satisfies the functional equation
and is also logarithmically convex. The proof of the latter result is identical with
that of the Bohr-Mollerap theorem and was included in Chapter 1. See Exercise
10.
From the definition of Tq{x) and the Bohr-Mollerap theorem, one may suspect
that lim^i- Fq(x) — Г(х), and this is indeed true. We derive it as a consequence
of the next theorem, the proof of which requires the following lemma.
Lemma 10.3.2 Iff(l) = /B) = g(l) = gB) = 0and0< f"{x) < g"(x)for
x>0, then f(x) < g(x) in [0, 1] U [2, oo) and f(x) > g(x) in [1, 2].
Proof. For x e [1, 2], it is easy to see that
/(*) = J h{x,t)f"{t)dt
with
AC*, *) = {?_
2)(jc - 1), 1 < x < t < 2.
Since h(x,t) is negative for x and t in [1, 2], it follows that f(x) > g(x) in the
interval [1,2]. We may assume without loss of generality that f(x) = 0. We have

10.3 The g-Gamma Function 495
just shown that, in that case, g(x) < 0 in [1,2] and g = 0 at the endpoints. By
the mean-value theorem, g' is zero somewhere in A,2), and since g' is increasing
we must have g'{x) < OforO < x < 1 and #'(¦*) > Oforx > 2. So g is decreasing
in @,1) and increasing in B, oo). This implies the result. ¦
Theorem 10.3.3 ForO < r < q < 1, we have
Гг(х) < Fq(x) < Г(х), 0<д:<1 or x>2,
and
Г(х)<Гч(х)<Гг(х), 1<д:<2.
Proof Observe that
d2 _^_ „x+n
j _ q*+nJ
We show that each term of the series
is increasing in @,1). Set a — n +x; then
aqa-\\ogq){\+qa)
To prove that h' > 0, it is sufficient to demonstrate that the expression within
the square brackets, which we denote by g(q), is negative. A simple calculation
gives
g'(q) = (l~q \2 > 0, q > 0.
q(\+qaJ
Since g(l) = 0, we have g(q) < 0 in @,1]. Thus h(q) is increasing and so
^2 log Г?(;с) is increasing for 0 < q < 1. This means that
d2
, , iv/6i qyx), 0 < r < q < 1, лг>0.
ил- dxl
Moreover,
logrV(l) = logr^(l) = logFrB) = logF9B) = 0.
The theorem is now a consequence of Lemma 10.3.2. ¦
Corollary 10.3.4 lim^i- Vq(x) = Г(х).
Proof. Theorem 10.3.3 implies that lim?^.i- Tq{x) = X(x) exists. Moreover,
X(x) satisfies the conditions of the Bohr-Mollerap theorem. Thus X(x) = F(x)and

496 10 Introduction to ^-Series
the corollary is proved for 0 < x < oo. For real nonintegral values, the functional
equations then give the same result. For complex x, the Stieltjes-Vitali theorem
completes the proof. See Hille [ 1962, p. 251] for this theorem. (For Gosper's proof
of Corollary 10.3.4, see Andrews [1986, p. 109].) ¦
There are several results about Tq (x) that are analogs of corresponding state-
statements about Г(х). The analogs of the Legendre duplication formula and the Gauss
multiplication theorem are given in the theorem below.
Theorem 10.3.5
(a) Г,Bдс)Г,2A/2) = A +<?J*-1Г>(х)Г>(х + 1/2).
(b) Letr =q". Then
Tq{nx)Tr{\/n)Tr{2/n) ¦ ¦ ¦ Гг((п - l)/n)
= A +? + •••+ ?"-1Г-1Гг(х)Гг(х + 1/n) • • • Гг(х + (n - l)/n).
The proofs are straightforward and left to the reader as exercises. The next two
formulas can be regarded as asymptotic formulas for Tq (x) for large x, but they are
of a different nature from Stirling's formula. They follow from Corollary 10.2.2:
{q\q)o
Re x > 0, A0.3.4)
Rex>0. A0.3.5)
Now recall that 1/ Г(х) has zeros at x =0,-1, -2,... whereas 1/ ГA - x)
has zeros at x = 1, 2, 3,.... Thus their product has a zero at each integer. This
fact is among the properties we see reflected in Euler's formula
Г(х)ГA -х) =jr/sin;rx.
Similarly, \/Tq{x) has zeros at x = — n due to the factor (qx; q)^. (In fact,
1/Г^(х) has zeros at x = —n ± 2л i k/ log q where к and n are nonnegative
integers.) Hence the function with the full range of integer points as zeros may be
taken as {qx\ qHO(q1~x; q)oo = (qx, q)oo(q/qx; q)oo- Replace qx with у to write
the function as (y; q)oo(q/y, q)<x- We expect this function to have interesting
properties. In fact, it is one of the theta functions discovered by Gauss and Jacobi
and is the topic of the next section.
10.4 The Triple Product Identity
The triple product identity expresses (x; q)oo(q/x; q)oo(q\ q)oo as a Laurent se-
series in 0 < |jr| < oo. One proof of this identity follows from the terminating

10.4 The Triple Product Identity 497
^-binomial theorem due to Rome given in Corollary 10.2.2. This proof was known
to Gauss [1866a] and Cauchy [1843a]. At the end of the previous section we re-
remarked that the infinite product (x;q)oc(q/x;q)oo arises naturally when we look
for a <jr-analog of the Euler reflection formula. We shall see that the Laurent-series
side of the identity also appears naturally from a particular Riemann sum approxi-
approximation of the normal integral. This raises the question of whether the triple product
identity could be approached from the series side. The answer is yes and this leads
to another proof of the identity. We end the section with a number of identities
that are important but simple consequences of the triple product identity. In later
sections we give applications to number theory and combinatorics.
Theorem 10.4.1 For \q\ < 1 andx e С - {0},
oo
(*;<?)ooO?/.*;<?)oo(<7;<?)oo = ^ (-i)V'2)**-
k=-oo
Proof. Take N = In in Corollary 10.2.2(c) to obtain
<y\k+n Ak+n)(k+n-\)/2xk+n
I П
k=—n
Then replace x by xq~" and rewrite (xq~"; qJn as
(xq-n; q)n(x; q)n = {-\)nxnq-n2+n(n~l)l2{q/x; q)n(x; q)n.
The above identity then becomes
{q/x;q)n{x;q)n = > -
i~n (я\ q)n+k(q\ q)n-k
When n -> oo, this gives
(x;qHO(q/x;qHO=
This limiting process can be justified by Tannery's theorem. The result in Theorem
10.4.1 is called the triple product identity. ¦
Remark 10.4.1 Replace q with e~2' and x with —e~'e'e in the identity. The result
is
oo oo
e-nheine = д (j
E
fc=-oo n=0
The left side is a solution of the heat equation
d2u du
1& = It'

498 10 Introduction to ^-Series
The right side is positive for t > 0 since
l+2rcos6>+r2 > l-2r+r2 = (l-rJ>0,
when 0 < r < 1. The positivity is not evident from the left side. Thus the two
sides give different properties of the function. Clearly, the right side also gives the
zeros of the function.
In view of the importance of the triple product identity, we look at it from another
point of view. First replace q with q2 and x with — qx to get
oo
(~qx;q2HO(-q/x;q2HO(q2;q2HO= J2 ^x"¦ (МАЛ)
n=—oo
The sum on the right of A0.4.1) comes from an important integral. Consider the
normal integral
e~x2dx.
Shift* by a/2 to get
Replace this integral with the approximation formed by summing over a discrete
one-dimensional lattice with space size 8:
00
г V^ -S2n2-aSn
n=—oo
It is natural to ask here how close the sum is to the integral when S is small. To
answer this, consider the formula in Exercise 2.26:
oo . oo
V e-m(n+aI = -= У е-"/l'e2nina'. A0.4.2)
n=—oo ' n=—oo
When t is small the terms on the left are close to one for small values of n but
all the terms on the right with one exception are very small. So the expression on
the right works very well for numerical calculation of the series for small 8. Later
we shall see that there is a deeper reason for the importance of the transformation
formula A0.4.2). It shows that X^oo Ч" is a modular form.
Let us now return to the sum in Theorem 10.4.1 when x = 1. The sum is
^ — Y^oo^^)n 4() -Apply the two changes in variables и -> — nandn ->¦ n+ 1
to get
OO OO
A = У (-1)V<«+1>/2 = У (-1)"+У<"+1>/2 = -А.

10.4 The Triple Product Identity 499
Thus A is zero. Write
H(x) =
n=—oo
Then
1 -°°. л(л+П 1
ti(qx) = — > (—1) о 2 дг = —
л: ^—' л:
n=—oo
or
H(x) = -xH(qx). A0.4.3)
This equation implies that if x is a root of H(x) = 0, then so are qx and ;c/<?. Since
we know that x = 1 is a root, it follows that qn for every integer n is a root. Thus
H{x) has (л:; q)oo(q/x; q)^ = T(x) as a factor. Without knowledge of Theorem
10.4.1 we cannot be sure that H(x) has no other zeros. However, it can be shown
by a simple calculation that
T(x) = ~xT(qx).
That is, T(x) satisfies the same functional equation as H. It follows that the
Laurent-series expansion of A0.4.1) is uniquely determined in a deleted neighbor-
neighborhood of x = 0 up to a constant factor. Consequently,
T(x) = C0(q)H(x).
Replace q with q2 and x with — qx to get
oo
(-qx; q2)oo(-q/x\ q2)^ = C0(q2) ^ q"'xn. A0.4.4)
n—~ oo
There are a number of ways of finding Co (q2). We apply a device due to Gauss and
Jacobi. Another method, also due to Gauss [ 1866b], using the arithmetic-geometric
mean is given in Exercise 13. There also exist some combinatorial methods but
we shall not describe them here. One such method is given in the next chapter.
Let x = i in A0.4.4). The result is
ОС
E
.«= — 00
(-1)"
ЭС
<E
n=—oo
(_1)«?B«+1J
Since the left side is real for q real, we get

500 10 Introduction to g-Series
The left side is identical to (—q2; qA)oo- Now set x = — 1 and replace q with q4 in
A0.4.4) to get
rt=—OO
The last two identities imply
C0(q2) (-q
C0(q*) (q^q^l (q2; q'
This gives
Note that (q; q)x and J2T=~ooqk x* are continuous functions of q in \q\ < 1.
Thus CoO?2) is also continuous, which implies that Co(O) = 1. Let n —> oo to get
C0(q2)(q2;q2)oo = 1.
This gives C0(^2) = l/(?2; ?2)oo and we have another proof of the triple product
identity.
Corollary 10.4.2
00
-?2")A - q2"-1J, (Gauss) A0.4.6)
(-l)"qni3n+1)/2 = J](l - q"), (Euler) A0.4.7)
OO OO
л=0 л=1
OO
-l)"Bn + l)^f"(n+1)/2 = JJA - 9"K. (Jacobi) A0.4.9)
n=0 n=l
Proo/ The identities in A0.4.5) and A0.4.6) are obvious from A0.4.4). Note
that A0.4.6) also follows from A0.4.5) by changing q to -q. To get A0.4.7)
replace q with q3/2 and set x = —^/q. For A0.4.8) and A0.4.9), write the identity in

10.5 Ramanujan's Summation Formula 501
Theorem 10.4.1, with x replaced by —qx, as
OO 00
\ qn{-n >i (x" + x ) = A + 1/jc) 11A — q )(l+<? /x)(\ + q x).
n=0 n=\
A0.4.10)
Setx = l.Then
n=0
This is A0.4.8). Now divide A0.4.10) by x + 1 and let x -* -1 to get A0.4.9).
The corollary is proved. ¦
The sequences of numbers {n2}, {n(n + l)/2}, {п{Ъп ± l)/2} are the square,
triangular, and pentagonal numbers, respectively. These are of number theoretic
interest and their appearance as powers of q in the series make the above identities
useful in combinatorial number theory.
Remark 10.4.2 Gauss and Jacobi independently discovered the triple product
identity. Jacobi's results including A0.4.9) are contained in his famous book,
Fundamenta Nova (Jacobi [1829]). See Gauss [1866] for A0.4.1), A0.4.5), and
A0.4.6). Identity A0.4.8) was published in Gauss [1808], a paper noted for con-
containing the first evaluation of the quadratic Gauss sum. See Exercises 5 and 6.
Euler's result A0.4.7) is the famous pentagonal number theorem. A discussion
of Euler's proof and references are given in Weil [1983, p. 281]. See also Euler
[1748].
10.5 Ramanujan's Summation Formula
The ^-binomial theorem evaluated a <jr-analog of the beta integral over @, 1). We
have seen that the ^-integral over @, oo) is a bilateral series. Therefore, a ^-analog
of the beta integral over @, oo), that is,
/oo ra~^
should be a bilateral series, but similar to the series in the q -binomial theorem.
The correct generalization was found by Ramanujan. He considered the bilateral
sum
—oo<*9>»
A0.5.1)

502 10 Introduction to ^-Series
which, as we shall see later, is a q -integral analog of
B(a, 1
4°
Jo
dx.
c01 Jo A + cx)a+P
We should first clarify the meaning of (a; q)n in A0.5.1) for negative n. Since
(a; q)n = (a; q)oo/(aq"; q)oo A0.5.2)
for n > 0, and the right side is meaningful for negative n as well, we take A0.5.2)
as the definition of (a;q)n for all n. If n = —m, then
(a;q)-m =
Ramanujan's evaluation of A0.5.1) in terms of infinite products is given next. It
contains the <jr-binomial theorem and the triple product identity as special cases.
Theorem 10.5.1 For \q\ < 1 and \ba~l\ < \x\ < 1,
yv (g;q)n n _ (ax;qHO(q/ax;qHO(q;qHO(b/a;qHO
n^0O (b\ q)n (x;q)oo(b/ax;qH0(b;qH0(q/a;qH0
First Proof. Use A0.5.2) to write the series as
+ ж Xlq;q)nfb\"
The first series converges for |jc| < 1 and the second for \b/ax\ < 1. So the
bilateral series converges when \ba~l\ < \x\ < 1.
Observe that
(a\q)n „ (a;q)oo ^ (bqn;qHO „
This proof of Ramanujan's identity depends on a functional relation satisfied by
f(b). To find it, note that

10.5 Ramanujan's Summation Formula 503
This gives the desired functional equation, namely
.... A -bid)
fib) = ——f(bq).
A — b/ax)
Now f(b) is an analytic function of b for \b\ sufficiently small. Iteration gives
fib) = |^%/@). (Ю.5.4)
(b/axiq)
It is not easy to sum /@), but f(q) can be obtained from the <jr-binomial theorem.
Note that
0?; q)oo v^ («; <?)« „ _ 0?; <
^ ) (
From A0.5.4)
/@) =
_ (g/ад:; q)ooiq; q)ooiax; q)^
iq/a\q)ooia\q)ooix\q)oo
Use this in A0.5.4) to get f(b) and then
(«;<?)„ „ _ jax; q)oo(q/ax; g)oo(q;
(b; q)n (x; qHO(b/ax; q)^; q)ooiq/a; q)oo
This argument assumed that we could take b = q. To prove the general case apply
analytic continuation on b and x. Ш
Second Proof (Venkatachaliengar). As in the case of the <?-binomial theorem,
we can start with the infinite product and get the Laurent series. Suppose that
F(x) = (**= *>*>(*/"'«>« = f Л.Х-. A0.5.5)
ix;q)oo(b/ax;q)x nf^
The Laurent series is defined in \x\ < 1 and \b/ax\ < 1, that is, \b/a\ < \x\ < 1.
We consider the Laurent expansion of F(qx), which we want to exist for \b/aq | <
x\ < 1; so assume for the present that \b/aq\ < \x\ < 1. Both F(x) and F(qx)
are defined in \b/aq \ < \x\ < 1. Thus it is possible to look for a functional relation

504 10 Introduction to ^-Series
between F(x) and F{qx) in this region. Now
oo
(ax;q)(X(q/ax;q)oo(\-\/ax)(\ - x) ,, /1Г>с^
F(^x) = 7 Г~^ ГП 771—^ ; = 2^ A»«x- (Ю.5.6)
(x; q)oc(b/ax; q)oc(l — ax)(I — b/aqx) f^
Therefore,
oo oo
q(l-x) J2 Anxn = (b-aqx) Y, Anq"xn.
«= — 00 «=—00
Equate the coefficients of x" to get
q(An — An_i) = bAnqn — aAn_\qn,
or
(?; q)n
This implies that
(ax;q)oo(q/ax;q)oo _
(x;qHo(b/ax;qHo
Multiply both sides by A — x) and letx —> 1~ to get
(a;q)cc(q/a;q)x = (^jOoo Aq A0.5.8)
(q\q)oc(b/a;q)oc (b\q)oc
Substitute this value of Ao in A0.5.7). Ramanujan's identity has the restriction on
b that \b/aq\ < \x\ < 1. This can be removed by analytic continuation. In the
derivation of A0.5.8), Abel's continuity theorem was used in the last step. This
theorem states that if Нт„_>ос о,п= a then
00
lim A — x) yjanx" = a. ¦
x^l~ «=o
See Appendix В for a discussion of this and related matters.
Third Proof (Ismail [1977]). Rewrite the g-binomial theorem as follows
(ax;^)oo ^(a;q)n °° '"¦-^ -
(a;g)N N y^ (aqN;q)n „
(q;q)N „_>,(qN+1;q)n

10.5 Ramanujan's Summation Formula 505
Replace a with aq~N to obtain
(a; q)n n _ (aq~Nx;q)oox-N(q;q)N
i(qN+l;q)n (aq~N;q)N(x;q)oc
(aq~Nx;q)N _N (q; q)oo{ax;q)Xl (q/ax;q)N{q;q)<Xl{ax;q)oo
-x
(aq~N;q)N (qN+1\ q)oo(x; q)oo (q/a\ q)N(qN+l; q)oo(x; q)oc
(ax; q)oc(q/ax; q)oc(g; q)n(qN+l/a; д)ж
(x\ q)oc(qN+1/ax; q)oc(qN+l\ q)oc(q/a\ q)x '
lfqN+l is replaced by b, both sides of this equality are analytic in b for b close
to 0, and they agree when b = qN+l. Zero is the limit point of this sequence, and
two functions analytic in an open set around b = 0 and that are equal at infinitely
many points in this set must be identically equal there. Analytic continuation gives
Theorem 10.5.1 for \b/a\ < \x\ < 1. ¦
Theorem 10.5.1 is called Ramanujan's i^i formula because bilateral <?-series
are denoted by \j/ and there is one upper and one lower parameter.
Remark 10.5.1 If the argument in the second proof is generalized to
(ax; g)oo(b/x; д)х
(cx;q)O0(d/x;q)O0'
a second-order difference equation arises. This reduces to a first-order equation
for d = q/c; the coefficients can then be determined. Unfortunately, the resulting
series is incorrect. This problem occurs because the change from F(x) to F(qx)
moves one region of the analyticity of F to an adjacent one, since the poles of
F(x) are at x = c~lqn, n = 0, ±1, Details are given in Askey [1987].
Remark 10.5.2 As remarked earlier, Ramanujan's formula is a ^-analog of a beta
integral. Rewrite the formula as
(bgn; <?)oq n_ (ax;qHc(qjax;qHc(q;q)oo(b/a\q)oo
_(.aqn;q)oo (a; q)oo(q/a; q)oo(x; q)oo(b/ax; q)x '
Set x = qa, a = -c, and b = -cqa+fi to get
^;<7)co „_, _ (-cqa; q)oc(-c-[ql-a; q)xrq(a)rq(P)
Finally, by Theorem 10.2.4,
ton C,q 'q_°° , C_.g_. 'q °° = A+сГ"A + \/c)a = с'01.

506 10 Introduction to ^-Series
10.6 Representations of Numbers as Sums of Squares
Ramanujan's formula for Y^L-oc(a'< <?)«x"/Ф'< Ч)п is useful in obtaining simple
and direct derivations of some results on the number of representations of a number
as a sum of squares. Three cases are worked out here: two squares (x2 + y2), three
squares where two are equal (x2 + 2y2), and four squares (x2 + y2 + u2 + v2).
For other results of this type consult Fine [1988]. At the heart of the proofs is the
following special case of Ramanujan's identity. Take b = aq in Theorem 10.5.1
and divide by 1 — a to get
у x = —
„tLl-aq" (x;
(ax; q)oo(q/ax; q)oo(q; q)\
\q)oo(q/x\qH0(a;qH0(q/a;qH0'
A0.6.1)
Now observe that
(OO \ * 00
Yl «" j =J2r^n)cin' A0-6-2)
«=-oo / «=0
where rs (и) is the number of ways n can be written as the sum ofs squares. Observe
also that rs (n) gives the number of points with integer coordinates on the sphere
in.?-dimensional space. Here the counting distinguishes between the order, so that
25 = 32 + 42 = 42 = 42 + 32 are counted, and negative integers are also used;
hence, for example, (—3J + 42 is counted. We consider s = 2 and s = 4. Now
note that
q2 =^s(n)qn, A0.6.3)
n=—oo m=—oo n=0
where s(n) is the number of ways of writing n as x2 + 2;y2.
The strategy for finding simple expressions for rs (n) and.? (n) is to use the triple
product identity to express the left sides of A0.6.2) and A0.6.3) as products and
then use A0.6.1) to express these products as a different sum.
Let djj(n) denote the number of divisors of и congruent to i mod j.
Theorem 10.6.1
(a) r2(n) = 4[dlA(n) - d3A(n)l
(b) s(n) = 2[dU8(n)
(c)
Proof.
(a) By A0.4.5) and A0.6.2),
n=0

10.6 Representations of Numbers as Sums of Squares 507
To apply A0.6.1) rewrite the product as
<g2;ffi(-«:g;;«. (ю.6.4)
Replace q with q2 in A0.6.1) and set x = q and a = — 1. Then
This implies
oo
n=0
2
(n)q -
00
=¦
= 1
This proves (a),
(b) As in (a), one can show
oo
«= —ОС
00
!E<
n=—oo
J2m
qn
00
n=—oo
00
+4E
00
+4E
n=l
that
on
+ q>
00
E<
m=0
(^1,4<
(-q;q2)lc(q2>q
(q;q2)U-q2;q
oo
2)oo
/00
oo
/ I 1
m=0
-3,,
A
This time replace q with q4 in A0.6.1) and then set a = — 1 and * = g to get
n=—oo Y
00 00
n=l m=0
00 ОС
Г_(8т + 1)л , _(8m+3)n ^,(8т+5)л ^,(8m+7)n]
[q + q — q ~ q j •
This proves (b).

508 10 Introduction to ^-Series
(c) Rewrite A0.6.1) as follows:
1
\-aqn I-a
1
I
xn(l-aqn)+aqnxn
aqn
-z^Y^V-
Let a = — 1 and combine with A0.6.1) to get
(~qx; q)oo(-q/x; 9H0(9;
{xq\q)oo{q/x\q)oo(-q;q
Then x -> -1 gives
= 1 +
2A - x) y, (q/x)n[\ - x2n]
n=\
1 +9"
A0.6.6)
It follows from A0.4.6) that
n=-oo v "'
:9H
n=—00
Change q to -g in A0.6.6) to arrive at
00 00 „
nq"
9)c
= 1
Enq"
. 1 - <7"
,„
Ln=i
00
CO
2"
+ 9
2"
E"9 V^ 4'"
n=l J n=l
,4n
A0.6.7)
The result follows as before.
The identities A0.6.5) and A0.6.7) and their interpretations were first discovered
by Jacobi [ 1829]. The number theoretic result in (b) was stated and proved by Gauss
for n prime.
10.7 Elliptic and Theta Functions
The theory of elliptic functions with its ramifications has been studied for two
centuries. Its founders are Euler, Gauss, Abel, and Jacobi. In this century the
theory of elliptic functions has been largely incorporated within the theory of

10.7 Elliptic and Theta Functions 509
elliptic curves, which was recently used by Wiles to prove Fermat's last theorem.
In this section we will simply give some definitions and show how Ramanujan
obtained the Fourier-series expansions of Jacobi elliptic functions from A0.6.1).
There are four theta functions of Jacobi. They are all really the same function,
just as sin z and cos z are the same, but it is useful to consider all four:
sinBn
00
(n+l/2feBn+l)iz = 0i(z + jt/2; q), A0.7.2)
00
02(z, q) = 2^^(n+1/2J cosBn + \)z
n=0
00
n=-oo
00 00
: 1 + 2 ]P g cos 2nz = ^2 qe2niz, A0.7.3)
n=l n=—oo
00
04(z, q) = 1 + 2 J^(-1)"^ cos 2nz
B=l
/ ТГ \
A0.7.4)
Here q = еЛ!Т with Imr > 0 so that \q\ < 1 for a given т. The value of qx for
some Л is determined from ел'Хт. The following relations, satisfied by the theta
functions, follow immediately from the definitions:
e3(z + n,q) =9A(z,q)\
and
0! (z + jrr, 9) = -«-'«"^^(z, 9),
7ГТ, ?) = «-^-^^(z, ?),
A0./.6)
) = ?-1е-2'г03(г,^),
+ 7ГТ, q) = q'le-2ize4(z, q).

510 10 Introduction to g-Series
The following formulas are obtained from the triple product identity:
Z, q) = V/4sinz(?2, q2)oo(q2e2iz; q2)cc(q2e-2iz; q1)^
q2; q2)oo(q2e2iz; q2)ac(e-2iz; q2)oo,
q)oo(q;q)oo(q;q)cc ?
= ql/4eiz(q2,q2)oc(-q2e2iz;q2H0(-e'2iz;q2)oc,
Z, q) = (q2; q2)oc(-qe2iz; q2)oo{-qe-'lil\ q2)oo,
94(z, q) = (q2; q2)cc(qe2iz\ q^ooiqe-**; q2)oc
For z = 0, these reduce to
в2 = 02@) = 02@, q) = 2qll\q2; q2)oo(~q2\ q2J^
03 = 03(O) = 0з(О, q) = (q2; q2)oo(-q\ q2J^ A0.7.8)
04 = ^4@H4@, q) = (q2, q2)oo(q; q2fx,
and
Proposition 10.7.1 в[ =
Proof. The right side equals
The factor (—q; q )oo(—q , q )oo(q', q ) equals one because it can be written as
, ч , 2ч (-q;q)oo(q,q2)(q2;q2)oo
i-q\ q)oc{q\ q ) = —г—г-
_ {-q\q)oc(q\q)oc _ (q2; q2)™ _ ,
This proves the proposition, which, it should be noted, is equivalent to A0.4.9). ¦
A meromorphic function is called an elliptic function if it has two periods whose
ratio is not a real number. To define the Jacobi elliptic functions, set
A0.7.9)
and let
030i(m/U2)
<10J10)

10.7 Elliptic and Theta Functions 511
It follows from A0.7.5) and A0.7.6) that sn(u, k) is periodic in и with periods
2л-Q\ and лтв2, and so it is an elliptic function. Other Jacobi functions are
сп(мД) =
ft
and
04
dn(«,)fc) = —
ft 04 (и/032)
There are also functions that correspond to esc z, tanz, and so on:
—, пс(иД) = —-——, nd(u,k) =
sn(M.ifc) , cn(u,k) , сп(мД) „ ^,,
sc(ii,jfe) = —^-(, cs(H,ifc)= cd(H,*)= ; / A0.7.11)
cn(u,k) sn(u,k) dn(u,k)
sn(u,k) dn(M.ifc) dn(u,k)
sd(u,k)= ) '/ ds(ii*) ; ' ' dc(MA:) У
dn(K)
сп(м, к)
Exercise 2.10 required us to show that
'' dt
= = K.
" Jo ^(l-t2)(l-k2t2)
SetM =2Kx/n.Then
в3 0i (x)
— • , A0.7.12)
02 0 W
and similar relations hold for the other functions. We omit the modulus к in sn(w ,k)
etc., and simply write sn и etc. It is easy to see that
_
- 2q2n~l cos 2x + q4
A0.7.13)
en и = 2ql/4ka/2k~l/2cosx TT < —; -.—, >,
1J-1 1-2a2'1-1cos2x + ?4'1 I
and
Here k' > 0 is denned by ka = I — k2. Again, by the exercise mentioned
above, klXl2 = 64/63. There we took 0 < к < 1, but the resultsJiold more
generally. One can also deduce from Exercise 2.11 that sn и has periods AK and
UK'.

512 10 Introduction to ^-Series
The Fourier expansion of the Jacobian functions is given in the next theorem.
Theorem 10.7.2 With и = 2Кх/л, we have
7Г 2л х^-ч qn cos 2nx
* ^ (Ш7Л5)
n=0
Proof. We prove A0.7.14) and leave the other two parts as exercises. In
Ramanujan's formula A0.6.1), first replace q with q1 and then set x = qe'2lx
and a — \/q. Then
ч2
_fl2n-l (np2ix- a2) (af-2ix- a2) (c- ~2\i '
/oo
Change « to n + 1 in the second sum; then combine with the first to get
" sinBn + l)x
о n
Now multiply both sides by —ielxql^27t/Kk and show that the product reduces to
A calculation almost identical with the one in Proposition 10.7.1 is needed. This
proves the theorem. ¦
There are similar series for cd, sd, and nd. The series for the other Jacobi
functions such as ns и are somewhat different. For example,
7Г 2ТГ ^ ^2« smBn + 1)X
+ —V- , , • A0.7.17)
n=0 ч
This can be proved directly or by first showing that
sn(M + iK') = k~ldc(u -K) = k~lns u.
See Whittaker and Watson [1940, p. 511].

10.8 я-Beta Integrals 513
10.8 tf-Beta Integrals
We have seen that beta integrals and their extensions and analogs are very useful
and important. There are several more extensions and it is worthwhile to look
at a few here. One is quite important as it is associated with a set of orthogonal
polynomials that is beginning to find applications in several areas of mathematics.
We noted earlier that Ramanujan's x t^i could be viewed as the integral of
xa~l{—ax; q)oc/(—x; q)^ with respect to the Fermat measure. Ramanujan also
integrated this function using the usual measure. He used a general interpolation-
type theorem to evaluate it. An account of this is given later. For now, we give
an evaluation using a functional relation since that fits in with the methods used
earlier.
Let 0 < q < I, Rea > 0, and \aq~a\ < 1. Assume for the moment that
0 < a < 1. Define
f(a) = Г Х" {~UX'q)codx.
Jo (-*; <?)oo
Then
f(a) =
о (-x;q)oo
= (l-a)f{aq)+aq'af(a).
This iterates to
1 — aq a \и.ц , c^oo
As in the evaluation of Ramanujan's i i/fi, it is hard to find /@). But
[°° x01-1 ж
f(q) = / -r-—dx = .
Jo 1 + x sin not
Thus
(aq-a; q)oo(q\ q)oo smita
when 0 < a < 1. The general case follows by analytic continuation in a, and by
continuity when a is a positive integer. The limiting case when a = к is a positive
integer is
Jo
(-«*; <?)cqdx = (-l)k+l(q/a)k(q;q)k-iuogq)
(-x;q)oo (a~lq;q)k

514 10 Introduction to ^-Series
To see that A0.8.1) extends
I
dx =
/1 I \(Y-t-n
0
take a = qa+fi in A0.8.1) to get
This formula lacks symmetry in a and /3. Symmetry is restored by the following
integral:
The details of the evaluation are left to the reader. First show that
f(a, b) = f(aq, b) + aq-cf(a, bq)
and then by a similar procedure that
f(a,b) = f(a,bq) + bqcf(aq,b).
The two equations together give
1 — ab (ab: q)oo
f(a, b) = f(a, bq) = *, q'°° f(a, 0).
A - bqc) (bqc; q)^
The value of /(a, 1) is given by A0.8.1), and the final result is
(ab; q)oo(qc\ q)oo(qX~c, q)oo7t
f(a,b) =
(bqc; qHO(aq'c; q)oo{q\ q)oo sin^c
When a = qa+c, b = q^c\ then
Г
Jo
(-¦*; q)oo(-q/x; q)oo
Г(с)ГA-с) Гч{а)Г„ф)
Г, (с) Г, A-е) TJa
To obtain other <?-beta integrals, let us again consider Ramanujan's i ^i formula
(see A0.5.3)). When \b/a\ < x < 1 and \q\ < 1,
(a; q)nxn (ax; qHo(q/ax; q)x(b/q; q)oo(q; q)oo
у
„fr^o (b><l)n (x;qHO(b/ax;qHO(q/a; q)oo(b; q)^

10.8 q-Beta Integrals 515
Set ax = ql/2ew andx = aew, \a\ < 1. Then
f (q e ''
; 9H
oo
¦7Г °°
'r,q)n
1^-2tz. A0.8.2)
9H0
To obtain a positive kernel in A0.8.2), take b = aq1?2. Then the integrand is an
even function of в, and becomes
,,
n=0 . __-, ,-а2?2" (a2; 9H0(9; 9)oo
To get a perspective on these integrals, replace a with qa~xl2 and b with ^ in
A0.8.1). The result is
{q«-W;q)O0{ql>-We-»;qH0™-'ai Tq{a)Tq{fi) '
and the limit when q ->¦ 1 is
/V.
*/ —7Г
A0.8.5)
In a similar fashion, A0.8.3) gives the special case a = /3 of A0.8.5), that is,
"(Z-lc^ey-'de = 2"-' f{\-x)«-\l-x2)-dx = ^ ~ 1}.
0 7-i Г (а) Г (a)
One simple extension of A0.8.2) can be seen when we multiply by e~'ke before
integrating. A more important extension is obtained when we reconsider the argu-
argument given in A0.8.2) and realize that the numerator could be replaced by other
products. The product
is one obvious choice. Two others are
(^;9)оо(е-1Й;9)оо and {-ew; qUi-e40; ?)«,.
Each of these has an extra factor from the theta product in the numerator of the i \j/\
sum, and there is no problem in evaluating a similar integral. One could get greedy

516 10 Introduction to ^-Series
and ask whether it is possible to evaluate the integral when all of these factors are
present simultaneously. The integral is
I(a,b,c,d)
; q)oc(e~'e; q)oo(q1/2e10; q)oo(ql/2e~'9', q)oc(—e'e; q)oo(—e~'e; q)oo
Г
Jo
(aeie; q)oo(ae-i»; q)oo(bei(>; q)^18; q)oo(ceie; q)^^^; qH
h(x, \)h(x,ql'2)h{x, -l)h(x, -ql'2)dx
/_i h(x, a)h{x, b)h{x, c)h(x, d)J\ - x2
where
00
h(x, a) = J|(l - 2axqn + a2q2n).
n=0
The condition on the parameters is that max(|a|, \b\, \c\, \d\, \q\) < 1. Instead of
directly evaluating the integral, let us first try to guess what its value should be.
We would like to discover some functional relations. Observe that
b a b(l - 2ax + a2) - a(l - 2bx + b2)
h(x,aq)h(x,b) h(x, a)h(x,bq) h(x,a)h(x,b)
_ (b-a)(\ -ab)
~~ h(x,a)h(x,b)
The reason we put the factors b and a on the left is to remove the x in the numerator.
This leads to
bl(aq, b, c, d) - al(a, bq, c, d) = A - ab)(b - a)I(a, b, c, d). A0.8.6)
This suggests that I {a, b, c, d) is a function ofab and the symmetric products. So
we try
I(a, b, c, d) = f(ab)f(ac)f(ad)f(bc)f(bd)f(cd). A0.8.7)
Apply this to A0.8.6) with с = d = 0 to get
(b - fl)(l - ab)f{ab) = {b- a)f(abq).
This gives
f(abq) f(abq") /@)
fiab) =
1 - ab {ab; q)n (ab; qH

10.8 q-Beta Integrals 517
To see if this is correct, use it in A0.8.7) and then put this conjectured value of the
integral in A0.8.6). The result is equivalent to
(abq)O0(acq)oo{adq)oo(bc)O0{bd)O0(cd)O0
a
(abqHO(acHO(adHO(bcqHO(bdqHO(cdHO
_ b-a
(abq)oo(ac)oo(ad)oo(bc)oo(bd)oo(cd)oo'
where we have written (х)х instead of (x; q)^ for convenience. This equation is
equivalent to
b(\ - ac)(\ - ad) - a(l - bc)(\ - bd) = b - a.
This is not true because the left side is (b — a){\ — abed). However, it suggests
that the error is a function of abed. So we write
h(abed)
I(a,b,c,d) =
(ab)oo(ac)oo(ad)oo(bc)oo(bd)oo(cd)o
Now use this in A0.8.6) to get
bh{abcdq)
(abq)oo(acq)oo(adq)oo(bc)oo(bd)(x>(cd)oo
ah(abcdq)
(abqHO(acHO(adHO(bcq)O0(bdqHO(cdHO
(b — a){\ — ab)h(abcd)
(ab)oo(ac)oo(ad)oo(bc)oo(bd)oo(cd)oo
Simplify to get
h(abcd) = A -abcd)h(abcdq).
Iterate this to arrive at
h(abcd) = (abed; q)ooh@) = (abed; q)xM(q). A0.8.8)
To identify M(q) we need to find specific values of a, b, c, and d where we can
evaluate both the integral and the infinite product in A0.8.8). The obvious values
(a, b, c, d) = A, ^/q, — 1, —^/q) work for the integral as well as the product. The
integral is
Г
d0 = л,

518 10 Introduction to ^-Series
and
(-ql/2; q)oo(-q\ q)oo(q1/2q)o
This implies that M(q) = 2n/(q; q)^. The conjectured value of I {a, b, c, d) is
in fact correct and we have the theorem:
Theorem 10.8.1 For max(|a|, \b\, \c\, \d\, \q\) < 1,
' h{x,\)h(x,Jq)h{x,-l)h{x,-Jq) dx
h(x,a)h(x,b)h(x,c)h(x,d) Vl - x2
2л(abed; q)x
(q> q)oc(ab; q)oo(ac; q)oo(ad; q)oo(bc; ^H0(Ы; ^)го(с^; q)oo '
when
00
h(x, q) = ЦA - 2axqn + a2q2n).
n=0
Proof. We have already seen that both sides of this relation satisfy the func-
functional equation A0.8.6). It is also clear that both sides are analytic functions
of a,b,c, and d when max(|a|, \b\, \c\, \d\) < 1. By the uniqueness of ana-
analytic functions, it will be sufficient to show that they are equal for (a, b, c, d) =
(qJ, qk+1/2, -qe, -qm+1/2), j, к, I, m = 0, 1, 2,.... We first show that they are
equal for a = 1, с = — 1, d = ^fq, and all b, \b\ < 1. In this case the integral is
simply
h(x; ^/q) dx
and this was evaluated in A0.8.3). Thus we have to show that
2?r(V/2;
2; q)c
But the right side is
; q)oo(bq1/2; q)M n{bq1'2; q)
2(q; q)oo(b2; q2)oo(b2q; q2)oo(-q; q)oo(q; q2)oo Ф2; q)x(qi q)oc '
This gives us a starting value, and the functional equation A0.8.6) then shows that
if the integral can be computed for some (a,b,c,d), then it can be computed when

10.8 q-Beta Integrals 519
one of the parameters is multiplied by q. This proves the formula for
(a, b, c, d) = (qj,b, -qe, -qm+1/2), j, I, m = 0, 1, 2, ...,
and all b, \b\ < 1, which completes the proof. ¦
If the restriction to max(|a|, \b\, \c\, \d\) < 1 is removed, then some discrete
mass points appear. This case was first treated directly via Cauchy's theorem by
Askey and Wilson [1985]. A direct extension from the case in Theorem 10.8.1 was
provided by Gasper and Rahman [1990, §6.6].
An extension of Theorem 10.8.1 is given below.
Theorem 10.8.2 Formaxi<i<5(|ai|, |o|) < 1,
h{x,l)h{x,Jq)h(x,-\)h(x,-Jq)h(x,Yl\ai) dx
/,
П1 *(*,*.-) VI-
27r(fl1fl2«3«4)oo(ai«2«3«5)oo
(«102H0 («1«з)оо(«1«4)оо(«1«5)оо(я2«з)оо
(a2«3«4«5)oo(«l«2«4«5)oo(«l«3«4«5)oo
(«2«4)oo(«2«5)oo(«3«4)oo(«3«5)oo(«4«5)oo0?)oo'
where (x)oo = (x; q)x-
This is proved in the same way we proved Theorem 10.8.1, except that a more
elementary initial result can be used. A more general integral was evaluated by
Nassrallah and Rahman [1985]. This special case was noted by Rahman [1986],
and the proof outlined here was given by Askey [1988].
Theorem 10.8.1 has an important limiting case, as we saw earlier. Write
and
in the integral and let q —*¦ 1 . The result is the de Branges-Wilson integral C.6.1):
00 Г (a + ix)T{b + ix)r(c + ix)T{d + ix) 2 J
dx
Г(Их)
Г (a + b)T(a + c)T(a + d)T{b + с)Гф + d)r(c + d)
~ T(a + b + c + d) '
Re(a, b, c, d) > 0, A0.8.9)
when either all the parameters are real or any complex ones appear in conjugate
pairs. When a is complex, | Г (a + ix) \2 should be replaced by Г (a + ix)T(a — ix).

520 10 Introduction to ^-Series
We saw that the integrand of A0.8.9) is the weight function for the orthogonality
of the Wilson polynomials
— n, n+a+b + c + d— l,a — ix,a + ix
; 1
a -\- b, a -\- с, a + d
There is a #-extension of these polynomials for which the integrand in Theo-
Theorem 10.8.1 plays the same role. See Gasper and Rahman [1990].
10.9 Basic Hypergeometric Series
The series in the #-binomial theorem takes the form Y2 cn with cn+\/cn a rational
function of qn. In this case
C/i+i _ 1 -aqn
<-« i q
and
cn = 7 Г"-* i CQ = 1.
(?; ?)«
Earlier, we started with the <?-binomial theorem and then proceeded to consider the
triple product and its generalization, the i ^i • This introduced us to theta functions
and elliptic functions, among other things. The above remarks on the ^-binomial
series suggest another direction. As a generalization of this series we consider a
series V cn for which
=
cn
This can be solved as
(a; q)n(b; q)n n
(c;q)n(q;q)n
The function Yl^Lo c« 's denoted by
а<Ь
;q,x
and is an example of a basic hypergeometric series. If we set a = qa, b — q@,
and с = qY and let q -> 1~, we have the ordinary hypergeometric function
2Fi(a,(i; y;x). The 2Ф1 series was studied by Heine [1847], who proved the
following theorem, which is an analog of Euler's integral formula for the hyper-
hypergeometric function:

10.9 Basic Hypergeometric Series 521
Theorem 10.9.1 For \q\ < 1, |x| < 1, and \b\ < 1,
a,b \ (b;q)oo(ax;q)oo (c/b,x \
\q,x = 201 ,q,b].
с ) (c;q)oo(x;q)oo V ax )
Proof. As one might suspect from the proof of Euler's integral formula, a proof
of the above transformation requires the <y-binomial theorem and a change of the
order of summation. Once again we write (а)„ instead of (a; q)n for convenience.
Then
,'a,b
201 f
(9). ^0 (9)-
c/b,x
;
which completes the proof.
To help understand this transformation, Thomae [1869] observed that it is a
^-integral analog of Euler's integral representation of the 2^1 hypergeometric
function. Set a = qa, b = q&, and с = qv in Theorem 10.9.1 to get
qr-P- q)m{x- q)mqP>n
х ^
{q)n{qy)n X (qy\q)oo(x;q)oo ^0 iq\ q)m(qax; q)m
{qP\ q^jq^; g)oo y^ (gm+1i q)oo{xqa+m\
^ -l. q)oo{xqm. q)x '
A0.9.1)
Use the notation
(a\q)a
(aqa;q)
to write A0.9.1) as the ^-integral formula
Tq{Y)
(xt;q)a ^
Euler's integral representation is one way of evaluating a 2 F\ at x = 1. Similarly,
we have the ^-analog of Gauss's sum due to Heine.

522 10 Introduction to ^-Series
Corollary 10.9.2 For \c/ab\ < 1,
a,b \ (c/a; q)co(c/b;
;q,c/ab =
с J (c; q)oo(c/ab; q)x
Proof. Takex = c/abin Theorem 10.10.1, assuming that |6| < Iand|c/a6| < 1.
Then
[( ' ;q,c/ab) = ——
V с J (c;q
{сjab; q)mb"
(q)m
(c; q)oo(c/ab; q)^ (b; q)^
The last step follows from the ^-binomial theorem. This proves the corollary except
for the assumption on b that \b\ < 1. This can be removed by analytic continuation.
¦
Bailey [1941] found an analog of Kummer's theorem that evaluates 2F\{a,b\
a+\ -b;-l).
Theorem 10.9.3 If\q\< min(l, \b\), then
л. ( a>b iu\ (aq>q)°c(q;q)oc(aq/b;q)oc
2<pi ; q, —q/o = .
\aq/b J (aq/b;q)oo(-q/b;q)oo
Proof. First assume that \a\ < 1. Interchange a and b in Heine's transformation
to get
/ a,b \ =
l\aq/b' ' /
(a;
(a;
(aqjb
(aq/b
q)oc
;q)a
q)oo
q)l
(-q\
(-?;
Л-q,
Л-q,
q)oo
/b; q)z
q)x
/b;q)c
/b;q)z
00
m=0
(a
X)
(q/ь
(q
(q2/i
(q2
(a;q
;q)m
; q)m
\q2\
0
(-q/b;
(-q;q)
>m m
m
q)m
m
(aq/b; q)oo(-q/b; q)x
Now remove the condition on a and the result is proved.
Corollary 10.9.4
00 an(n-l)xn i
, (Cauchy), A0.9.2)
= {aq; q2)oo(-q; q)oo. A0.9.3)

10.10 Basic Hypergeometric Identities 523
Proof. For A0.9.2) set a = I/A, b = 1/B, and с - x in Heine's for-
formula (Corollary 10.9.2) and then let A -+ 0, В ->• 0. To obtain A0.9.3), set
b = l/В in Bailey's formula (Theorem 10.9.3) and let В -> 0. This proves the
corollary. ¦
The case x = q in A0.9.2) gives
This particular case of A0.9.2) was known to Euler. Similarly, the special case
a = q in A0.9.3) gives Gauss's formula
°° °° 1 Jim
1 -I- X^an(n+l)/2 = TT ~q
2-^<4 11 2 — q2m-\ ¦
We obtained this earlier from the triple product identity.
It should be clear by now that many results on hypergeometric series have
extensions to basic hypergeometric series. We develop this theme a little more in
the next section, where we derive some basic hypergeometric identities. For much
more on this subject, see Gasper and Rahman [1990]. Identity A0.9.2) is in Cauchy
[1843].
We terminate this section with a generalization of the 2ф\ series that will be needed
in the next section and later. An r<ps basic hypergeometric series is defined by
ai,a2,
n ПО 9 4)
where q ф 0, when r > s + 1. An г+\Фг series is called ^-balanced if x = q and
b\b2---br — qka\a2 ¦ ¦ ¦ ar+\. A0.9.5)
When к = 1, the series is called balanced. The series A0.9.4) is called well poised
if s = r — 1 and
qa\=b\ai = ••• = br-\ar. A0.9.6)
10.10 Basic Hypergeometric Identities
An iteration of Heine's transformation in Theorem 10.9.1 gives
'a,b \ (c/b;q)oo(bx;q)oo fabxlc,b \
' •--> w 'HJOoy 'HJ°° + I ' ;q,c/b), A0.10.1)
с J (c; q)oo(x', <?)oo V bx
and a second iteration gives

524 10 Introduction to g-Series
Theorem 10.10.1
'а,Ъ
с
This is a ^-analog of Euler's transformation formula
'a,b \ „ „-„-h „ fc-a,c-b
> \ (abx c;q)x (с а, с b \ ПП1П~
;q,x) = — 20i ;q,abxc). A0.10.2)
In A0.10.2), expand the infinite product on the right by the ^-binomial theorem
and equate the coefficients of x". This gives
(a; q)n(b; q)n A (c/a;q)k(c/b;q)k k(ab/c;q)n-k
(c;q)n(q;q)n f^ (c;q)k(q;q)k (q\q)n-k
(ab/c; q)n -A (q~n; q)k(c/a\ q)k(c/b; q)kqk
(q;q)n frZ (ql-nc/ab;q)k(c;q)k(q;q)k
After renaming parameters, we get
~",a,b, \ (c/a;q)n(c/b;q)n
(c;q)n(c/ab;q)n '
This extends the Pfaff-Saalschiitz identity for the balanced 3F2. The 302 in
A0.10.3) is balanced because the product of the numerator parameters q~"ab
times the power-series variable q equals the product of the denominator parame-
parameters. Recall that the q used in the definition of balanced appears as the power-series
variable. For Heine's 2^1 sum in Corollary 10.9.2, the variable is c/ab, and the
same type of condition holds, that is, the product of the numerator parameters
times the power-series variable is the denominator parameter (a ¦ b ¦ (c/ab) = c).
The following is a more general result due to Sears [1951]. It gives the transfor-
transformation between two terminating balanced 403 series and extends Whipple's 4F3
transformation.
Theorem 10.10.2 For a positive integer n,
(q~n,a,b,c, \ n(e I a\ q)n(fI a; q)n
403 ; q.q ] = a
1 J ~ r 4] (e;q)n(f;q)n
/ n~n n И/h die \
A0.10.4)
i,aq~n+l/e, aq~n
when def = abcql~n.
Proof. Take the transformation A0.10.2) twice each time with different param-
parameters; then take their product such that the function on the left is multiplied by the

10.10 Basic Hypergeometric Identities 525
function on the right in the other identity. This yields
a,b \ (dex/f; д)ж (f/d, f/e dex\
;q,x\— 20i , ;q,—r)
с ) (x;q)oo \ f f J
d,e \{abx/c;q)oo , (c/a,c/b abx\
;q,x]— 201 \q, •
Reduce this to the product of two series by taking
ab de
The equality of the coefficients of x" gives
/e\ g)nk (dc/f)
A (a; q)k(b; q)k (f/d; q)n-k{f/e\ g)n-k (dc/f)n-k
„м) ^с; ^к^' q^>k (f' i)n-k(q; q)n-k
_ ^ (d; q)k(e\ q)k (c/a; q)n-k(c/b; q)n_k
~^ (/; q)k(q; q)k (c; q)n-k(q\ q)n-k
After some rearrangement and renaming of parameters, it can be seen that this is
equivalent to A0.10.4). ¦
It should be observed that both sides of A0.10.4) are balanced series. There are
many interesting limiting cases of this transformation given in the next theorem.
Before we state it, note that if с = d in A0.10.4), we get A0.10.3).
Theorem 10.10.3
'q-\a,b_
q-",a,b \ (e/a;q)n(f/a;q)n „ ( q~n,a,abql~n/ef
A0.10.6)
(a,b,c \ (e/a;qH0(de/bc;q)oo , (a,d/b,d/c . \
302 ; q, de/abc = -—302 , , ,, ; q, e/a ,
V d,e J {e;q)o0{de/abc;qH0 \ d,de/bc J
A0.10.7)
fa,b,c , , , \ (a;q)oo(de/ab;q)oo(de/ac;q)o
302 , ;q,de/abc =
d,e J (d;qH0{e\q)x{de/abc\qH0
'd/a,e/a, de/abc
, ,4,» i- A0.10.8)
de/ab, de/ac '

526 10 Introduction to g-Series
Proof. To prove A0.10.5), let / and с tend to zero in A0.10.4), keeping f/c
fixed. Similarly, for A0.10.6), let d and с tend to zero with d/c = q~"+1ab/ef.
Finally, to prove A0.10.7), set f — ql and let n -»¦ oo with n + к fixed so that
qn+x _ abcq/dg.
Now A0.10.7) is a generalization of Kummer's transformation
a,b,c \ r(e)r(d + e-a-b-c) / a,d-b,d-c
fa,b,c \ _ r(e)r(d + eabc) / a,db,dc
3 2\ d,e ' У ~ r(e-a)r(d + e-b-cK 2\d,d + e-b-c'
A0.10.9)
When A0.10.7) is applied to itself, the resultis A0.10.8), witha and с interchanged.
Remark 10.10.1 It is worth mentioning that if we set x = c/e and let с -> oo,
Kummer's transformation becomes
"•''-* " ¦ A0.10.10)
In contrast, if x = a/e and a —>• oo it becomes
¦^i-o-xj-W-V-^i.
We have already seen a g-extension of the last formula. Let us determine a q-
extension of A0.10.10). In formula A0.10.7), let de/abc = x and с and e tend to
zero. The result is
a, .qx\ _ Уах'У>°° y^ (a'q>n( ' 'q)"q [——фх)п.
d ) {x\q)oo ^ (d; q)n{ax; q)n{q; q)n
A0.10.12)
When a, b, and с are replaced by qa, qb, and ^e respectively and we let q —> 1~,
this reduces to A0.10.10). This is an instance where the ^-extension is nicer than
the q = 1 case in the following sense: The left side of A0.10.10) is analytic for
|x| < 1, while the right side has two factors, the first analytic forx in С — [1, oo)
and the second analytic for Rex < 1/2. In A0.10.12), the 2<pi on the left is analytic
for |x| < 1, while on the right, l/(x; q)^ has poles when x — 1, q~l, q~2,
The other factors are entire functions since {ax; q)oo/(ax; q)n = (axq"; q)^ is
entire while the series converges uniformly for |x| < A for each A.

10.11 g-Ultraspherical Polynomials 527
10.11 4-Ultraspherical Polynomials
In Section 6.11, we introduced the (continuous) g-ultraspherical polynomials de-
defined by
Cn(x; p | q) = У^ —:—-—'-—— cos(n — 2k)9
f^ {q;q)k(q;q)n-k
^'i)n JM , , , ,r o-^-iw i A011 Л)
(<?; q)n
(Because in this section we discuss only the continuous g-ultraspherical polyno-
polynomials, the word continuous will be dropped.) When the generating function found
in Chapter 6 is combined with the g-binomial theorem, the result is
It is also easy to conclude from F.11.5) and F.11.6) that the polynomials Cn(x;
/5 | q) satisfy the recurrence relation
2A - Pq")xCn(x; /5 | q) = A - qn+l)Cn+l(x; /3 | q)
+ (l-p2qn-l)Cn-i(x;P\q). A0.11.3)
An implication of Theorem 6.6.2 is that [Cn(x; ft \ q)}, x — cos0 is an orthog-
orthogonal polynomial sequence with respect to the distribution
2
cop(cosd)dd —
lw;q)o
dd. A0.11.4)
(Pe2if>; q
Since we did not prove Theorem 6.6.2, we verify this fact directly here.
Theorem 10.11.1 When \q\ < 1 and |/8| < 1,
/ Cn(cos6>; p | <?)Cm(cos6>; p \ q)a>e(cos0)d9
Jo
^2n(\-p) (p2; g)n (Р;д)х(Рд;д)оо5
A - pq") ' (q; q)n ' (p2; ?)«,(?; q)K mn'
where cop(cosd) is given by A0.11.4).
Proof. Take p and q real. Since the integrand in A0.11.5) is unchanged by
the transformation в ->¦ —в and Cn(cos#; p \ q) has an expansion in terms of
ei(n-ik)B^ start wjtll tlie integral
i-n („2i0. _\ („-Ив. „\
= em %,' Ч)°°(е 'q)°° dB.
J-n (Pe^-q^ipe-^-q)^

528 10 Introduction to q-Senes
For | f} | < 1, the g-binomial theorem gives
When k is odd, the integral is zero. When к = 21,
г-ж "
/ •""
J —11
n=0 (q;q)n(q\q)e+n
к ¦ V qe+l
Apply A0.10.1) to the above 2Ф1 to get
This 20i has only two terms, and so we have
nee . ал 2лР(Г; g)<d + ge)
e (°(COS9)
Le
A second way to evaluate this integral is to use Ramanujan's 1^1 sum A0.5.3).
Let m < n. Now consider the integral
/ cos(m - 2к)в Cn(cos6>; /8 | q)со^(cos в)dd
Jo
= I Г em-W у (Р-'ЯШ'ЯЪ-'^-трв (cose)de_
2 j-п j^ (<?; q)dq\ q\-t
Since m -Vn — 2k — 21 has to be even for the contribution of the integral to be
nonzero, let m — n — 2k = — 2s orm—2k = n—2s. Then the integral is
у (Р;яМР;я)»-1 Г ш-i-,» (cosQ)dQ
J P
a}fi(cos0)d0
~-°(\+qs-'
3s(l+qs
q; q)e(q; q)n-
(Pq;q)oo(P;q)oo (Р;я)п(Р~1;я),
A (g-"; яНя"Р~и, яМР; яМ-я1-'; q)d
' j^o (q; q)t(ql-"p-l\ q)dqx~sP\ q)d-q~s\ q)e '

10.11 g-Ultraspherical Polynomials 529
The second equation in the above calculation is obtained by changing t to n — I
and в to —в. The last sum, which is a balanced 4^3, when transformed by Sears's
formula (Theorem 10.10.2) becomes
Pn(ql-s\q)n(qX-n/P2;q)n (q~\ P, ~P, q~[.
V ?"' Ч'~п P2
(qt-'P; q)n{qx-nIP; ?)« V -?"', Ч'~п, P2
This 40з has only two nonzero terms and the above expression becomes, after a
simple calculation,
(P2;q)n(ql-s\q)n-iu-qn-2s)
(P;q)n(ql-'P;q)n(l+q-s)
The factor (qx~s; q)n-i is zero for s = 1, 2,..., и — 1. This gives
/ cos(« -2sN>Cn(cos6>; fi \q)cop(cosd)de
Jo
{0 for.s = 1,2, ...,n- 1,
(P;q)oo(Pq;q)oo(P2;q)n . „
n— fors = 0 or n.
(P2;q)oc(q;q)oc(Pq;q)n
Jo
By A0.11.1) and the above relation we obtain the orthogonality
Cm(cos6>; p I <?)C,,(cos6>; p | ^)«/j(cos6>)u?6>
(<8; q)n (<82; g)n(<8; g)oo(j6g: q)"° j
(q;q)n (Pq;q)n(P2\q)oo(q;q)oo
A-/8) (I2; <?)„ (P;g)oc(Pq;q)oos
n(\-Pq") (q;q)n {q; q)x(P2\ q)
The theorem is proved. ¦
The polynomials Cn(cos#; p\q) satisfy a difference equation. To state it, we
need the #-difference operator Dq defined by
P with Sgg(ew) = g(q1'2ew)-g(q-^2ew), x=
bqx
A0.11.6)
It can be shown from the generating function for Cn(cos#; p\q) that
DqCn{x; p\q)= ^ J ^'""^Сп-К*; Pq\q). A0.11.7)
The ^-difference equation is
A - q2)Dq[(oN(x)Dqy(x)] + V~n(l - qn)(\ - p2qn)coft(x)y(x) = 0,
A0.11.8)

530 10 Introduction to g-Series
with y(x) = Cn(x; p | q). As a first step to the proof of A0.11.8), one can show
that
Dq(wp(x)Ca(x;p\q))
2q-n/1(l - qn+l)(l - p2qn~l)
One can use A0.11.7) to give a proof of the connection coefficient formula:
Cn(x; у | <?) = > ^— — —Сп-2к(х; P I q).
f^ (q;q)k(Pq;q)n-k 1-/8
A0.11.9)
The proof follows the same steps as that of Theorem 7.1.4'. Formula A0.11.9)
was first given by Rogers [1895]. Rogers also found the following linearization
formula:
Cm(x;p \q)Cn(x;P \ q)
min(m,
= E
mm(m'n) (<?; g)m+n-2k(P; g)m-k(P; q)n-k(P; q)k(P2\ q)m+n-k
0 2' q)m+n-2k(q; q)m-kiq; q)n-k(q; q)kiPq\ q)m+n-k
- pqm+n-2k)
-Ст+п„2к(х; р \q). A0.11.10)
A - Р)
Proving this by induction is easy. It is likely that Rogers first computed the formula
for a few small values of m and then guessed the general result. The simplest direct
evaluation of the formula may be the one similar to the proof of Theorem 6.8.2,
which uses the ^-analog of Whipple's transformation given in Chapter 12. For
details see Gasper [1985]. The proofs of A0.11.7) to A0.11.10) are left to the
reader.
When we set p = 0 in Cn (x; p \ q) we get the (continuous) g-Hermite polyno-
polynomials. They are defined by
Н„(х | q) = (q; q)nCn(x; 0 | q). A0.11.11)
The following properties of the g-Hermite polynomials are now immediate:
#n(cos6> \q) = ir (Jtlh cos(n _ 2к)в
trt(q;q)k(q;q)n-k
They satisfy the orthogonality relation
Hn(cose | q)Hn(cose | q)\(e2w; q^fde = / m" ¦ A0.11.13)
/"
Jo
(qn+u,q)o

10.11 g-Ultraspherical Polynomials 531
Since the weight function for the Hermite polynomials is e~x and
we may regard the integral
f71
f
J-a
/0
as an extension of the normal integral
/¦00
e'^dx =
-00
The generating function for Hn(x \ q) is given by
J2 n(x\1>rn = —_ __ ) x = cos6>; A0.11.15)
and the three-term recurrence relation is
2xHn(x | q) - Hn+i(x \q) + (l- qn)Hn-x(x \ q). A0.11.16)
By A0.11.10), the linearization formula is easily seen to be
Hm(x | q)Hn{x | q) _ sr-^ Hm+n-2k(x | q)
(q;q)m(q;q)n ~^ {q\ q)k{q\ q)n-k(q\ q)m-k
A direct proof is indicated in Exercise 41. From A0.11.17) and the ^-binomial
theorem, it is possible to derive a formula for the Poisson kernel of the ^-Hermite
polynomials:
2;<?)oo
A0.11.18)
where r is real, —1 < r < 1. The derivation is left to the reader as an exercise.
Finally we note that the integral I (a, b, c, d) of Section 10.8 can be written in
terms of #-Hermite polynomials as
v^ akbecmdJ
I(a,b,c,d)= }
A0.11.19)
i >o ^' tiM* 4)kiq\ q)e(q\ q)
• Г Hk(x | q)Ht(x | q)Hm(x \ q)Hj(x | q)\(elw;,
Jo
where x = cos#. This follows easily from the generating function A0.11.15). By
means of the linearization formula A0.11.17), another evaluation of the integral is

532 10 Introduction to g-Series
obtained. This observation is due to Ismail and Stanton [1988]. They also pointed
out that A0.11.17) and A0.11.18) are equivalent. Al-Salam and Ismail [ 1988] used
the connection coefficients and linearization for (continuous) g-ultraspherical and
Hermite polynomials to prove Theorem 10.8.2.
10.12 Mellin Transforms
The integral of xa~l(—ax; q)oo/(-x, <?)oo, evaluated in Section 10.8 by a func-
functional equation, is also a particular case of an interesting Mellin transformation
formula. This formula, given by Ramanujan, has many important applications,
some of which will be presented in the exercises. The Mellin transform connects
up in an important way the transformations of some ^-series with functional equa-
equations satisfied by certain Dirichlet series. Earlier we mentioned a transformation
for
n=\
that was useful in computing a Riemann sum approximation of the normal integral
f°° e~c'2dt. In Exercise 2.28, the reader was asked to use this transformation to
obtain the functional equation for the Riemann zeta function, ?(s). This relation-
relationship between ^-series, which arise from elliptic functions, and Dirichlet series is
particularly important in number theory. We discuss a few examples, especially
those involving ^-series, as considered in the previous sections.
Ramanujan stated his formula as
/
_
- хф{\) + х2фB) }dx =
sin sn
In this form, one must put some strong restrictions on 0C$). Hardy [1940, pp.
189-190] gave fairly general conditions for the validity of this formula. See also
Berndt [1985, p. 299]. Hardy's theorem is stated below.
Let H(S) denote the half plane и = a + it, a > -8, 0 < S < 1. Suppose
A < n; let К (A, P, S) denote the set of all functions ф, holomorphic on H (<5),
that satisfy
\ф(и)\ = O{ePa+m). A0.12.1)
Take 0 < с < S, and define
1 fc+i°° n
Ф(х) = — / ф{-и)х~ийи. A0.12.2)
2ni Jc-iao &mnu
The integrand is

10.12 Mellin Transforms 533
thus that the integral converges uniformly in any interval 0 < xo < x < X. There-
Therefore, the integral represents an analytic function Ф (x) for x > 0. An application
of Cauchy's theorem gives
1 f n
2ni y_Ar_i_,-00 smnu
= 0@) - x0(l) + • • • + (-lfxN(p(N). A0.12.3)
If we take 0 < x < e~p, then the series 5^°(-1)п0(и).хп converges and the
integral in A0.12.3) goes to 0 as N -> oo. So, for 0 < x < e p,
Ф(х) = 0@) - 0(l)x + 0B)x2 .... A0.12.4)
Theorem 10.12.1 LetO<Res<S.IfФ(x) is given by A0.12.2), then
/ xs~x^{x)dx — —'¦
Jo sir
Sin 57Г
Proof. Choose c\ and C2 so that 0 < C\ <Res=or <сг < S. Compute the
absolutely convergent double integral
i A rci+ioo —
r- / /
in two different ways. One way it equals
o— / ^"M / <H-u)x-udu)dx = / х^1
2л-1 Jo \Jc-ioo Sin^M / Jo
The other way it is
Xs ldx)du = -—l -. du.
) 2J
,_I00 sin;rM \J0 ) 2niJCl_i00 &innus-u
Therefore,
Jo
2ni JCl_100 sin^M s — и
Similarly, use the double integral
1 Г ri+io° л- . . . .....
/OO ЛС2+1
Л-2-ioo
to get
Г
Г .-u/ ч^ ! Г2+'°° ^ ^z"L
/ x Ф(х)с?х = / du.
J\ 2ni Jc i00 smnu s-u

534 10 Introduction to ^-Series
An application of Cauchy's residue theorem gives
fC2+i00 N д.
/o
ZTll \Jci—ioo Jc2—ioo
П
smns
Corollary 10.12.2 If0<q<l,s>0,and0<a< qs, then
/¦00
Гxs-i(-ax'
Jo (x;q
;^)oo sinsn (q;q)x(aq s;q)oo
Proof. Take
ш = (a;q)^,q)Xj
and check that ф satisfies the conditions of Theorem 10.12.1. ¦
Another corollary is Carlson's theorem, which we proved and used in previous
chapters for ф bounded.
Corollary 10.12.3 Suppose ф(и) is analytic in a half plane H(8) = [и \ Rew >
-8], 0 < 8 < 1 satisfying A0.12.1), and ф(п) = Ofor n — 0, 1, 2,.... Then
ф = 0.
This corollary shows that Ramanujan's formula is actually an interpolation for-
formula. In fact, Newton's interpolation formula
k(-s) = A@) + ^
where
ДЛ@) = Ml) - МО), Д2М0) = ЯB) - 2ЯA) + МО),
can be derived from Theorem 10.12.1, under conditions on X. For this and other
applications the reader should see Exercises 34-36.
When we discussed Mellin transforms earlier, it was mentioned that Mellin
transforms are just Fourier transforms with a change of variables. Thus theorems
about Fourier transforms have analogs for Mellin transforms. Therefore, there is a
uniqueness theorem for the Mellin transform. Suppose xa~lf(x) is integrable on
@, oo) for a < о < p. Then
/¦OO
F(s)= xs~lf(x)dx A0.12.5)
Jo
exists and is analytic for a < Re s < p.

10.12 Mellin Transforms 535
Lemma 10.12.4 IfF(cro + it) = Ofor all t e (—oo, oo) and fixed go in («, P),
then f(x) = 0 almost everywhere.
Proof. Let x = e" in A0.12.5). Then
/¦OO
F(flr0+i0= / eituea«uf(e")du
J — 00
is the Fourier transform of ea°uf(e") = 0 almost everywhere. Since ea°u never
vanishes, the lemma is proved. ¦
Note that if, in the lemma, / is continuous at a point, then / must be zero at
that point. This implies that if the Mellin transforms of two continuous functions
are the same, then the two functions are equal. Here is an application of this fact:
Theorem 10.12.5 For и > О
I e-(utwlAt)^t =
Proof It is sufficient to prove this for и = 1. Denote the integral by f(x). For
Re s > 1, the double integral
/°° f°° г dt
Jo ft
is absolutely convergent. Thus it is equal to
5 dt 1 f°°
—p \ax — I x j(x)ax.
wt \ Jo
Jo [Jo
Now change the order of integration to get
/•00 — t Г /-00  /-00
e-~\ xs-le-x2'4tdx\dx=2s-lr(s/2) t
Jo -ft \_Jo J Л
The last step follows from Legendre's duplication formula. We have
/•OO
/ xs-lf(x)dx = f^T(s).
Jo
This means that f(x) and fjre~x have the same Mellin transform and hence they
are equal. This concludes the proof of the theorem. ¦
This proof follows Bellman [ 1961, p. 30], which contains an interesting discus-
discussion of theta functions.

536 10 Introduction to ^-Series
There is another related integral transform that is important, the Laplace trans-
transform, defined by
/*oo
F(s) = / e-stf(t)dt.
Jo
If / is integrable and satisfies \f(t)\ = O(ebt) as t -> oo, then F(s) is analytic
for Re s > b. There are uniqueness theorems for Laplace transforms as well. We
can use this uniqueness to prove the transformation formula for theta functions
mentioned earlier.
Rex > 0.
(A more general result can be proved by the Poisson summation formula, but this
result shows that the theta function defined by the series is a modular form. See
Remarks 10.12.1 and 10.12.2 at the end of the section.)
Proof. For Re s > 0,
-oo °° 1 °° 1
Theorem 10.12.6
oo
и=—oo
frc
CO
n=—oo
/
Jo
e
n=\
By Theorem 10.12.5, when Re s > 0
/¦0
/
Jo
n=l
n=l
ж 2тг е'1п^ л-
+
According to A.2.5), we have
л cot лх = —1-2
X n=\
After proper identification, this shows that
л 1 + <
^/c 1 _ p~2n^s c. /L^i „2 _]_ г
and the lemma is proved by the uniqueness of the Laplace transform. This proof
is due to Hamburger [ 1922]. ¦
When we take the Mellin transform of У^„ е~п x, the Riemann zeta function
is obtained. This gives a connection between ^-series and Dirichlet series. An

10.12 Mellin Transforms 537
exercise in Chapter 2 asks the reader to use the result of Theorem 10.12.6 to obtain
the functional equation of t;(s). We give the details here because the result and
proof are important.
Theorem 10.12.7 The expressionn~s/2T{sj2)C,{s) is invariant under s -> \—s.
The following proof goes back to Riemann [1859].
Pmof. For Re s > 1, and ф(x) = J^Li e~"lnx-
/00 °°
/¦1 roo
= / x(s/2)~l\j/(x)dx + / x(-sl2)-lf{x)dx.
Jo J\
Change x to l/x in the first integral to get
I X-il/2)-lf(l/x)dx. A0.12.6)
By Theorem 10.12.6
+2ф(х) = ~(
Thus A0.12.6) becomes
— 1 1 Г°°
= —- --+ x-^
l-s s Jx
We can conclude that
1
+
s l-s Ji
A0.12.7)
Because the integral is an entire function of s, the expression on the left is defined
for all s. In particular, ?(.?) is defined for all s except s = 1 where it has a pole of
order 1. The integral is invariant under s -> 1 — s. This proves the theorem. ¦
We have shown that the functional equation for the zeta function, the transfor-
transformation formula for the theta function, and the partial fraction expansion of the
cotangent function are equivalent results. Of course, we still need to show that the
theta transformation is a consequence of the functional equation. This can be done
by Mellin inversion. To illustrate the technique, we apply it to a different though

538 10 Introduction to ^-Series
related function, that is,
00
tl(T)=ql'24Y[(l-q"), \q\<l, A0.12.8)
n=l
where q = elnix. This is called the Dedekind ^-function.
Before we state the theorem, note that the integral in A0.12.7) is bounded in
vertical strips, a < Re s < p. So the same is true for
n-s'2T(s/2)l;(s) + -
s 1 — s
Theorem 10.12.8 The Dedekind function r)(x) satisfies the relation
T](-l/T) = ^ TJ(T).
Proof. We follow Weil [1968]. Since t]{t) has no zeros for Im r > 0, we prove
that
logi?(-l/T)=logij(T) + ilog(T/i). A0.12.9)
From A0.12.8)
Let
ю = E
m,n>l
00 Jlnimnx
m
m,n=l
The Dirichlet series corresponding to / can be found by taking its Mellin transform
as before:
— Inmnx
/o
^ m
Now
m=\ n=l
It follows from Theorem 10.12.7 that
A(s) - Bff)-T(j)C
remains invariant under s -> —s. It is clear that A(s) has simple poles at s — ±1
with residues ±7г/12 respectively, since f (—1) = —1/12. This last fact can be

10.12 Mellin Transforms 539
verified from the functional equation for ? (s). At s = 0, however, Л (s) has a double
pole. These are the only poles of A(s). Since ?'@) = —1/2, we conclude that
/ Л (Ю1210)
12(s-l) 12E+1) ' 2s2 x '
is entire and bounded on every vertical strip. The last observation follows from
the remarks made before the theorem. By Cauchy's theorem, the Mellin inverse
of r(,s) is «Г?, that is,
2 /-c+ioo
e~y = r— / T{s)y-Sds, с > 1.
From this we get
i /¦c+ioo
fdy) = T~ / Ms)y~'ds, с > 1. A0.12.11)
Observe that t;(s)t;(s + 1) is bounded on the line Re s = с > 1, because the series
for ?(s) converges absolutely. By Stirling's formula
r(.s) ~ V2Jr\t\a-{1/2)e-*W2, s=a + it, \t\ -> oo,
in any vertical strip a <a < /3. These two facts imply that for any ц, > О
\A(s)\ = O(\t\-'*), \t\ -> oo, Res = ol. A0.12.12)
Now choose c\ so that —c\ > 1. Thus, for any д > О,
\A(s)\ = \A(-s)\ =
Since the expression in A0.12.10) is bounded in every vertical strip, it follows
from the Phragmen-Lindelof theorem that A0.12.12) holds in (ci,c) for large t.
All this implies that we can move the line of integration in A0.12.11) to — с < — 1,
while picking up the residues at s = ±1 and s =0. The expansion for y~s is
1 - s log у Н ,
so the residue at s = 0 is (log v)/2. At s = ±1, we have residues n/\2y and
-тгу/12. The result is
r-c+ioa
^: /
2ni J-c-ico
1 r— c+ioo
= — / A{-s)y-sds + tt/12v + (log v)/2 - ttv/12
2ТП J-c-ioo
2 /¦c+ioo
= — / A(s)y'ds + ir/12y + (logy)/2 - тгу/12
2тгг Jc_i00
(\ogy)/2 - nyj\2.

540 10 Introduction to ^-Series
We may therefore conclude that the relation A0.12.9) holds on the imaginary axis.
Since the functions involved are analytic, the result is proved. ¦
Remark 10.12.1 Note that g(r) = r?8(r) has the following two properties:
g(-l/T)l/T4 = g(T).
The interest in the two transformations
r -> r + 1, r -> -1/r
stems from the fact that they generate the modular group G, which consists of
all the fractional linear transformations (at + b)j(cx + d), where a,b,c, and d
are integers with ad — be = 1. A function / on the upper half plane is called a
modular form if for some integer к
f(Ax)A'{x)k = /(г)
for all A e G. This holds for g(x) = t]8(x) with к = 2. Subgroups of G of finite
index also play a very important role. Consider the theta function
where q = enix. The function h{x) = 6>4(r) satisfies
A(r+2)=A(r),
й(-1/гI/г4 =
The transformations r -> r +2 and r -> —1/r generate a subgroup of the modular
group. A problem of interest to number theorists is to find properties of the Fourier
coefficients of modular forms. For example, the coefficient of q" in the expansion
of h(t) is the number of representations of и as a sum of four squares, which was
studied in Section 10.6 by nonmodular methods. For a proof using modular forms
and Hecke operators, see Koblitz [1984, p. 174].
Remark 10.12.2 Theorem 10.12.8 has a simple proof that considers the integral
of cotzcot(z/r) over a suitable contour in the complex plane; it was given by
Siegel [1954]. Rademacher [1955] obtained a more general result by a similar
method. Let h, ti, and к be integers and let

10.12 Mellin Transforms 541
with gcd(h, k) = 1, к > 0, Re z > 0, and hti =e -l(mod k). Then Rademacher's
result states that
i°8 tf \ и i —о ¦< i i. ' ' 2
where the Dedekind sum s(h,k) is given by
k-l
It is, however, worth noting here that the transformation formula for tj(t) is
implied by a similar formula for theta functions, a particular case of which is
contained in Theorem 10.12.6. We have
OO 00
J2 e-s(n+xI = s~l/2 J2 ennl/se2ltinx A0.12.13)
n=—oo n=—oo
(Appendix D, Equation (D.4.2)). Set q = enix and p = е~л'/г. By the definition
of #з in A0.7.3), the above formula implies
l, p) = */VlW/t03Grz,q). A0.12.14)
Now the infinite product for #3 contained in A0.7.7) gives
Я-+1 1 +g-^' J-1V Ч У V У
n=l
and
f n=\
An easy calculation shows that A0.12.14), A0.12.15), and A0.12.16) imply
-.3
тК-1/гK = - W-
i V г
or
where с is a cube root of unity. Set r = i to see that с = 1.
Before closing this chapter, we also note that quadratic Gauss sums are related to
theta functions. Cauchy evaluated these sums from the formula in Theorem 10.12.6
by taking x = e + 2ni/N and letting e -> 0, where N is an odd integer. A more
general reciprocity relation for Gaussian sums can also be obtained by this method.

542 10 Introduction to ^-Series
Bellman [ 1961, p. 40] attributed this proof of the reciprocity relation to Landsberg,
but it was published earlier by Henry Smith in 1859. Smith's [1859-1866] report
contained the reference to Cauchy. For the connection of the reciprocity of Gaussian
sums to Tauberian theory, see Bochner [1952]. The reciprocity formula is given in
Exercise 43.
Exercises
1. Let р„ (к) denote the number of permutations of 1, 2,..., n with к inversions.
Prove that
n\q = Y^Pn(k)qk- (Rodrigues)
/t>o
Then find the total number ofinversions for all the permutations of 1,2,..., n.
(Stern)
The problem on the total number of inversions was posed by Stern and a
solution was given by Rodrigues. An interesting history of results on inversions
is given in W. Johnson's unpublished notes, some old and new results on
inversions.
2. Give a proof of A0.0.10) similar to the proof of the noncommutative binomial
theorem given in the first part of the chapter.
3. Let p(m,n,k) = number of partitions of к into < n parts, each part < m.
Show that
i,n,k)q ¦
k>0
Deduce that
p(m, n, k) = p(n, m, k).
4. Prove the following identities:
m
The last identity is a ^-analog of the Chu-Vandermonde identity.
5. Let
\\ , /(9,0) = 1.
Prove that
f(q,m) = (l-qm~l)f(q,m-2).
Deduce that
,( ч j(l-q)(l-q3)--(l-qm-1), m even,
f(q,m) = < (Gauss)
0, m odd.

Exercises 543
6. Let n be an odd integer and x a primitive nth root of unity,
(a) Use the result in Exercise 5 to show that
k=l
(b) Deduce that
n-1
/fc=O
= (-1)"г (x2 - x-2)(x4 -x~4) ¦ ¦ ¦ (xn~l - x-n+l).
(c) Show that
(±л/п torn = 4k + I,
1 ±iy/n for n = 4k + 3.
(d) Set x = e2niln in (b) to obtain
„ ,-.,s=i . 2л- . 6л- . 2(и-2)л-
G = Bi) 2 sm — sm — • • • sm .
n n n
Note that when n = 4k +1, there are /t negative factors in the sine product.
Conclude that
Do a similar analysis for n = 4k + 3 to show that, in this case,
G = i-s/n. (Gauss)
7. Define inv(mi, mi, ¦ ¦ ¦, mr; n) to be the number of permutations x\x2,...,
хШ]+Ш2+...+ш„ of {lmi2m2,..., rm'\ in which there are exactly n inversions.
Show that
(a) inv(mi,...,mr;n) = ]?"=oinv(mH hmr_bmr;
(b) Use induction and (a) to show that, for r > 1,
\mi+m2-\ Ymr
„>0 L mum2,...,mr Jq
(q\ q)mt ¦¦•(#; q)mr
This is a result of MacMahon. See Andrews [1976, p. 41].
8. Prove that the Gaussian polynomial
G{m,n; q) =
is reciprocal, that is, G(m, n\ q) = qmnG(m, n\ 1/q). Deduce that
p(m, n, k) = p(m, n, mn — k).

544 10 Introduction to <?-Series
9. Prove the following version of the q -binomial theorem:
bkia;q)kib;q)n-k-
State and prove a multinomial extension of this formula.
10. (a) Prove the following analog of the Bohr-Mollerap theorem: For 0 < q
< \,Tq (x) is the unique logarithmically convex function that satisfies the
functional equation
fq(x + 1) = -~-fq{x) with fq{\) = 1.
(b) Let g(x) be defined for x > 0, and let
lim «W = 0.
*->-oo x
Prove that any two convex solutions of
f(x + 1) - f{x) = gix)
differ at most by a constant. Use this to derive the result in (a). (See John
[1938].)
11. Prove the ^-analogs of Legendre's duplication formula and Gauss's multipli-
multiplication formula contained in Theorem 10.3.5.
12. Forq > 1, define
Show that
(a) Tqix) satisfies the functional equation in Exercise 10.
(b) Иш^1+Г,(х) = Г(х).
(c) The residue of Tq (x) at x = — n is
iq - l)"+iqn(n+i)/2
iq;q)n\ogq
(d) If q > 1 and
qx - 1 d3
/(x+l) = ^ -fix), /A) = 1, and --^
q — 1
for x > 0, then
Я*) = Г,(дО forx>0.
(e) If q > 1 and
^ =1, and J-j
for x > 0, then
fix) = Tqix) forx>0.
(See Moak [1980].)

Exercises 545
13. Compute C0(q2) in A0.4.4) as follows:
(a) Note that
n=l
and
n=l
(b) Show that
n=\
(Hint: Use 2
(c) Show that
Deduce that
00
Co(q2rl = Y[(l-q2n). (Gauss)
14. Prove the quintuple product identity:
00
tf (x) = Д(! - «")(! ~ V)d " q"~l/x)(l ~ Л2""')A - q2"~l/x2)
П = — 00
Deduce that
00 OO
n=l n=—oo
One method is to take H(x) — JZ^l-oo c(«)x". To find c@), compute Щ^-
and ^У? to determine c(n) in terms of c@). Then specialize x.
15. Prove that the quintuple product identity in the previous problem is equivalent
to the identity
For Exercises 16-24 and related results and references, see Gasper and
Rahman [1990].

546 10 Introduction to ^-Series
16. Prove the following q -analog of Dougall's formula for a 2-balanced, very well
poised 7^6 :
/ a,q*Ja,—q*Ja,b,c,d,e,q~N
V л/а, —л/а, aq/b, aq/c, aq/d, aq/e, aqN+1'
(aq; q)N(aq/cd;q)N(aq/bd;q)N(aq/bc; q)N
(aq/b;q)N(aq/c;q)N(aq/d; q)N(aq/bcd;q)N '
when bcde = a2qN+l and TV is a positive integer. One proof goes as follows:
(a) Write / instead of q~N and express the formula as
/ a,q*Ja,—q*Ja,b,c,d,e,f
\-Ja, -л/а, aq/b, aq/c, aq/d, aq/e, aq/f'
(aq)oa(aq /cd)oo(aq /bd)oa(aq /bc)o
(aqlb)oo{aqlc)oc,(aq/d)OCl(aqlf)oo(aqlbcd)oo
when a2q = bcdef.
(b) Suppose the formula is true for / = 1, q~l, q~2,..., q~N+1 and take
/ = q~N. By symmetry the result is true if с or d = a2q/bcef is equal
tol,*-1,...,*-^1.
(c) Observe that if the original formula is multiplied by (aq/c; q)iy and
(aq/bcd; q)iy, then the formula gives the identity of two polynomials
in с of degree 2N. From (b) the two sides are equal for 2N values of с
Now set с = aqN and verify the equality in this case. (Jackson)
17. Use the formula in Exercise 16 and that method of proof to show that
. -,,„-., -ly~,-,d,e,f,g,h,j
*' л/а, —л/а, aq/c, aq/d, aq/e, aq/f,aq/g,aq/h,aq/j'
k, q*/k, -q'/k, kc/a, kd/a, ke/a, f, g, h, j
лД, -лД, aq/c, aq/d, aq/e, kq/f, kq/g, kq/h, kq/j 4'4
when к = a2q/cde and cdefghj = a3q2, and /, g, h, or j is of the form q~N
where TV is a nonnegative integer. (Bailey)

Exercises 547
18. Derive the following formula from Bailey's formula given in the previous
exercise:
/ a,q^/a,q^/a,c,d,e,f,g 2 2
\<Ja,-*/a,aq/c,aq/d,aq/e,aq/f,aq/gy ' Ч
(aq)oa(aq /fg)oo(aq / ge)oo(aq /ef)oo
{aq I e)oo(aq I f)oo(aq I g)oa(aq I efg)o
aq/cd,e,f,g
•403 , . , ,,;?-?,
\efg/a,aq/c,aq/d )
when e, f, or g is of the form q~N. This is a ^-analog of Whipple's transfor-
transformation.
Deduce Sears's transformation formula in Theorem 10.11.2. (Watson)
19. Let c, d, e, f, and g = q~N tend to oo in Watson's formula given in Exercise
18 to get
у (-1) WE")/2O - aq2n){aq'' q)n~l
Simplify when a = 1 and a = q.
20. Derive the following identities from Jackson's formula in Exercise 16:
(a) 6051 r~ /-' ,, ; q,aq/bcd)
_ (aq; q)oa(aq/bc; qHO(aq/bd; q)oo(aq/cd; q)^
(aq/b; q)oo(aq/c\ q)oo(aq/d; q)oa(aq/bcd; q)^
provided \aq/bcd\ < 1.
(b) The g-Dixon formula
/ a,-qy/a,b,c _ \
403 r- ,, . ;q,qVa be)
\-<Ja,aq/b,aq/c * )
__ (aq;q)oa(aq/bc;q)
(aq/b;q)oo(aq/c;q
provided \q*Ja/bc\ < 1.
21. Prove the following identities and also deduce (b) from (a):
. ч . fq-",b,c \ (de/bc;q)n (bc\n fq-",d/b,d/c
(a) 302 , ; q, q I = —:—-— -r 302 , , ., ;
\ d,e ) (e;q)n \d) \ d,de/bc

548 10 Introduction to ^-Series
(Ь) 2<i
/\ л(ч~П>а>Ь de(in\ (Ф^дЬ, fg'na,d/b .
(c) 302 , , q, —j— = —,—-—3021 , , , ; g, g •
V d, e ab ) (e\q)n \d,aql-n/e >
22. Prove Watson's g-Barnes integral formula
fa,b
20i \q,z
J-ioo (aqs;q)oo(bqs\q)ocismns
\z\ < 1, |arg(-z)| < 7Г.
23. Let Re с > 0, Re d > 0, and Re(x + y) > 1. Prove the following analog of
Cauchy's beta integral:
J_ f°° (-csq^qUidsq^q)^^
2xi J_ioo {-cs; q)oo(ds; q)^
Fq(x + y-l) (-cqx/d;q)oo(-dqy/c;q)oo
rq(x)rq(y) i
when 0 < q < 1. (Wilson)
24. Verify the following analogs of Barnes's first and second lemmas.
(a) _I_ /"'°° (з'~С+*: g)oo(g'"^; <7)oq nq'ds
2ni Jioo (qa+s; q)oo(qb+s; q)oo sin^(c - s) smn(d - s)
qc (q\q)o<.
d)" (qa+c;q)oo(qa+d;q)oc(qb+c;q)oc(qb+d;q)oo '
(Watson)
_L Г(qi+s' ^°°tid+s< q)™^' 1H0 *qsds
2ni J_ioo (qa+°; q)x(qb+s; q)oo(qc+s\ q)<*> sinns sinn(d + s)
(g; gU{qd\ gHO(g'"'; gHO(ge~a; gHO(g'-''; g)oo(ge"c; g)<x,
f', q)ooiqb', q)oo(q1', q)co(q]+a d;q)oo(q1+b d;q)<x(q1+c~d;q)<x
when d + e=\+a+b + c, 0 < q < 1. (Agarwal)
25. Prove the theta relations A0.7.5), A0.7.6), and A0.7.7).
26. Prove the product formulas for the Jacobi elliptic functions in A0.7.13).
27. Prove that when и = 2Кх/п,
ж 2л x-^ qn cos 2nx
(a) dn и = 1—
n=\

(b)
28. Show that for к
Also prove that
and
Note that
(-
I
en и ¦¦
Exercises
2n -^-\ qn+ll2 cosB« + l)x
К ^ 1 +q2n~1
ж 2тг ^ q2n+l sinBn + l)x
«=o
/0,±l,
sn(M,
du
—sn(M, к) = сп(м, А:)ёп(м, к),
du
sn2(u,k)+en2 (u,k) = 1
2
/
29. Use the ideas of Section 10.8 to prove that
poo
h
_c_! (-ax; q)oo(-qb/x; q)oo d ,
(-x;q)oo(-q/x;q)oo
(ab; q)oo(qc; q)oo(q^c; q)ooTC
549
(bqc\ q)oo(aq~c; q)oo(q; q)
30. Prove Theorem 10.9.2, that
for max (|a,-|, |o|) < 1,
i<;<,5
h(x, l)h(x, y/q)h(x, -l)h(x, -Jq)h(x, A) dx
J-i
Х\\Цх,щ) VI -x2
where A = i
31. Prove the formulas (where 0 < q < 1 and Re a > 0 and Re b > 0):
-xqb;qHO(—qa+l/x;qHOdqx _ Yq{a)Yq(b)
о (-x;q)oc(-x/q;q)oc x Vq(a+b)'
-xqb; q)oo(—qa+l/x; q)oc dx logq Tq(a)Tq(b)
(-x;q)oo(-q/x;q)x x \-q Tq(a+b)'
(a)
(b) /
Jo

550 10 Introduction to g-Series
(c) Extend the formula in (a) to
700 ,
dqX
о (-x;q)oo(-qlx;q)oc
_ (-qc; q)oo(-ql~c;q)oo Tq(a + c)Tq(b -.
(-i;q)<x>(-q;q)oo rq(a+b)
where Re(a + c) > 0 and Re(b — c) > 0 and 0 < q < 1.
32. Prove that when max(|ai |, \a2\, \щ\, \a4\) < 1,
33. Use Theorem 10.12.1 to prove the following results:
(a) If 0 < s < min(a, fc), then
Jo
s_i P, , л, Г(с) r{s)T(a-s)T(b-s)
Г(а)Гф) T(c-s)
(b) IfO< j < l.then
00 7Г
Sini'TT
34. Show the formal equivalence of the formulas
f°° -i 2 л-
(a) / x @@) — хф(\) + x d>(z) — ¦ ¦ -}dx = —
Jo
/•oo v v.2
(b) / xs-
Jo
35. Suppose s > 0, and
«=o
converges for all x. Show the equivalence of Newton's difference formula
s
X(-s) = X@)
Ak@) +
where AX(n) = X(n) — X(n + 1), to formula (b) in the previous problem.
36. Suppose a > 0, b > 0, and
iy(xey.
T^o nl
Use formula (b) of Exercise 34 to prove that X(n) = a(a + nb)"~l.
The results in Exercises 34-36 are due to Ramanujan. See Hardy [1940, Chap-
Chapter 11].

Exercises 551
37. Use Exercise 5 and the q -binomial to prove that
^ 2, q2)n t,
38. (a) Use Heine's transformation in Theorem 10.9.1 to show that
fb2,b2/c:q2,cq/b2'
201
(b) Take b = q " and let с -> оо to obtain
« /„2. „2\ „m
^0(g2;g2Ug2;g2)«-™ v ^'"
The last identity was obtained by Gauss [1808, §9] by a different method.
39. From Exercise 38(b) and the ^-binomial theorem derive the following identi-
identities:
2 2
= {-aq \q )oo
40. Show that the recurrence relation A0.11.3) follows from the generating func-
function A0.11.2), and conversely.
41. (a) Prove the linearization formula A0.11.17) by first observing that for
x=cos#,
n^hmCn fW
Hk(x\q)Hm(x\q)Hn(x\q)\(e2ie;q)\2de
/
27Г
(q; q)oo(ab; q)oo(ac; q)oo(bc; q)c
2n ^ ar+sbr+'cs+t
(g; g)oo rf^0 (g; g)r(g; g)s(g; g)»
(b) Complete the evaluation of the integral in A0.11.19).
42. Prove the formula A0.11.18).
43. Take x = e — nim/ln in Theorem 10.12.6 and let e -> 0 to show that
1 2n — 1 ^ . . m —1
1 nik2m/2n _ l ~T ' V^ ~2nik2n/m

552 10 Introduction to ^-Series
Deduce that for odd q,
1=0 '
44. Consider a more general divided difference operator
~y\(x)
If D takes polynomials of degree n to polynomials of degree n—l, show that
3>i (x) and уг(х) satisfy a quadratic equation
Ay1 + Bxy + Cx2 + Dy + Ex + F = 0,
and show that two solutions of this equation can be used to define D so that
it takes polynomials of degree n to polynomials of degree n — l. Specific
choices of уг(х) and y\ (x) and limits of them give the operators used in this
book, and the operators used in this book are essentially the standard forms
to which y\{x) and уг(х) can be reduced. See Magnus [1988].
45. Show that
46. If
/
Jo
and / is smooth then
00
/ (cos в) = У^ at cos кв.
к=\
If /(cos в) is a polynomial in cos в or the coefficients ak decrease sufficiently
rapidly, then
Define
Iqgix) = f(x).
Show that
1 Г
Iqg(x)=— g(cos<p)K(e +<p) sin <pd<p,
Л J-n
where
See Brown and Ismail [1995].

11
Partitions
The theory of partitions is a subject that, on the one hand, fits naturally into the
subject of ^-series, and on the other, is highly combinatorial in its methods. This
provides for a variety of treatments of this subject. P. A. MacMahon, one of the
pioneers in the study of partitions, titled his seminal two-volume work, Combi-
natory Analysis. It is clear that he saw a major role for analysis in the study of
partitions. We shall follow his lead and examine partitions by means of the an-
analytical technique he developed: partition analysis. This method is used to find
the generating functions of various kinds of interesting partition functions. Exam-
Examples of a few other ways of developing the theory of partitions will be given in
passing.
11.1 Background on Partitions
The theory of partitions concerns representing integers as sums of positive integers.
Thus there are five partitions of4, namely 4, 3 + 1,2+2, 2+1 + 1, and 1 + 1 + 1 + 1.
Note that the order of the summands (or parts) is not considered: 1 + 2 + 1 is the
same partition of 4 as 2 + 1 + 1.
One object of study is p(n), the number of partitions of n. Other examples
of interesting partition functions are pm{n), the numbers of partitions of n into
< m parts and p$\n), the number of partitions of n into distinct parts. Thus
pD) = 5, ргD) = 3, and p^D) = 2. An explanation for the last notation is
given below.
The theory of partitions dates back to Euler. The generating function of a given
partition function has turned out to be one of the most fundamental objects in the
study of partitions. Euler's basic observation lay in the introduction of the geometric
series in treating generating functions. Suppose A is some set of positive integers.
A partition of n into elements of A is a representation of n as a sum of elements of A
(where the order of summands can be disregarded.) Thus n = a\ + п2 + • • • + ar,
553

554 11 Partitions
a, e A, and to make the representation unique we may require ai > a^ > ¦ ¦ ¦ > ar.
Let Pa (n) denote the number of partitions ofn into elements of A. The generating
function for pA (и) is given by
00
?) PA(n)q" = JJA +qa+q2a+q3a + -- •)• A1-1.1)
n=0 aeA
The equality in A1.1.1) becomes clear once we multiply the terms together and
see that the general exponent on q that arises is
/lfll + /2«2 H h fjuj Л .
This last expression is an arbitrary partition into elements of A, where a\ appears
/i times, аг appears fi times, and so on. Consequently,
n=0 aeA
If we were to require that each part appear < .? times and to define p^ (n) to be
the number of these partitions of n, then we would see as before
n=0 aeA
aeA
Note that when s = I, p^ (n) is the number of partitions of n into distinct elements
of A and
n=0 aeA aeA
These observations on generating functions allow us to prove one of Euler's
striking, albeit elementary, theorems.
Theorem 11.1.1 The number of partitions ofn into elements of в (the set of odd
numbers) equals the number of partitions ofn into distinct parts (i.e., the parts
taken from N, the set of all positive integers.) More succinctly,

11.2 Partition Analysis 555
Proof. This is now easily seen because
1
л=1 Ч i=0
Here A1.1.2) was used in the first step and A1.1.4) in the last step. The theorem
is proved. ¦
At this point the reader may try to prove the following result by the method of
Theorem 11.1.1:
The number of partitions ofn into summands not divisible by 3 is equal to the
number of partitions ofn where no summand occurs more than twice.
One of the difficulties encountered in studying the theory of partitions lies in
the fact that each new result seems to require some new trick. Although this fact
may seem charming to insiders it is somewhat discouraging for noncombinatorial
outsiders. In this chapter we hope to present a systematic derivation of a variety of
elementary results. Our focus will be MacMahon's partition analysis for obtaining
generating functions for a number of interesting partition functions.
11.2 Partition Analysis
To illustrate this method, we start with the following problem: What is a closed
form for the generating function YlT=o Pm(n)qnr> Recall that pm(n) is the number
of partitions ofn into < m parts.
It is not difficult to see that we can write this generating function as a multidi-
multidimensional sum. Thus
1=0 П1>П2>--->Пт>0
The requirement ni > П2 > ¦ ¦ ¦ > nm > 0 comes from the fact that order is
disregarded in a partition. Thus each partition can be written uniquely as a sum
of a nonincreasing sequence of numbers. MacMahon's idea is to introduce new
variables Х\,Хг, ¦¦¦, A.m-i that handle the inequalities satisfied by n, while the nj
themselves become free. Consider the sum
E-B1+B2H Hm 1  ~ т. 12-13 I, 1m-l -1m
q kx A2 •¦•Am_i
11,12,...,1m>0

556 11 Partitions
If we select only terms in this sum with nonnegative exponents on the X, then the
corresponding exponent will be a partition of n into < m parts. For example, when
m = 2 and n = 4 the exponents that result are 4 + 0,3 + 1, and 2 + 2. The method
of partition analysis applies a linear operator ?2> to such multiple Laurent series
in Л],. •., A.m-i ¦ The operator annihilates terms with any negative exponents and
in the remaining terms sets each X,¦ = 1. Hence
oo
Рт\П)Я — " /_^ У 1 Am-1
n=0 ~~ ni,...,nm>0
~ «i>0 12>O 1m>0
= (l лоA
The next step is to produce an algorithm to evaluate the effect of ?2>. To this end
we state and prove the next lemma.
Lemma 11.2.1
Proof. The left side equals
x"ym-
~ n,m>0 n>m>0
Set к = n — m so that the last sum becomes
*,m>0 *>0 m>0 U Д ^^
and the lemma is proved. ¦
Repeated application of Lemma 11.2.1 gives the closed form for the generating
function of pm(n).
Theorem 11.2.2
00
^2pm(n)qn = 1/A - <?)A - Ч2) ¦ ¦ ¦ A - <Л-
n=0
Proof. One application of Lemma 11.2.1 and A1.2.1) gives
„=0

11.3 A Library for the Partition Analysis Algorithm 557
A second application gives
1
It is now clear that the result of the theorem follows after applying the lemma to
each of Ai, X2,..., Am_i. The theorem is proved. ¦
Before developing partition analysis further, we consider a simple example to
further illustrate the power of this method. We need an extension of Lemma 11.2.1,
which will be useful for other purposes as well.
Lemma 11.2.3 If a is a nonnegative integer,
X-" x"
?2-
The proof of Lemma 11.2.2 will work here as well. The reader may check
this. The example is the following: Let A(n) denote the number of noncongruent
triangles of perimeter n with positive integer sides. What is X^o Д(и)<7"?
Suppose n\,n2,n3 are the sides of the triangle in decreasing order. We must have
n2+щ > n i +1. MacMahon's partition analysis gives us the answer automatically:
00
П=0 П[уП2,Пз>0
~ 4
4 A - qki Дз)A - ЧХ2Хг/Хх){\ - q
k^
~ A /X)(l - q2X2)(l - qX3/X2)
Аз
3
Therefore, A(n) = the number of partitions of л into twos, threes, and fours with
at least 1 three.
11.3 A Library for the Partition Analysis Algorithm
The examples of the preceding section illuminate the technique of partition analy-
analysis. From a variety of simple and simply proved evaluations of the operator ?2>
(such as Lemma 11.2.3), it is possible to apply the operator to numerous rational
functions in several A,. We list a few more results similar to Lemma 11.2.3. These
and others were originally given by MacMahon.

558 11 Partitions
Proposition 11.3.1
1
(a)
> A-Лх)A-л A)(i-j2A) ¦¦•(!-
1
(b) Q7r l l~xyz
- ky)(l - z/k) A - x)(l - j)(l - xz)(l - yz)
1 1 + xyz — x2yz — xy2z
Proof. Each such result (and countless others) can be proved in several ways.
The method of partial fractions will reduce most to applications of Lemma 11.2.3.
The proofs of (a) and (b) are given below and (c) is left to the reader.
Note that for j = 1, the result in (a) is just Lemma 11.2.1. Using induction,
suppose the result is true up to j — 1. Observe that
1 1 ( У 1-х yj
A - yj.i/Л)A - yj/k) yj-i -yj\l- yj-i/k 1 - yj/k
Then the expression in (a) can be written as
1 J
j-i ~ yj* Ld -^)d - y\A) •
yj
A - ЛХ)A - J, A) • • • A - yj-
1 -x)(l ~xyi)---(l -
- xyx) ••
1
A - x)(l - хУ1) ¦ ¦ ¦ A - xyj)
The proof of (b) is very similar. We have
x
=?A-xA)A-jA)A-zA) х-у1>\\-хХ l-ykj l-z/k
x у
(x - j)(l - x)(l - xz) (x - j)(l - j)(l - jz)
1 — xyz
This proves the proposition.

11.4 Generating Functions 559
11.4 Generating Functions
In this section we apply partition analysis to find the generating functions of some
important partition functions.
Theorem 11.4.1 Let Qm (n) denote the number of partitions ofn into exactly m
distinct parts. Then
00 qm{m+\)/2
Proof. If n = n\ + n2 + ¦ ¦ ¦ + nm, then we require n\ > n2 + 1, n% > n3 + 1,
¦ ¦ ¦. nm > 1, because the parts are distinct and exactly m in number. So
00
(JmWq — S2 2^ ^ Л1 Л2 •¦•Лш-1 Лш
п=0 ~ п\,...,пт>0
• • • A - X
Apply Lemma 11.2.3 to Ль А.2,..., Я„ with a = 1 and obtain
q ¦
q-q2---qm
which proves the theorem. ¦
In exactly the same way, we may consider Q^'e) (n), the number of partitions of
n into m parts where each part differs from the next by at least к and the smallest
part is >?. The closed form for the generating function is given in the next theorem.
Theorem 11.4.2
alm+km(m-l)/2
Proof Reasoning as before, we can see that
\-k\-k
Л1 Л2
и>0 ni,.-.,nm>0

560 11 Partitions
The only change is that in the first m — 1 applications of ?2>, a —k, and in the final
(trivial) application, a — t, у = 0 in Lemma 11.2.3. This proves the theorem. ¦
We can also introduce further variables that keep track of other facts about the
partitions. For example, we have the following theorem.
Theorem 11.4.3 Suppose pm(j, n) (respectively Qm(j, n)) denotes the number
of partitions ofn into <m parts (respectively exactly m distinct parts) with largest
part j. Then
Y^ A»0\ «)ZV = 1/d - Zq)d - zq1) • ¦ • A - zqm)
j,n>0
and
J2 Q*V- »)zV = Zmqm(m+i)/2/d - zq)(l - zq2) ¦ ¦ ¦ A - zqm).
j,n>0
Proof. From the definition of pm(j, n) it is clear that
Pm\J,n)Z q — Ы 2_^ Z Я Al A2 A
j,n>0 ~ ii),...,nm>0
Пт-,-Пт
m-1
1
- zq)(l - Zq2X2)(l-
1
1
For the other generating function, the argument is similar. Thus,
j,n>0
= П
~ ni,...,nm>0
К Л2
A2 ' ' 'Am-1
zm m(m+l)/2
The theorem is proved. ¦
In what follows we use the notation [zJ] YlT=oanZn — aj, that is, the operator
[V] applied to a power series gives the coefficient of z' ¦ An observation needed

11.4 Generating Functions 561
in the proof of the next theorem is that
N N oo
Л=0 Л=0 л=0
= [zN][(\ + z + z2 + ¦ ¦ -)(ao + an + a2z2 + •••)]
[N]^° " A1.4.1)
L M ',
[z] ¦
1 -z
Theorem 11.4.4 Suppose p(N, M, n) (respectively Q(N, M, n)) denotes the
number of partitions ofn into <M parts (respectively exactly M parts), each <N.
Then
and
л=0
Here \nm ]q is the q-binomial coefficient defined by A0.0.5)
Proof. By Theorem 11.4.3 and the observation A1.4.1),
oo N
^T p(N, M, n)qn = ^[z*] Y
n=0 h=Q j,n>0
N
Note that Corollary 10.2.2(d) was used to get the second-to-last equation. For the
second part, we have
oo N
У^ Q(N, M, n)qn = y\zh] 2_. QmU> n)zJ'<?"
n=0 h-0 j,n>0
= \7N~[7MaM{M+Y)l2 Y^ I ^ ~*~ ^\ zk = OM<M+1)/2 I ^
f^0 [ м \q [m
This proves the theorem. ¦

562 11 Partitions
Corollary 11.4.5
Q(N> M> n)zMqn = A + Zq) ¦ ¦ • A + ZqN).
n,M>0
Proof. By Theorem 11.4.4 and Corollary 10.2.2(c)
п,М>0 М>0 \ л>0
(-D/2 |
М
and the corollary is proved. ¦
The next theorem gives limiting cases and other consequences of some previous
results.
Theorem 11.4.6 Let p{n) denote the total number of partitions of n and let
p(m, n) denote the number of partitions ofn into exactly m parts. Then
00
(a) Y
n=0
n,m>0
(c) Y,
m,n>0
(e)
Proof.
00
?
n=0
n,m>0
n,m>0
By Theorem 11.2.2,
00
p(n)qn = lim Y^Pm(n)t
m—юо ^—^
n=0
m=0
00
m=0
j" = lim 1/[A —
m—>oo
= \/{q\ q)oo-
-<Л]

11.5 Some Results on Partitions 563
To prove (b), first observe that p(m,ri) = pm(n) — pm_i(n). Then
? P(m, n)zmq" = J2 z>n Hfo»(«) - Prn-x (n))qn
n,m>0 m>0 n>0
1 1
= у z
(q;q)m-i
z>nim
where the last step follows from Corollary 10.10.4. Formula (c) is obtained from
Corollary 11.4.5 by letting ./V —> oo. To derive (d), observe that, by Theorem
11.4.2,
? Q%l)(n)zmq" = E^ E Q^
n,n>0 m>0 n>0
The final formula is obtained in a similar way. The theorem is proved. ¦
The series on the right side of (d) and (e) occur in the Rogers-Ramanujan
formulas, which will be stated and proved in the next chapter.
11.5 Some Results on Partitions
In Section 11.1, we showed that the number of partitions of n into odd parts
equals the number of partitions into distinct parts by showing that their generating
functions are equal. This is a very powerful way of obtaining results on partitions.
In this section, we give some applications of the theorems derived in the previous
section. For example, a simple consequence of Theorem 11.4.4 is the next result.
Theorem 11.5.1 The number of partitions ofn into <M parts each <N equals
the number of partitions ofn into <N parts each <M, that is,
p{N,M,n) =p(M,N,n).
Proof. By Theorem 11.4.4, the generating function for p(N, M, n) is
~N + M~
M
which is clearly the generating function for p(M, N, n) as well. This proves the
theorem. ¦

564 11 Partitions
An immediate consequence of this theorem is the following:
Corollary 11.5.2 The number of partitions ofn into <m parts equals the number
of partitions ofn in which each part <m.
A direct proof of this corollary can also be given by showing that the generating
function of b(m, n), the number of partitions where the parts are <m, is also
i/(q\q)m- Let Ik denote the number of times к occurs in some partition of n. Then
л=о и em>o
In,
1
(l-q)a-q2)---(l-qm)'
Here Ik is the number of times к occurs in some partition ofn.
The results of the previous section can also be used to give a different deriva-
derivation of some identities obtained in Chapter 10 from the ^-binomial theorem. For
example, it is intuitively clear that
A + zq + Z2qi+1 + ¦ ¦ 0A + zq2 + Z2q2+2 + ¦ ¦ -)A + zq3 + Z2q3+3 + ¦••)¦•¦
p(m,n)zmq",
m,«>0
where p(m, n) is the number of partitions ofn with exactly m parts. However, by
partition analysis (Theorem 11.2.2)
(Pm(n) ~ pm-l(n))q"
n>0 «>0
1 1 qm
(q;q)m (q\q)m-i (q;q)n
Thus
1 _ ^ g^z*
(zq;q)oo ^ (q\q)m
or
V z l
^ (q\q)oo (z;q)x'
a result of Euler (Corollary 10.2.2).

11.6 Graphical Methods 565
In a similar way,
(~zq; <7)oc = A + zq)d + zq2)(l + zq3) ¦ ¦ ¦
„m(m+l)/2 m
m,n>0 m=0 ^>^m
where the last equality follows from partition analysis (Theorem 11.4.1). This
implies the other result of Euler in Corollary 10.2.2, namely
We end this section with the statement of the Rogers-Ramanujan identities
and their partition theoretic interpretation. A more complete discussion of these
identities and their proofs is given in the next chapter. The identities are
m=oKq,q)m n=0
and
>. A1.5.2)
m=0 4"J' ~2"" n=0
The right-hand side of A1.5.1) is obviously the generating function for the number
of partitions in which the parts are identical to 1 or 4 mod 5. The left-hand side,
by Theorem 11.4.6(d), is the number of partitions where the parts differ by at least
2. A similar interpretation holds for A1.5.2).
Theorem 11.5.3
(a) The number of partitions ofn in which the difference between any two parts
is at least 2 equals the number of partitions ofn into parts = 1 or 4 (mod 5).
(b) The number of partitions ofn in which the least part > 2 and the difference
between any two parts is at least 2 equals the number of partitions ofn into
parts = 2 or 3 (mod 5).
This interpretation of A1.5.1) and A1.5.2) is due to MacMahon [1917-1918].
11.6 Graphical Methods
A powerful way of studying partitions is by representing them graphically. This
method was discovered by N. M. Ferrers in the 1850s. It has since been used exten-
extensively in partition theory, unlike MacMahon's partition analysis, which has begun
to gain some prominence only recently. This may partly be due to MacMahon's
admission that partition analysis failed to give significant results in plane partitions

566 11 Partitions
Figure 11.1
for which he had initially developed it. As a contrast, Sylvester, who published Fer-
rers's graphical method, gave this method much positive publicity. In a short paper
titled, "Note on the graphical method in partitions," published in 1883, Sylvester
wrote,
The discovery of this process is due to Dr. Ferrers, who informs me that he himself never
published it but left it to me to do so in his name in the London and Edinburgh Philosophical
Magazine for 1853.1 may mention that I have never missed an opportunity of expressing
my sense of the great importance of the discovery and bringing it under the notice of my
pupils
Ferrers's graphical representation of a partition is a collection of lattice points
where each row of points (or dots) corresponds to a part of the partition. For
example, the graphical representation of 5 + 2 + 2+1 is shown in Figure 11.1.
The conjugate of this partition is another partition obtained by reading the
columns of the above partition. In this case the conjugate is 4 + 3 + 1 + 1 + 1.
According to Sylvester, the result that Ferrers proved by this method is the
following: The number of partitions of n into exactly m parts equals the number
of partitions of и with maximum part m. This is intuitively obvious from Ferrers's
graph. Each partition with exactly m parts has a conjugate with largest part m and
conversely. In a footnote Sylvester [1853] makes the following interesting remark:
"I learn from Mr. Ferrers that this theorem was brought under his cognizance
through a Cambridge examination paper set by Mr. Adams of Neptune notability."
The reader should check that Theorem 11.5.1 and its corollary also follow
immediately from an application of Ferrers's method. In this section, we consider
a few applications of this method to see some of its scope and power.
Consider the result contained in Theorem 11.4.1:
where Qm(n) is the number of partitions of n into exactly m distinct parts. We
have seen that the generating function for the number of partitions with <m parts is
I/O?; <?)m- To understand the factor m(m +1)/2, take the partition 7 + 6+4 + 2+1
where m = 5. Graphically this is depicted in Figure 11.2. There are |E)F) dots

11.6 Graphical Methods
567
Figure 11.2
Figure 11.3
inside the triangle. The remaining nodes form the partition of some number with
<5 parts.
Now consider Euler's identity contained in Corollary 10.10.4:
-qJ(l-q2J-"(l-qnJ
n=\
For each partition n we find the largest square of points (starting at the upper
left-hand corner) contained in the Ferrers graph. Such a square is called a Durfee
square, named after a student of Sylvester who used this idea. Suppose n is given
by 6+4+4+2+1 + 1; then its Ferrers graph and the 3 x 3 Durfee square are shown
in Figure 11.3. In general each partition n of m has a Durfee square of side и, for
some n, and we can write n = n2 + 7t\ + тхг, where п\ is the partition made by
the points below the square and m is the conjugate partition of the points to the
right of the square. Since the partitions with parts <n are generated by l/(q; q)n,
it follows that the set of all partitions with Durfee square of side n is generated by
1
1
4 (q;q)n (q-q)n A - qJ(l - q2J ¦ ¦ • A - q»J "
Now the generating function for p(n) is \/{q\ q)oo, and thus
E
n=0
1
(q;q)n(q\q)n
n=\

568
Figure 11.4
As an exercise the reader should prove the following identity:
00 7ПП
У Li =
f^0 (q; q)n(zq; q)n
using the Durfee square.
As a final example, we take another look at the triple product identity,
1
(q;q)c
к" = (-xq;q)oo(-x l;q)oo.
We have seen that the real difficulty in some proofs is to show that the term
independent of x on the left-hand side is in fact I/O?; <?)oo-
One way to show this fairly easily is to use the Frobenius symbol for a partition.
To describe this idea, consider the partition 6 + 4 + 3 + 3+1 whose Ferrers graph
consists of the points in the graph shown in figure 11.4.
Associate the points as indicated in the Figure and read off the Frobenius symbol
for n:
'5 2 04
2
The top row represents the horizontal lines to the right of the diagonal and the
bottom row the vertical lines below the diagonal. Clearly the sum of the numbers
in the Frobenius symbol plus the number of columns gives the number being par-
partitioned. More generally, every partition n of n can be represented by a Frobenius
symbol
a\ a2... ar
b2...br
where a\ > a2 > ¦ ¦ ¦ > ar > 0, b\ > ¦ ¦
Now let us prove that the constant term in
> br > 0, and n = r +
xqn) f[A + дт~1/х)
Y,bt.
n=\
m=\

11.7 Congruence Properties of Partitions 569
is l/(q;q)oo, which will imply the result we want. Observe that a contribution to
the constant term is obtained whenever r terms of the xqn are multiplied with r
terms of the form qm~1/x, that is,
_oi+O2+-+ar _ „Ь1+Ь2+-+Ьг
where the as are positive and distinct and the bs, are nonnegative and distinct. This
is a partition represented by the Frobenius symbol
i — 1 u2 — 1 ... ar —
b\ b2 ... br
Hence the constant term is the generating function for p(n), which is l/(o; q)oo-
The result is proved.
11.7 Congruence Properties of Partitions
Congraence properties of p(n), the number of partitions of и, were first discovered
by Ramanujan on studying the table of values of p(n) constructed by MacMahon
from n = 1 to 200. Ramanujan gave simple proofs of the theorems
pEn+4) =0(mod5), A1.7.1)
p(ln + 5) = 0(mod7). A1.7.2)
He also found expressions for the generating functions of pEn +4) and p(ln + 5)
as products (or a sum of two products). These are given by the formulas below:
pD) + p(9)q + pA4)q2 + • ¦ ¦ = 5
- q)(\ - дг)(\ - q^) ¦ ¦-f '
A1.7.3)
/?E) + pA2)q + p(\9)q2 + • • • = 7'
+ 49q-
{A - q)(l - q2)(l ~ q3) ¦ ¦ ¦}* '
A1.7.4)
Ramanujan [1927, paper 25] sketched a proof of A1.7.3) promising to give details
in a later paper. A year later he died and the promised paper never appeared. How-
However, he gave the necessary details in an unpublished manuscript. See Ramanujan
[1988, p. 238]. We reproduce this proof here. It has also appeared in an interesting
unpublished book manuscript by Thiravenkatachar and Venkatachaliengar.
Observe that the congruences A1.7.1) and A1.7.2) follow immediately from
A1.7.3) and A1.7.4) respectively. These generating functions can also be used to

570 11 Partitions
prove congruences modulo 52 and 72. First observe that for any prime p,
A -q)p = \-qp(mod p)
or
^sHmodp). A1.7.5)
Thus A1.7.3) and A1.7.5) with p = 5 imply that
pD)q + p(9)q2 + • ¦ ¦ q A - q5)(l - q10) ¦ ¦ ¦
_^10)...}4 A
(mod 5). A1.7.6)
(\-q)(\-q2)---
Since
the coefficient of q5m is divisible by 5. The coefficient of q5m on the left of A1.7.6)
is pB5m — 1) and hence
pB5m -l) = 0(mod25). A1.7.7)
Similarly A1.7.4) implies
pD9m - 2) = 0(mod49). A1.7.8)
To prove A1.7.3), start with Euler's pentagonal number theorem (Corollary
10.4.2(c)):
Partition the series into five parts according to n sO,±l, ±2(mod 5). For example,
the subseries of terms with n = 5m — 1 is
00
/_1уПдEт-1)A5т-2)/10 _ _ 1/5 V^ t-\\m„5-mCm-l)/2
=—oo m=—oo
00

11.7 Congruence Properties of Partitions 571
Thus
00
Па-*
n=\
«/5)
00
m=—oo
_\yn mEm+l)/2 _|_
00
^ (—I) q
m=-oo
00
E
m=—oo
i-2)E»j+
/
D/2
^m Cm-l)Em-2)/2
00
_ V^ (_l)m^m(l5m+7)/2
m~—oo
Divide by the infinite product Щ1] A - q5n) to get
л=1
where f and |i are power series in q. Our claim is that %%\ = 1. We recall that by
Corollary 10.4.2(e) and A1.7.9),
Since the power of q, given by n(n +1), is either 0, 2, or 6 (mod 10), it follows that
no power of q is of the form 2/5 + an integer. So the term 3^2/5^ — 3fj2f q2/5 =
3<?2/5fi(l - f?i) on the right side of A1.7.10) must be zero. This implies that
§i =§~1. Therefore,
i
=: 1
The denominator on the right-hand side is
Consider the expression AT1 — A. — 1, where к — %qllbco, and со is a fifth root
of unity. Observe that if A. — A. = 1; then a simple calculation shows that
A.-5+A.5 = 11. Thus
Г5 - И? -*V = П^"' -?1/V -^2"<o2k), (П.7.12)
where со = е2я'/5. It is now easy to check by long division that A1.7.11) can be

572 11 Partitions
written as
1 - q5"
Пт?
A1.7.13)
(In fact, Ramanujan [1927, paper 25] starts the sketch of his proof of A1.7.3) by
observing that A1.7.13) can be shown to be true.) Now multiply across by ql/5
and replace g1/5 withql/5e2jnk/5, where A: = 1,2, 3,4. Add the five identities after
using the fact that
n=\ n=\
to obtain
= —; T-i- (П.7.14)
t — 1 \Q — Q Ё
Replace q with qe2ltik, where к = ±1, ±2 in A1.7.11) and multiply the five
equations. Note that
By A1.7.12), we now get
Combine this with A1.7.14) to get
PEn + 4),- = 5{
About this identity, Hardy wrote that if he were to select one formula from Ra-
Ramanujan 's work for supreme beauty, he would agree with MacMahon in selecting
this one.
Remark 11.7.1 Ramanujan remarked that
й=оA-^+1)A_^5„+4)-

Exercises 573
Note the connection of this function with the Rogers-Ramanujan identities, which
Ramanujan used to express f-1 as a continued fraction. Formula A1.7.15) can
be proved by first observing that A1.7.11) and the pentagonal number theorem,
stated immediately after A1.7.8), imply that
00
1 П/1 -<?5") = S (_i)Y»/io>C»+i)p
n=l n=0,3
(mod 5)
Now apply the quintuple product identity (Exercise 10.14) to the last sum and
rearrange the resulting product to get A1.7.15). The continued fraction for i-~l
can be found in Hardy [1940, p. 99].
Remark 11.7.2 Several proofs of A1.7.3) and A1.7.4) have been given. For a
relatively simple one, which is similar to Ramanujan's proof in some respects, see
Kolberg [1957]. For a proof that uses the machinery of modular functions, see
Knopp [1971, §8.3].
1. Verify the
(a)
(c)
(d)
(e) ft
(f) О
following
п
>
>
о
>A-А
formulas:
1
1
Exercises
-у л2)
A-А2х)A-уД)
1
A-A.xXl
1
1
1
-Ху)A-
1 — хуш
1
1 +
1
-у/А/) A-х)A
1 +хуA -у'
V/X)
1 - Z/X)
Z/X) A
1
xy
-xy2)'
-x'y)'
1 + xy + xz + xyz
1+xz
)A -Xz)(l -со/Х)
— xzco — yzco + xyzco
-xy2)(l-xz2)'
- хуг - хуг2
y)(l-yz)(l-xz2)'
+ xyzco2
- yu>){\ - zco)
2. Prove that the number of partitions of n into parts not divisible by 3 equals
the number of partitions of n where no part occurs more than twice.
3. Show that the number of partitions of n in which only odd parts may be
repeated equals the number of partitions of n in which no part appears more
than three times.

574 11 Partitions
4. Generalize Exercise 3 by showing that the number of partitions of и in which
only parts ^ 0(mod 2m) may be repeated equals the number of partitions of и
in which no part appears more than 2m+1 — 1 times.
5. Prove that the number of partitions of n in which each part appears 2,3, or 5
times equals the number of partitions of n into parts congruent to 2, 3, 6, 9,
or 10 modulo 12.
(This result is due to Subbarao; see Andrews [1976, p. 15].)
The next three exercises are from Ramanujan [1927, paper 25].
6. Prove that pEm + 4) = 0(mod 5) as follows:
(a) Show that
-. OO OO
_ _
?=—oom=—oo
(b) Show that if the exponent 1 + ?(? + l)/2 + т(Ът + 1)/2 is a multiple of
5, then the coefficient 2? + 1 is also a multiple of 5.
(c) Show that ^^ = ^(
(d) Use (c) to observe that
П°° (\
9
n=l
(e) Deduce that the coefficient of ^5m+5 in f(q) is a multiple of 5.
(f) Conclude that pEm + 4), the coefficient of q5m+5 in 9/П^=1 C1 - #").is
a multiple of 5.
7. Show that pGm + 5) = 0 (mod 7). Use the identity
n=l (=—oom=—oo
and the method employed in Exercise 6.
8. Use A1.7.4) to prove that pD9m - 2) = 0(mod 49).
9. Prove A1.7.4).
10. Show that the number of partitions of и that are self-conjugates, that is, iden-
identical with their conjugates, equals the number of partitions of n with distinct
odd parts.
11. Let Mi (n) denote the number of partitions of n into parts, each greater than
1, such that consecutive integers do not appear as parts. Let M2(n) denote the

Exercises 575
number of partitions of n in which no part appears exactly once. Show that
The result in Exercise 10 is due to Sylvester and that in Exercise 11 to
MacMahon. See Andrews [1976, p. 14].
12. Use Euler's pentagonal number theorem, namely
n~l m=oo
to show that
"^ ifn = mCm±l)/2,
•- 0 otherwise.
Here Ре\п) (respectively Р^1)(п)) is the number of partitions of n into an
even (respectively odd) number of distinct parts.
13. Prove the following relation that gives an efficient algorithm for computing
p(n):
p(n) - p(n - 1) - p(n - 2) + p(n - 5)
+ pin — 7) + ¦ • • + (— \)m(p(n — mCm — l)/2)
+ (-l)mp(n - mCm + l)/2) H =0,
where p(M) = 0 when M is negative. [Hint: (q; q)oo, Л = 1-1
14. Use the Durfee square discussed in Section 11.6 to prove mat
OO n П2
— = (xq; q)^.
n (q\ q)n{xq; q)n
15. Show that
~ nyn n2 °o X1n n 2n2 <» X2n+1 n+1 (n+l)Bn+l)
> — = > V +> — •
^ (xq;q)n(yq;q)n ^0 {xq; q)n(yq; q)^ ^ (xq; q)n(yq; qJn+\
For the left-hand side, use the idea of Exercise 14 to show that the coefficient
of ymxrqn is the number of partitions of n into m parts with the largest part
equal to r. Do the same for the right side, where instead of a Durfee square
consider the largest rectangles that can be of size n x 2n or (n + 1) x Bn +1).
Note that the largest rectangles of these dimensions cover all the possibilities.
16. Use the fact that n2 = 1 + 3 + 5 -I 1- 2n - 1 to see that ?^0 q/(q; q)n
is the generating function for the number of partitions in which the difference
between the parts is at least 2.
The next six results are the partition theoretic interpretations of the six
identities of Rogers given in Exercise 12.6. Prove them. For references see
Andrews and Askey [1977].
17. The number of partitions b\ + b2 + ¦ ¦ •+br of n where b\ > b2 > &з > • • • and
each bi is odd or = ±4 (mod 20) equals the number of partitions c\+c2-\
of и where c\ > c2 > c3 > c4 > c% > • • •.

576 11 Partitions
Hint: Note that ra2 = 0+l + l+2H \-(n-l)+n and that l/(q; qJn
is the generating function for partitions in which there are at most 2n parts.
(Gordon)
18. The number of partitions i>i + b2 + ¦ ¦ ¦ of и where b\ > bt > ?>з... and each bi
is odd or = ±8 (mod 20) equals the number of partitions c\ + c2 + ¦ ¦ ¦ + c2k+\
of n into an odd number of parts where c\ > c2 > cj, > с*, > cs > с в > cj >
Hint: n2 + 2n-l + l+2 + 2-\ <r (n - 1) + (n - I)+n +n + n.
(Connor)
19. The number of partitions b\ + b2 + ¦ ¦ ¦ + br of n where b\ > b2 > b-i > • ¦ •
and each fe,- is ^ ±1, ±8, ±9, 10 (mod 20) equals the number of partitions
c\ + c2 + ¦ ¦ ¦ + C2k of n into an even number of parts where c\ > c2 > cj, >
Q > C5 > • • ¦. (Connor)
20. The number of partitions b\ + b2 + ¦ ¦ ¦ + br of n where b\ > b2 > Ъг >
• ¦ • and each b, ^ ±3, ±4, ±7,10(mod20) equals the number of partitions
c\ + c2 + ¦ ¦ ¦ + q of n where c\ > c2 > cj, > c,\ > c$ > eg > • • •.
(Connor)
21. The number of partitions of и with distinct parts and with each even part larger
than twice the number of odd parts equals the number of partitions of n into
parts = 1 or 4 (mod 5).
Hint: The left-hand side of Exericise 12.6 (e) can be written as
„1+3+5+-+2л-1
q
22. The number of partitions of n into distinct parts each larger than 1 in which
each even part is larger than twice the number of odd parts equals the number
of partitions of n into parts = 2 or 3 (mod 5).
Hint: 3 + 5 + ¦ • • + Bи + 1) = n2 + 2n.

12
Bailey Chains
L. J. Rogers is the pioneer of the work leading to the Rogers-Ramanujan identities
and beyond. His idea, published in Rogers [1917], provides the starting point for the
work of this chapter. We shall recount his seminal idea in Section 12.1. In the 1940s,
W. N. Bailey began a systematic study of identities of the Rogers-Ramanujan type.
See Bailey [1949]. He saw great generality in the methods introduced by Rogers.
This greater level of generality provides for a wide variety of applications well
beyond those considered by Rogers.
Motivation for the techniques presented here is scant. As you will see, Rogers's
original idea seems almost magical in its construction. Since the advent of computer
algebra we can better see how to make sense of Rogers's fortuitous discoveries.
However, it is still not evident why one would initially expect that this method
would bear fruit.
A systematic account of Bailey's ideas leading up to Bailey's lemma is given
in Section 12.2. As an application of these ideas, the important 8</>7 transforma-
transformation formula of Watson is derived in the next section. A few consequences of
this formula are also included. The last section makes passing mention of other
applications of the ideas in Section 12.2.
12.1 Rogers's Second Proof of the Rogers-Ramanujan Identities
The Rogers-Ramanujan identities were first discovered by Rogers [1894]. Rogers
made considerable contributions to several areas of mathematics but surprisingly
his work went largely unnoticed and did not have the influence it should have had.
Part of the surprise comes from the fact that Rogers's early work in invariant theory
was noticed by Sylvester and given a prominent place in his "Lectures on the The-
Theory of Reciprocants" [1886]. The long neglect of invariant theory after Hilbert's
discoveries may have played a role here. Rogers did not receive credit for his dis-
discovery of the Holder inequality, and his papers containing the Rogers-Ramanujan
identities, among the most beautiful formulas in mathematics, went unheralded.
577

578 12 Bailey Chains
These identities were later rediscovered by Ramanujan. Ramanujan's first letter
to Hardy in 1913 contains some continued-fraction formulas that are consequences
of these identities. Ramanujan did not have a proof of the identities and he posed
them as a problem in 1914 in the Journal of the Indian Mathematical Society.
MacMahon stated them without proof in the second volume of his Combinatory
Analysis. He also noted the connection with partitions.
What happened next is best described in the words of Hardy [1940, p. 91]:
The mystery was solved, trebly, in 1917. In that year Ramanujan, looking through old
volumes of the Proceedings of the London Mathematical Society, came accidentally across
Rogers's paper. I can remember very well his surprise, and the admiration he expressed
for Rogers's work. A correspondence followed in the course of which Rogers was led to a
considerable simplification of the original proof. About the same time I. Schur, who was
then cut off from England by war, rediscovered the identities again. Schur published two
proofs, one of which is "combinatorial" and quite unlike any other proof known.
In this section we discuss Rogers's second proof published in Rogers [1917].
As noted before, the Rogers-Ramanujan identities are
CO „2
? _*!_ = ___L__
= —'- °° ' ч °°—- A2.1.1)
and
y> <?"(n+1) i (<?; <?5)co(<?4;
2
¦ \ ¦ ¦ )
n(q\q)n (q2;q5)oo(q3;q5)<x> (q\q)<x>
The products on the right side can be transformed by the triple product identity
(Theorem 10.4.1),
(x; q) ]?
k=-oo
Replace q with q5 and x with q2 and then with q to get
k=-oo
and
(q; q5)oo(q4; q5)oo(q5; q5)oo = ? {
?=-00

12.1 Rogers's Second Proof of the Rogers-Ramanujan Identities 579
Thus it is sufficient to prove that
and
' - (n- n\
S<*«>«
A2.1.3)
A2.1.4)
Rogers's second proof of the Rogers-Ramanujan identities depends on the fol-
following lemma.
Lemma 12.1.1 Let
1
(_1)t
k=-e
~k
k=-i
(q)e+i-k(q)e+k
Then
S2e = S2e+i — ——, where (q)k = (q; q)k.
(q)e
Proof. The proof consists in rearranging the terms in the sums. In S^, combine
the terms corresponding to the indices — к and к + 1. (Note that when к = I, the
term corresponding to I + 1 is 0.) We get
(q)t+k+iiq)e-k~\
1-
It is readily seen that this is also the expression obtained in combining the terms
corresponding to indices — к and к + 1 in the sum S^+i- Thus •%< = SWi-
We now prove that
S2(+i = (I - qt+l)S2e+2,
A2.1.5)
which together with S2i = S2i+\ will prove the lemma by induction. Consider the
sum of the terms in ^2^+1 corresponding to +k and —k. For к ф О,
1 +
= A - ql

580 12 Bailey Chains
This expression is A — qe+]) times the sum of the terms in S2e+2 corresponding to
the indices ±k, when к ф 0. When к = 0, the corresponding terms in S21+1 and
S2t+2 are
{q)t+\iq)i+\
This proves A2.1.5) and the lemma.
and
The idea of Rogers's second proof of A2.1.1) and A2.1.2) is to expand the
function (-^/qe'e; q)oo(—->/q~e~'e\ #)co as a Fourier series in two different ways.
One way is to use the triple product identity and the other is to apply the q-
binomial identity to each of the products. Since the Fourier expansion is unique,
the corresponding Fourier coefficients in the two expansions are identical. Hence
the exponentials е'пв can be replaced in the two expansions by any numbers as long
as the series are convergent. Rogers shows that the Rogers-Ramanujan identities
are obtained if this replacement is done appropriately. The details follow.
By Euler's result in Corollary 10.2.2 (b), we have
and
So the product is
7[n(n+l)+m(m+l)]/2 m+n J(n-m)l
{-xewq;qHO{-xe-ieq;q)O0= ]T 2
_eiBn-p)e
Break up the last sum into two parts for p even (p = 21) and p odd (p = 2? + 1)
and set n = I + к. Then
f
„к2-к „(lk~\)ie
q

12.1 Rogers's Second Proof of the Rogers-Ramanujan Identities 581
When qx = q"l/2, this relation becomes
„к1 „2Ш °° e+l „k2-koBk-l)ie
+ /я yV(<+1) V
^ (q; q)e+k(q;q)i+i-k
qk2e2kW V
е=к (<?; ?)*+*(?; ?)*-*
This gives one Fourier expansion. To get the other, start with the triple product
identity
OO
(q, q)oo(x\ q)oo(q/x\ q)oo = Yl (-1)*?*(*-1)/2х*
k=—00
and replace x with —^/qe'e to get
k2/2 Jk6 v^oo 2k2 2ki6
(9; 9H0 (q;q)°c
A2.1.7)
Thus the right sides of A2.1.6) and A2.1.7) are equal to each other. Now replace
e2kw with (-1)V№~1)/2 andeB*-1)ie withO. Apply Lemma 12.1.1 to get A2.1.3),
which is equivalent to the first Rogers-Ramanujan identity. The second identity is
obtained by taking e2kw = 0 and e^'^6 = (-l/^+D/2 and applying Lemma
12.1.1.
Remark 12.1.1 Rogers's argument provides oneofthesimplestproofsofA2.1.3)
and A2.1.4). It appears to depend on formulas from the theory of theta functions.
These formulas, however, are used only for motivation and are not necessary for
the proof. Observe that, in the above argument, the equality of the coefficients of
e2kie in A2.1.6) and A2.1.7) are a consequence of the uniqueness of Fourier series.
Thus we have
> = — . A2.1.8)
?-? (?; q)i+kiq\ q)i-k (q\ q)x
But this follows directly from Cauchy's formula in Corollary 10.9.4, which can be
written
V — = . A2.1.9)
^ (q; q)n(xq; q)n (xq; q)x
In fact, we can give a very simple graphical proof of A2.1.9), using Durfee squares,
where the exponent of x counts the number of parts of a partition. Then A2.1.8)

582 12 Bailey Chains
follows by taking x = q2k. Next, A2.1.3) is obtained by multiplying both sides of
A2.1.8) by (—\)kqk(k~l)l2, summing over all k, and applying Lemma 12.1.1. In
the next section, we shall see that the scope of this simple argument can be greatly
extended.
Remark 12.1.2 Note that we get Euler's pentagonal number theorem
k=—oo
when we let I ->¦ oo in Lemma 12.1.1.
12.2 Bailey's Lemma
In a series of two papers in the 1940s, W. N. Bailey elucidated the underlying
stracture of Rogers's proof. He began with Rogers's replacement of 2 cos пв —
ешв +е~1пв in certain Fourier-series expansions and quickly observed the following
simplified version of the relevant expansion.
Lemma 12.2.1 (weak Bailey lemma) Suppose an and fin are two sequences
related by
n
Pn = У" ; A2.2.1)
f-? (q; q)n-r(aq; q)n+r
then, subject to convergence conditions (which in most applications boil down to
\q\ < U
oo ,
X ^ и2 A.
^ (ftq; ?)«,
Proof. The proof of this is quite short:
A2-2.2)
00 00 „ „2
a" q" <xr
= ??
?; q)n(aq; q)n+ir
OO OO n(n4-2r) _f
where the last step follows from Corollary 10.9.4.

12.2 Bailey's Lemma 583
The proof given by Rogers can be subsumed by this result. For example, to get
the first identity, take a = 1 and
1, n = 0,
a" ~ \ (-1) v*3""'^! + qn), n > 0.
By Lemma 12.1.1,
ff (q\ q)n-riq\ q)n+r ^ (q;q)n-r(q\q)n+r (q,q)n'
so that Bailey's result implies
y> q
; q)n (q; q)
(q\ q)oo
The reader should work out the details for the second identity.
Seeing this, it is clear that the method may be greatly extended by using more gen-
general summations than the very special limiting case of the ^-analog of Gauss's sum.
Bailey observed in his second paper that one could indeed invoke the full force
of the <7-Pfaff-Saalschiitz summation. He carefully described the proof of such a
generalization. However, seeing that it would look quite complicated he chose not
to write it down. This omission caused him to miss the full power of what we now
call Bailey's Lemma.
We start with the statement of "Bailey's transform."
Lemma 12.2.2 Subject to suitable convergence conditions, if
r=0
and
Yn =
then
л=0 л=0

584 12 Bailey Chains
Proof.
2_^anYn = 2.^1^ a"°rUr~n Vr+"
n=0 n=0 r=n
00 T
r=0 n=0
r=0
and the lemma is proved. ¦
The above argument is purely formal and the suitable convergence conditions
are those necessary to make all the infinite series converge and to validate the
change of order of summation. We now state and prove Bailey's Lemma.
Theorem 12.2.3 If for n > 0
И
ar
r=0 yi>i;n~r(aq; q)n+r
then
" a'
# = V , A2.2.3)
f^ (q\ q)n-r(aq\ q)n+r
where
, _ (Рь q)r(P2', q)r(aq/PiPiYoir
r (aqIp\; q)T{aqIрг\ q)r
and
a' _ y^ JP^'DjiPi, q)j{aqlP\Pi\ q)n-j(aq/piP2)}Pj A2 2 4)
j^0 {q\q)n-j{aq/puq)n{aq/P2\q)n
Proof. Take Un = l/(q; q)n and Vn = l/(aq; q)n and
_ (Pu q)n{pi\ q)n{q~N\ q\qn
(PiPiq~N/a; q)n
in Lemma 12.2.2. To compute yn, we need the g-Pfaff-Saalschiitz identity A0.11.3),
which we restate here for the reader's convenience:
_ (c/a; q)n(c/b; q)n
(c;q)n(c/ab;q)n '

12.2 Bailey's Lemma 585
Then
00
Yn = У~]8гиг-„Уг+п
-,-N
\ к/1 w? t
/a; q\(q; g)r-n(ag; q)r+n
— (Puq)r+n(P2, q)r+n(q~N; q)r+nqr+n
= T
^ (PiPig /a\q)r+n(q;q)r(aq;q)r+2n
= (Pug)n(P2-,g)n(g-N;g)ngn^ fpig", m", g-(N~n); g, g
(piP2q~N/a;q)n(aq;qJn \ PiPiqn~N/a, aq2n+1
_ (P\\q)n{p2\ q)n(q~N'; q)nq"(aqn+l /P\\ q)u-n(aqn+l I Pi\ q)N-n
(P\Piq~N/a; q)n(aq; q)in(aq2n+u, q)N-n(aq/'р\рг\ q)N-n
\q)N (-\)n(puq)n(p2\q)n(q~N;q)n
(aq; q)ы{aq/p\pi\ q)N (aq/рп q)n(aq/p2; q)n(aqN+l; q)n
We now turn to the proof of A2.2.3). Note that
r=o v"^"' i/i\-ii r2\ q)r(q\ q)N-r(aq; q)N+r
r=o (ag/pi>g">r(aq/P2;q)r(aq\q)N+r
(aq/'р\рг\ q)N
. ,
(aq/pi;q)N(aq/p2; q)N jzt
(aq/pip2;q)N v~- a t /u т п-пч
fir8r (by Lemma 12.2.2)
(aq/pu q)N(aq/P2\ q)N ~
(дд/Р1р2\д)ы y-y (p\\q)r(pi\q)r(q~N\q)rqrpr
(aq/pu q)N(aq/pi\ q)N f^> iPiPiq~N/a; q)
where the last equation follows after an algebraic simplification that produces the
expression given in A2.2.4). The theorem is proved. ¦

586 12 Bailey Chains
Remark 12.2.1 The Weak Bailey Lemma (Lemma 12.2.1) follows from the full
Bailey Lemma by letting n, pi, рг -> oo.
We call (an, pn) a Bailey pair if they are related as in Bailey's lemma. The
power of the full Bailey Lemma is that, given a Bailey pair (а„, Р„) a new Bailey
pair is produced. Consequently, from one such pair one can construct an infinite
sequence (а„, р„) ->¦ (а'„, P'n) -*¦ {a'n\ P'n') ->¦ • ¦ ¦ of Bailey pairs by successive
application of Bailey's Lemma. This sequence is called a Bailey chain. The simplest
conceivable chain starts with
„ fl ifn = 0, ,,„,>
Pn = <5„,о = < A2.2.5)
l-O if n > 0 .
It is an simple exercise to show that the corresponding an is
{l-aq2n){a;q)n{-lfqii)
a« = tj г; : • A2.2.6)
A -a){q;q)n
Remark 12.2.2 The fact that (an, pn) is a Bailey pair follows immediately from
a formula of Agarwal [1953],
A ; q)j(a; q)jqn> _ (aq; q)MqnM{д'~п\ q)u 2
^ 1;9)y(9;«)y (q;q)M(aq"+i;q)M '
a result that can be proved directly by induction on M. Another way is to use the
"inversion" formula: If pn is given by A2.2.1), then
a. = (.^)?(^)-+'-'(-''"-'«'':')ft A2.2.8)
12.3 Watson's Transformation Formula
In this section we use the Bailey chain to derive some important formulas. In
particular, we obtain a <?-analog of Whipple's transformation due to Watson [1929].
This transforms a terminating very well poised %ф1 to a terminating balanced 4</>3.
The phrase "very well poised" is defined below.
We start with a simpler result. Recall that (а„, рп) in A2.2.5) and A2.2.6) form
a Bailey pair. By Theorem 12.2.3, the next pair in the chain, namely (a'n, p'n), is
given by
a, {aq/piP2\q)n
Pn =
QL =
(q; q)n(aq/pi; q)n{aqlpi\ q)n '
(pi; q)n(p2; q)n(aq/p1p2)na - aq2n)(a; q)n{-
; q)n(aq/p2\ q)n0- ~ a)(q\ q)n

12.3 Watson's Transformation Formula 587
And the relation
П i
when written out in full, is
(aq/p\p2)n
(q)n{aqlp\)n{aq/pi)n
^ (q)n-r(aq)n+r(aq/p1)r(aq/p2)r(l-a)(q)r
where (x)r = (x; q)r. Note that
1 - aq2r = A - V^?r)(l + ^qr) _ (Vflg; q)r(-^q; q\
and
{aq)n+r = (aq)n(aqn+1)r.
Thus A2.3.1) is equivalent to the formula
u p2,q~" aqn+1\ (aq)n(aqIp\pi)n
(ЩI P\)n(aq I pi)n
A2.3.2)
The 6</>5 is well poised because the product of a numerator parameter with the
corresponding denominator parameter is aq. Additionally, the presence of the
factor A — aqlr)/(\ — a) now makes it a very well poised 6</>5.
Watson's transformation is obtained by moving up the Bailey chain to (a^, P'J).
The relation
v— a"
f^ (q)n-r(aq)n+r
is seen to be equivalent to
(aq/Xq)r(aq/k2)r(aq/Pi)r(aq/P2)r(l - a)(q)r(q)n-r(aq)n+r
j(k2)j(aq/k\k2)n
_ v^ (A.Qj(k2
y^(q)n-j(a
q/*.\)j(aq/X2)j(q)J(aq/pl)j(aq/p2)J'
3 34

588 12 Bailey Chains
By the formulas used to obtain A2.3.2), it is easy to show that A2.3.3) can be
written as follows:
/ a,q^/a,q^/a,kuk2,pi,p2,q~n . a2qn+1 \
1 \/a, --Ja,aq/ki,aq/k2,aq/p1,aq/p2,aqn+1' 'k1k2p1p2j
(aq)n(aq/p1p2)n ( aq/kik2, Pu P2,q \ „,,,..
*3 __„ __„ _„, \q,4 • A2.3.4)
n{aq/P2)n \aq/ku aq/k2, p\piq n/a
This is Watson's formula. When aq/k\k2 = p\Piq~n/a, the 40з becomes a bal-
balanced 302, which can be summed by the g-Pfaff-Saalschutz formula A0.11.3).
The result in this case is the ^-analog of Dougall's formula due to Jackson [1921]:
\^/a,-^/a,aq/k1,aq/k2,aq/p1,aq/p2,aqn+r
(aq)n(aq/pip2)n(aq/kipi)n(aq/klk2)n 2
= , when a q = k{k2p\p2q .
(aq/k1)n(aq/k2)n(aq/p1)n(aq/k1k2pl)n
A2.3.5)
Let n ->¦ oo in A2.3.5). This gives
/ a,q*ja,-qja,ki,k2, pi aq \
605 I y- y- ., ., . ;^'7~; I
\Va' -Va'a(l/^uaq/b2,aq/pi kxk2px )
(aq)oc(aq/pip2)oc(aq/kipi)oo(aq/kxk2)oc
= . (Iz.3.6)
(aq /ki)oc(aq /k^^aq / pi)oo(aq /ki^pi)^
This is a nonterminating form of A2.3.2).
Now observe that the 807 in A2.3.4) is symmetric in Ль к2, pi, p2 and hence
(aq/p1p2)n ( aq/kik2,pup2,q-"
403 I ! ^! i
(aq/pi)n(aq/p2)n \aq/kuaq/k2, pip2q-nja
(aqjk{k2)n ( aq/p1p2,ki,k2,q~'n .
-4*3 __,. _.,. ,,._„,_; 9.9 • A2-3.7)
(aq/k1)n(aq/k2)n \aq/pu aq/p2, k\k2q n/a
A ^-analog of Dixon's formula for a well-poised ?,F2 is obtained by setting p\ =
Vain A2.3.6):
/ a, -q^/a,kuk2
403 I r ,, ., ; q,
\Jaq/kaq/k
-Ja,aq/ki,aq/k2 kxk2 J
(aq)oo(aq/kik2)
(aq/ki)oc(aq/k2)oo(q^/a)oo(q^/a/k1k2)c

12.4 Other Applications 589
The two Rogers-Ramanujan formulas can also be derived from Watson's trans-
transformation A2.3.4). Let A.i, A.2. Pl P2 ->• °° to get
y^(fl;g)j(l-
h (q;q)
Now let n —>- oo and apply the dominated convergence theorem. After simplifica-
simplification, the result is
l +
E i
For a = 1, this is the first Rogers-Ramanujan identity and for a = q it is the
second one.
If one wants to generalize a <?-series identity that has quadratic powers in it, it is
natural to try to replace each #B> with (a; q)k times something else that will give
q( 2 'using
]im(a;q)k/{-a)k=qti).
a—>-oo
Since there are two Rogers-Ramanujan identities and five factors q(?\ the mini-
minimum extension of the Rogers-Ramanujan identities of this type should have six
free parameters. Formula A2.3.4) is this extension.
12.4 Other Applications
The summation formula for a very well poised 6</>s and Watson's transformation
were fairly straightforward consequences of Bailey's Lemma. Further interesting
results are possible by starting with the Bailey pair A2.2.5) and A2.2.6) and moving
further along the chain. For example,
v- q"
' -> (n\ (n\
П
n#0,±(fc+l)(mod 2/1+3)
This identity reduces to the first Rogers-Ramanujan identity when к = 1. As noted
before, it can be proved by examining (a*+1), Д^+1)) in the Bailey chain starting
with the pair A2.2.5) and A2.2.6).

590 12 Bailey Chains
It should be stressed that this is not the only Bailey chain of interest. In a totally
different context, another of Ramanujan's series,
("+1)
„=0 <¦ q'q)n n=0
was studied. The relevant Bailey pair is
P" ~ (q2; q2)n
and
From
r=0 (q; q)n-r(q\ q)n+r
we can deduce
S(q) = ^ (-l)n+Jqn^n+1)/2-j\l - q2n+1).
и>0
From this result it can be shown that almost all sn equal zero and that for any
integer M (positive, negative, or zero), there exist infinitely many n so that sn = M.
For details, see Andrews, Dyson, and Hickerson [1988]. This paper also contains
references to the work of Slater and Bressoud in Exercises 3, 4, and 5.
Exercises
1. Obtain the second Rogers-Ramanujan identity A2.1.2) from the weak Bailey
Lemma 12.2.1.
2. Prove Agarwal's formula A2.2.7), namely
A A - agV)(q-n; q)j(a; q)mqnj = (flg; gbg"M(g'""; q)u
l;q)j(q;q)j (q; q)M(aq"+1; q)M
3. Show that the (а„, fin) denned below is a Bailey pair with a = 1:
' -qf«2-^+\ m = Ъп - 1 > 0,
q6n2~n+q6n2+n, т = Зи>0,
.1, m =0,
(Slater)

Exercises
591
4. Suppose
Urn = <
6n42n
-q
1,
6n2+2n
m = 3n-l>0,
m = 3n>0,
т = Зи + 1>0,
m = 0,
and
Prove that (an, fin) is a Bailey pair with a = 1.
5. Prove that
(Slater)
и = odd,
is a Bailey pair with а = 1. (Bressoud)
6. Apply A2.3.8), Exercises 10.37 and 10.39, and Jacobi's triple product formula
to obtain the following six identities of Rogers:
(a)
6; q2°
; q2°)ool
+
(C)
(d)
(e)
(f)
33 A
n^±l,±8,±9,10(mod20)
oo
П (]
n=l
n^±3,±4,±7,10(mod20)
?__п I
+
7^п = П A _ ^„
Whipple's 7^6 transformation was obtained in Chapter 7 as a byproduct
of the solution of the connection coefficient problem for Jacobi polynomials.
There are several ^-analogs of the Jacobi polynomials. The next set of four

592 12 Bailey Chains
problems shows how the little <?-Jacobi polynomials can be used to derive
Watson's 8</>7 transformation. See Andrews and Askey [1977].
7. Define the little <?-Jacobi polynomials by
pn(x;a,P :q) = 2</>i I ^ ;q,qx
(a) Show that
lim pn(x; q\ /:<?) = P^\\ -
(b) Derive the orthogonality relation
Ealqi(ql+1;q)oo ¦ i
,„ ,-+1. . pn(q';a,P :q)pm{ql\u,P : q)
0 ifm/и,
anqn(q; q)oc((xPqn+1; q)oo(q; q)n
Wqn+u,q)oc{uq\q)oc{aq;q)n{\-aPq2n+l)
[Hint: First obtain the following identity:
ifm=n
(=0
0 if 0 < т <п,
l)oo(q\q)n .f n
ifm=n.]
8. Suppose that
n
pn(x; y,8 : q) = ^У~\акпрк{х; a, fi : q).
k=0
Show that
,q)k(q n;g)k(oiq;q)k
(aPqk+u,q)k(q;q)k(yq;q)k
q~n+k,y8qn+k+\aqk+i
9. Note that the identity in Exercise 8 is a polynomial identity in x. Deduce that
with the same akn, we have
(q-n,y8qn+\ax,...,ar
г+гФг+i \q,qx
\ yq,bu...,br
акпг+2Фг
k=0
(q k,aPqk+1,au...,ar
aq,bi, .. .br
10. Take r = 2, ft = 8, a\ = aq, x = 1, and b2 = q2a8a2/b1 in the previous
exercise. Observe that the з</>2 in akn becomes balanced and may be computed

Exercises 593
by A0.10.3). Deduce that
q-n,y8qn+l,aq,a2 \ anqn(8q; q)n(y/a; q)n
yq,bi,q2a8a2/bl ' ' / (a8q2; q)n(yq; q)n
~n,q^a&q, -q^/aSq,aSq,aq,aSq2/bu ySqn+l,b\/a2-
The standard form of Watson's transformation is obtained from this formula
after proper identification.
11. The big g-Jacobi polynomials are defined by
{q; q)n{-q; q)n
/q-n,qn+a+P+1,xqa+ld/c
ъФг „,, „,, ,. ,q,<l
Prove that the orthogonality relation is
Г
y_rf « x,c, . m . q
О т ф n,
(-qH+lc/d; q)n(-qa+ld/c; q)n{\ - t
¦ , m=n,
(-q\ q)n(-q\ q)n
where
M _ cd{\ - q)(q; q)oc(qa+li+2; q)co{-d/c; q)oo(-c/d; q)oc
(c + d)(qa+1; q)oc(qP+1', q)oo(—qa+ld/c; q)oo(—qP+lc/d; q)oo
Note that when с = d = 1, the weight function in the <?-integral tends to
A -х)аA+х)Р asq -+ Г.
12. A set of g-Laguerre polynomials is defined by
- q)k(qn+a+lx)k
(a) Replace x with -A - q)q~{P+1)x and let /J ->¦ -oo in the little ^-Jacobi
polynomial of degree n. Show that the result is
(q«+l;q)n " *4"
(b) Prove that lim^,- L(na)(x; q) = Lan(x).

594 12 Bailey Chains
(с) Prove the discrete orthogonality relation
ka+k
q
0,
(Use Ramanujan's i^i sum.)
(d) Prove the continuous orthogonality relation
dx
x;q)
(-A -q)x\ q)
О, тфп,
m — n.
Yq(-a){q\q)nq«
(Moak)
13. A multiplicative shift of the variable x in Exercise 12 gives a set of g-Laguerre
polynomials defined by
Lan(x;q) =
(a) Show that
q)x (~q x , qH
m = n,
(-x;q)oc(-qx~l;q)oc
g-n{q\q)n{q\q)oo Г (a + 1 - 0)Г@ - а)
= I (qa+];я)п(qa+]; q\ r,(a + 1 -p)T4(p-a)'
О, тфп.
(b) Set a - p = с (fixed c) and let a ->• oo in L"(jc; #). Show that the result
is the Stieltjes-Wigert polynomial
/i=u ()n-k
(c) Prove that a weight function for the Stieltjes-Wigert polynomials is
xc
w(x) - .
(-x;q)oo(~q/x;q)x
For Exercises 12 and 13, see Gasper and Rahman [1990, Chapter 7].

Infinite Products
A.I Infinite Products
For readers unfamiliar with infinite products, a brief introduction is given here.
Definition A. 1.1 Let pn^ = Yili=k^~^am)- Vthere is akforwhich рп^ converges
to a nonzero value p as n ->• oo, then we say that the infinite product ]^=1(\+ап)
converges. We write it as p = ГС^ЦA + an)- The reason for not taking к = 1 is
to allow an finite number of zero factors. The convergence of the product is said
to be absolute if fl^li A + \an\) converges.
The following basic theorem reduces the problem of convergence of a product
to that of a series. For simplicity, assume that Re an > -1, n — 1, 2, If not,
start the product after this holds.
Theorem A.1.2 The product fl^iO + an) converges if and only if the series
J2T=\ loiA + a») converges.
Proof. Suppose that Sn = J2"m=i l°g(l + am) converges to S. Since exp is a
continuous function,
n oc
Yl A + am) = ехрF"и) converges to es = JJA + an).
m = l и=1
To prove the converse, let 1 + am = AmeWm and П^Д1 + am) = В„е'ф». The
convergence of ПA + а„) implies am ->• 0 so that 9m ->• 0 and фп can be chosen
so that, say, фп ->• ф. We also have
ву+в2 + \-6n =ф„ + 2пкп,
where kn is an integer. Therefore,
0n+i = </>n
595

596
A Infinite Products
Since kn+i — kn is an integer, вп+х ->• 0, and since фп+\ — фп -> 0, we must have
kn = k, a constant for sufficiently large n. Thus, for sufficiently large n,
n +27rfei.
Let n —>• oo to get
= log p + 2n ki.
This proves the theorem. For absolute convergence the condition is simpler and is
contained in the next theorem. ¦
Theorem A.1.3 The product Y\^=1(l + an) converges absolutely if and only if
^2T=i a" converges absolutely.
Proof. Convergence of the series or the product implies that an -> 0. For suffi-
sufficiently large n we must have \an \ < 1/2. Suppose an ф 0 and n large. Then
1 -
an
< -[|tf«l + KI
1 la, I
21-|ая|
So
or
1 ^ log(l+an)
an
-K|<|log(l+an)|<-K|.
These inequalities together with Theorem A. 1.2 imply the result and also the fact
that absolute convergence of a product implies its convergence. ¦
Definition A.1.4 The infinite product
n=l
where x is a real or complex variable in a domain, is uniformly convergent if
n
Pn(x)= ]J(l+am(x))
m=k
converges uniformly in that domain, for each k.

Exercises 597
Theorem A.1.5 If the series Y2T=i \an(x)\ converges uniformly in some region,
then the product Yl™=l(l + an(x)) also converges uniformly in that region.
The proof of this result is left to the reader.
Corollary A.1.6 If an(x) is analytic in some region of the complex plane and
П A +an (jc)) converges uniformly in that region, then the infinite product represents
an analytic function in that region.
Exercises
1. Prove directly from the definition that the product {\ - \){\ + \){\ - \) ¦ ¦ ¦ \s
convergent.
2. Prove that an absolutely convergence product is convergent.
3. Show that if Еа„, Ha^,..., Sa*, ?|а„|* are all convergent, then ПA +ап)
is convergent.
4. Discuss the convergence of the product

в
Summability and Fractional Integration
B.I Abel and Cesaro Means
The following theorem, the first part of which was proved by Abel, is often en-
encountered in a first course in real analysis.
Theorem B.I.I Suppose that the series Y2T=o ^« converges to B. Then Y2T=o b"x"
converges uniformly on [0, 1], and in particular
lim 2^bnxn = В. (В.1.1)
X^ л=0
Moreover, if Bn = YH=o bk> then
lim — l- — = B. (B.I.2)
и^ос n + 1
The example of the series 1 — 1 + 1 — ¦ • ¦ shows that the limits in (B .1.1) and
(B.I .2) may exist even when the series does not converge. Thus (B. 1.1) and (B. 1.2)
may be used to assign a sum to divergent series.
The series YlT=o bn is Abel summable or summable A to В if (B. 1.1) holds; and
it is Cesaro summable or summable (C, 1) to В if (B.I .2) holds. We also speak of
(B. 1.1) as the Abel mean and (B.I.2) as the Cesaro mean of the series J2T=o ^n¦
The Abel and Cesaro means of the series 1 — 1 + 1 — • ¦ • are both 1/2. The
next theorem shows that Cesaro summability is a stronger requirement than Abel
summability.
Theorem B.I.2 If the series YlT=o^n is (C, 1) summable to B, then it is A
summable to B.
599

600 В Summability and Fractional Integration
Proof. Use summation by parts to get
00 ОС
*„*" - В = A - *) ? Bnxn - В where Bn=
л=0 л=0 V k=0
ОС
— X ) > (on Т* ' * ' Т* Вп)Х — &
ос
= A - хJ^2{(В0 + ¦ ¦ ¦ + Вп) - (п + })В}хп. (В.1.3)
я=0
Now (В. 1.2) implies that, for e > О, there exists an integer N such that
|(Bo + ¦ ¦ ¦ + Bn) — (n + l)B\ <e(n + l) forn > N.
Use this in (B. 1.3) to arrive at the necessary result. ¦
The example of the series 1 — 2 + 3 — • • ¦ shows that the Abel mean may exist
(in this case it is 1/4) but the Cesaro mean may not. The Abel and Cesaro means,
together with other summability methods, are very useful in analysis and analytic
number theory. As an elementary example, consider the following theorem of Abel
on the product of two series:
Theorem B.1.3 Suppose J2T=o an and J2T=o ^n are convergent series. Let cn =
121=0 аФп-к and suppose that J2T=o c" *s convergent. Then
л=0 и=0 и=0
Proof. Since
cn =
k=0
ос ос
к=0 j=0 k=0
and Theorem В. 1.1 completes the proof. Ш
The Cesaro and Abel means play a very significant role in the theory of Fourier
series and transforms. We take a very brief look at how they appear in Fourier
series since these ideas will play a role elsewhere in the book.
Suppose f(x) is an integrable function of period 2л. Let J2°°oc а"е'"х be its
Fourier series. An important question is: When does the Fourier series of a function
converge to the function? Once again it is easier to deal with the Cesaro mean or

В. 1 Abel and Cesaro Means 601
the Abel mean. The Abel sum is given by
Since
an = — / f(t)e~(n'dt,
2я Jo
the sum in (B.1.5) is equal to
1 Г2л
2л- Jo
The sum inside the integral is called the Poisson kernel and it is a straightforward
calculation to show that it is equal to
1-r2
= Pr(x-t).
1 - 2r cos(jt - t) + r2
The following properties of Pr(x) are worth noting:
(i) Pr(x) > 0,
(ii) ±fin Pr(x)dx = I, md
(iii) for 8 > 0, max Pr(x) -> 0 as r -> 1".
S<x<2n-S
These properties can be used to give a proof of the following theorem:
Theorem B.1.4 If f is periodic and integrable on @, 2л) then the Abel means
of the Fourier series converge to \{f(xo+) + f(x0—)} at every point x0 where the
right and left limits /(xo±) exist.
A similar result exists for Cesaro means. The nth partial sum of the Fourier
series is given by
sn(x) = 2^ a*« = -
k=-n " "'" i=-n
The sum inside the integral is
A sin(n + 1/2)(jc - t)
1+2 > cos/t(jc —f) = . (B.1.7)
*-^ sin((jc - 0/2)
k=0
This expression is called the Dirichlet kernel. One drawback of the Dirichlet kernel
is that it is not always positive. However, if we take the (C, 1) mean of the Fourier

602 В Summability and Fractional Integration
series, then we get
so(x) + si(x) H \-sn(x)
an(x) =
n+l
^ sin((x - 0/2)
The sum in the integral denoted by Kn(x — t) is called the Fejer kernel. The sum
is equal to
{, ч 2
svnUn + l)(x - t)
2. i > > 0. (B.I.8)
The Fejer kernel has the three properties of the Poisson kernel mentioned above.
One may then deduce the following theorem:
Theorem B.1.5 For f as in Theorem B.I.4, the Cesaro means of the Fourier
series of f converge to \{f(xo+) + f(xo—)}at every point xq where the right and
left limits /(*o±) exist.
If we assume / to be continuous on [0, 2л], then the Abel and Cesaro means
converge to /. An important consequence of Theorem B.1.5 is the following
corollary:
Corollary B.I.6 A continuous function on [0,2л] can be uniformly approximated
by trigonometric polynomials, that is, polynomials of the form Yl"-n ake'kx.
Since eikx is uniformly approximated on [0, 2л] by partial sums of its Taylor
series, we get another proof (see Exercise 1.40) of the Weierstrass approximation
theorem.
B.2 Cesaro Means (C, ex)
In the previous section, we saw that the (C, 1) means do not assign a value to the
sum 1-2 + 3—4H .To handle this and some other situations, we define Cesaro
means of higher order.
For the series У"Ч?° bn, set
where
k=0

B.2 Cesaro Means (С, а) 603
The limit of the sequence {^tj} gives the (C, 1) mean. Now, set
ВР = В + В
and
The limit of the sequence {^5-} is the (C, 2) mean of the series Y^ bn.
Similarly,
в<р = в™ + в™ + --- + вр
and
jtP) 1 , * l j (" + DC» + 2) (и+1)(и + 2)(и + 3)
Thus, the (C, 3) mean is given by
Define B^*' and ?^ inductively by
Bf =
and
?« = ?() + ?(« + ... + ?»»).
It is possible to express B* explicitly in terms of Ь„ as follows. Note that
n=0 n=0
and
n=0
So
?=0
and
±!>1. (В.2.3)

604 В Summability and Fractional Integration
We define the (СД)теап of Ym bn as the limit of the B^/E^a&n -» oo.Note
that these quotients are meaningful even when к is not a positive integer. We take
к to be real and greater than — 1.
Definition B.2.1 The series J^° bn is (C, a) summable to В (for a > —1) if
^r^^^B. (B.2.4)
Remark B.2.1 Note that the limit in (B.2.4) is equal to
lim ?^±22вИ
B.3 Fractional Integrals
It is possible to extend the summability definitions to integrals. Suppose f{t) is
integrable on finite intervals @, x). We say, for example, that / is Abel integrable,
that is, /0°° f{t)dt exists in the Abel sense, if
lim /
lim / e~x'f(t)dt exists.
A— n'
We also write this as
/•00 /.00
/ /(r)c?f = lim / e'ktf{t)dt (Abel).
To define Cesaro integrability (C, ?), where ? is an integer, we first have the integral
analog of Blk) (see (B.2.2)) as
/(*>(*) = /o Ak-i)(Odt = Jox(x - t)f(k-2)(t)dt = • •. = 1 /^(дс -
(B.3.1)
Following the remark after (B.2.4) we say that /0°° f{t)dt is (C, fc) integrable if
ГГук+11 Л* / f \ *
lim v 7 /*(*) = lim / I 1 - - I /@^ exists.
Observe that the final expression for /*(jc) in (B.3.1) is meaningful for all real
к > — 1. Thus we have a definition of (C, a) integrability for a > — 1.
Note also that the formula for /да in (B.3.1) expresses /№) as a (jfc + l)-fold
integral of /. Thus, we define Iaf, the a-fold integral of / for Re a > 0, by the
formula
f\x - t)a-x
Jo
» fix) = 7^- f\x - t)a-xfit)dt. (B.3.2)
Г (a) J

В.4 Historical Remarks 605
It is easy to check that the operator Ia satisfies the relation
The operator Ia is called a fractional integral of order a. Euler's integral expression
for the 2.F1 hypergeometric series can now be interpreted as a fractional integral.
Write the formula as
r(b)r(c-
b)J0
where fit) = tb-l(l -t)~a.
Remark B.3.1 Abel and Cesaro integrability can be used to study Fourier inte-
integrals just as the corresponding summability is used to study Fourier series.
B.4 Historical Remarks
Euler had used Abel means and other methods for associating a sum to a divergent
series. In particular, he obtained the functional equation of the zeta function in the
form
- 2
5
1 _ 2- + 3-» B'-i
When Re s > 1, the series in the denominator converges but the series in the nu-
numerator is interpreted in the Abelian sense. It can be proved that
oo
lim ^(-ly-V-1*" = f(l -s)(T - 1);
thus Euler had the functional equation correct. He verified it only for integer values
of s and for s = j and |. Abel's name is attached to this method of summation
since he proved formula (B.I.I).
Leibniz used (C, 1) means to evaluate 1 — 1 + 1 — • • • and somewhat later
D. Bernoulli employed the same method to consider the more general series
J2T=o^n, where bn+p = bn for all n and Ylo~l bk = 0. Neither Leibniz nor
Bernoulli explicitly stated that he was giving a new definition of convergence.
This was done by Cesaro. For more on the history of Abel and Cesaro means the
reader should consult the very interesting treatment given in Hardy [1949].
Abel was apparently the first mathematician to use fractional calculus, though
the concept had been considered by others before him. Abel used fractional calculus
in his solution to the problem of finding the tautochrone - the curve down which

606 В Summability and Fractional Integration
a particle slides freely under gravity with zero initial velocity and reaches the
bottom in the same amount of time regardless of the starting point, provided that
the starting and lowest points are distinct. This problem had already been solved
by Huygens; the motivation was the construction of a clock in which the period of
oscillation does not depend on the amplitude of the pendulum.
Suppose that the particle slides down from the point (a, b) and that the bottom
of the curve is the origin of coordinates. If s is the length measured from the origin,
then conservation of energy implies that
velocity = v = ds/dt — ^/2g(b — y)
at a point on the curve with ordinate y. Set ds/dy — f(y) to see that the time
taken to reach the bottom is given by
, Гь
T(b)
Abel observed that
T{b) = J^(
Since T{b) is independent of b, use (B.3.2) to get
/о Jb-y ж
Now take the derivative of both sides to obtain
Since
we get a differential equation that can be solved. The tautochrone curve turns out
to be a cycloid. An English translation of Abel's [1826] paper on this topic is also
available.
The above calculations suggest that the concept of a fractional derivative would
also be useful. For smooth enough functions, in particular analytic functions, it is
possible to define fractional derivatives as follows:
Daf(x) = — /„_«/(*), (ВАЛ)
where 0 < Re a < n, and n is an integer. Observe that we took the integral first
and then the derivative. If the calculations are done in the opposite order, a different

Exercises 607
function arises since the first n — 1 terms are removed by the nth derivative. For
an analytic function
oo
0
the fractional integral has the form
00
k\
as is easily verified. This shows that the value of n in (B.4.1) may be arbitrarily
chosen as long as n > Re a. In fact,
DkIa — DnIa+n-k.
However, the inclusion of fractional derivatives with fractional integrals does not
give a group, for although
DaDfi = Da+fi
when a, fi and a + fi are not integers,
Dl/2D ф D3/2
for the reason mentioned above.
Exercises
1. Prove Theorem B. 1.1.
2. With the notation of Theorem В. 1.3, prove that if J^° an converges absolutely
and Y^ju bn converges, then Ya^ cn converges.
3. Prove that if the series Y^ bn is (C, a) summable to B, and fi > a, then the
series is (C, fi) summable to B.
4. Prove thefollowing extension ofTheoremB. 1.2: If the series Y™ bn is (C, a)
summable to B, then it is Abel summable to B.
5. Show that
00
Deduce that
1 - 22k + 32k = ¦ ¦ ¦ = 0,
1 _ 22* + 32* - • • • = (-1)* ——Bk
2k

608 В Summability and Fractional Integration
in the sense of Abel summability. This result is due to Euler. See the historical
introduction in Hardy [1949].
6. A sequence Sn converges to S in the sense of Borel if
OO о
Show that if Sn converges to S then Sn -» S in the Borel sense. Also prove
that if Sn (x) = 1 + x н \-xn, then Sn (x) -* 1/A - x) in the Borel sense
for Rex < 1.
7. Show that if fi > a > — 1 and the (C, fi) means of Еа„ are nonnegative, then
the (C, a) means of Hanrn are nonnegative for 0 < r < (a + \)/ф + 1).
Hint: Observe that
(l-fflJ-'-'EvV = (l-a>r"-lu-rw)f>+l(l-rw)-f>-li:anrnaf
and show that the power series coefficients of A — a))~a~x(l — rw)P+1 and
those of A — rvS)~P~xYjanrnaf are nonnegative. This proof is due to Bustoz
[1974].
8. A series Y^,T an is summable (L) (Lambert summable) to S if
oo „
.. „ , v^ na"x с
lim A - x) 2_^ = S.
Prove that if Т,а„ = S(C, a), for some a, then Еа„ = S(L).
9. Prove Dirichlet's theorem that if x is a quadratic Dirichlet character, then
L(l, x) = YZ=\ X(n)/n ф 0 as follows:
(a) Show that J2d
(b) Show that
(c) Let
f<x\ _ x ' ' лч"' XKn)x
Show that b\ > Ьг > ¦ ¦ ¦ > К > bn+i > ¦ ¦¦¦
(d) Note that | J2n=i X(n)\ < M, where M is a constant that does not depend
on m. Use summation by parts to prove that f(x) < ^~ = M.
(e) Deduce that L(l, /) ф 0.
This proof is taken from Monsky [1994].
10. Prove the following theorem of Hardy and Littlewood: If an > 0 and
oo
lim A -x)Y^anxn = 1,
x~* л=0

Exercises 609
then
hm — y^ Щ = 1.
Karamata's proof is sketched below.
(a) Show that
oo
lim A — x) 2^onxng(x") -
wheng(f)
(i) is a polynomial,
(ii) is a continuous function, or
(iii) has a discontinuity of the first kind.
(b) Take g(t) = 0 @ < t < l/e) = \/t(\/e < t < 1) and complete the
proof of the theorem.

Asymptotic Expansions
C.I Asymptotic Expansion
Let x be a real or complex variable in an unbounded region D and let Y^ anx~n
be a formal power series that may be convergent or divergent.
Definition C.I.I The series Y^ anx~n is an asymptotic expansion of a function
/(*) if
n-\
iX'k + Rn(x), (C.I.I)
where Rn(x) = 0(x~") as x ->¦ oo in D. Usually {C.I.I) is written as
f(x) ~ ao + a\X~x -\-aix~1 + ¦ ¦ ¦ as x-^ oo in D. (C.I.2)
This definition is due to Poincare [1886]. As a simple example we have the
following asymptotic expansion of A + x):
1 111
1 + X X X2 X3
as x —>¦ oo.
(C.1.3)
In this case the series is convergent. The more interesting situation occurs when the
series is divergent. Consider the complementary error function of a real variable
x >0,
Successive integration by parts gives
2 \e-*2 [°°e-r
erfc x = —= — / ——
y/n\2x Jx It2
2 [°° ,
erfc x = —= / e dt.
dt
(C.I.4)
k=\
611
*.¦ г* г--:^

612
С Asymptotic Expansions
where
Rn(x) = (-1У+1
It is easy to see that
1 • 3 • • ¦ Bи + 1)
2"
-xe
Jx t
<
¦ 3 • ¦ • (In
Bx2)«+i
(C.I.5)
Hence
1 ¦ 3 ¦ ¦ ¦ Bk - 1)
Bx2)k
is an asymptotic expansion of the complementary error function. It is clear that
the series diverges for all x > 0. However, if a fixed number of terms is taken,
then for large enough x a good approximation of erfc x is obtained. On the other
hand, for a given x, taking more and more terms of the series does not improve
the approximation, since the series diverges.
C.2 Properties of Asymptotic Expansions
The following theorem follows almost immediately from the definition.
Theorem C.2.1 A function f(x) has an asymptotic expansion Т,а„х~" if and
only if for each n,
л-1
fix)-
--*
k=0
as x —>¦ oo in D,
(C.2.1)
uniformly with respect to arg x when x is complex.
Proof. It is easy to see that (C.2.1) implies (C. 1.2). To reason in the other direc-
direction, observe that
x"Rn(x) = xn[anx
and the theorem is proved. ¦
Rn+i(x)]-+an asx-*oo,
A consequence of this theorem is that a function has at most one asymptotic
expansion in D. In a different unbounded region the asymptotic expansion may
be different. However, two different functions may have the same asymptotic
expansion in some region. For example, the function e~* in | arg x\ <\n—&< \ж
satisfies x"e~x -» Oasx -» oo. By (C.2.1),
n 0 0
'0 + - + — + ¦
x x2
|argx| <-n-
00.
So the zero function and e x have the same expansion in this region.
The next theorem gives some algebraic properties of asymptotic expansions.

C.2 Properties of Asymptotic Expansions 613
Theorem C.2.2 Suppose that f(x) ~ T,anx~" in D\ andg(x) ~ ~Ebnx~" in D2.
Then:
(i) For constants A. and д
Xf(x) + iig(x) ~ T,(Xan + iibn)x~n in Dx П D2.
(ii) f(x)g(x) ~ T,cnx~n in Dx П D2, w/геге
си = ao^o + ai^/i-i H h anbo-
(Hi) Ifa0 ф 0, then
1 "' л/
= У^-4 + O(x~") asx -> ooinD],
f(x) f^Q x
where a.Q+ldk is a polynomial in uq, a\,..., ak. The dk can be obtained from
the relations
aod() = 1, aQdk = —(a\dk-\ + a2dk_2 + • • • + akdo), к = 1, 2,....
The proof of this theorem is left to the reader.
Asymptotic series can be integrated over the interval л: < t < oo,ifao = a\ = 0.
In this case
XL XJ
and
Г0° tl u «2 ЙЗ
Integration is possible here because f(t) = a2t~2 + O(t~2) for t large. If a0 and
a\ are not zero, then
— an — a\t ]dt ~ 1 + • • •, x —> oo.
x 2x3
Differentiation of an asymptotic expansion may not always be valid. A standard
example where differentiation fails is /(*) = e~Jtsine* where x > 0. The
derivative f'(x) = cose* — e~* sine* oscillates as x —*¦ oo and hence does not
have an asymptotic expansion by Theorem C.2.1. But
0 0
f(x) ~0H 1—- + ¦¦¦, лг-4-оо.
If f'{x) is continuous and has an asymptotic expansion, then it can be obtained
from term-by-term differentiation of the expansion for f(x). This follows from
the result on integration and uniqueness of the asymptotic expansion.

614 С Asymptotic Expansions
C.3 Watson's Lemma
In this section, we discuss a method of obtaining the asymptotic expansion in some
region of a function expressible as a Laplace integral. Section 1 has an example of
an integral whose asymptotic expansion was obtained using integration by parts.
The failure of this method in one case was discussed in Chapter 2. Also see Wong
[1989, p. 18].
The next theorem, called Watson's lemma, gives the asymptotic expansion of
f™e~x'f(t)dt. See Watson [1918].
Theorem C.3.1 Let f(t) be analytic in \t\ < a + S, where a > 0, S > 0, except
possibly for a branch point at 0; and let
m=\
when \t\ < a and r > 0. Suppose also that \f(t)\ < кеы, where к and b are
positive numbers independent oft, when t > a. Then
/•О
/
as x -» oo (C.3.2)
for |argx| < \n - S < \n.
Proof It is clear that for any fixed integer M, a constant С can be found such
that for t > 0
M-l
f(t) - У* amt{m/r)-
Hence
e-x'f(t)dt = V / e-^flmr(m/r)-^r + RM
»=i ^
M-l
= \J атГ(т/г)х~т/г + Rm,
where
/•OO
I^mI < / \e~xt\Ct(Mlr)-lebtdt
Jo
= СГ(М/г)/[Кех-Ь]м/г,
provided Rex > b. Since |argx| < \ж — 8, it follows that Rex > b if |x| >
b esc S. Thus,
This proves the theorem.

C.4 The Ratio of Two Gamma Functions 615
Watson's lemma can be generalized to the case in which the integral in (C.3.2)
is a contour integral, /^+> e~xtf(t)dt and (C.3.1) is replaced by an asymptotic
expansion for f(t). See Olver [1974, pp. 112-115] and Wong [1989, p. 22].
C.4 The Ratio of Two Gamma Functions
We give an application of Watson's lemma to obtain the asymptotic expansion of
a ratio of gamma functions. This expansion is due to Tricomi and Erdelyi [1951].
Observe that for Re(x + a) > 0 and Re(b — a) > 0, we have
Г(х
1 f°°
-!— / e-xte-at{\-e-<)
' - a) Jo
Г(х + Ь) Гф
Define the generalized Bernoulli polynomials B° (u) by
taeut _y> a tn
^ n=0
Then for a = a + 1 - b,
jT^Fr = ? ^TB« (a)r""CT' |r| <2;r-
Watson's lemma now implies that
T{x + a) _ y, (-1)" Гф-а+п) 1
n! "v F(ft-a) x*-a+»'
^-«, |x|->oo. (C.4.3)
If instead of the integral representation (C.4.1), one were to use
Г(* +a) _ ГA +a - b) [^ ,x+a)u, .^
¦ I e<-x+a^t(et —
J —ooe'a
l)b-a'ldt, (C.4.4)
T(x+b) 2ni J_oo.e,,
Re[(x + a)eia] > 0, |ot| < я/2, and for small \t\,
a - тс < arg(e' — 1) < a + ж;
then one could extend (C.4.3) to |arg x\ < я — 8.
There is an improvement of (C.4.3) from the computational standpoint due to
Fields [1964]. Note that if и is replaced by a - и in (C.4.2), then B%{o - u) =
{-\)kB%(u). This implies that В%к+1(а/2) = О. Now write the integrand in

616 С Asymptotic Expansions
(C.4.1)as
л!
«=о
°° DJ /
_ -(x+a-a/2)t ST^ a2n\a
?—< ПпЛ
This implies
„=о W-
- a + 2n)
Bи)! Г(й-а) (x + a - CT/
for |arg(jc + a)| < \n — S, \x\ ->¦ oo.
Exercises
1. Let Re x > 0. Show that as x -> 0,
OO -( ^
dt~Y{-\)nn\xn
by two methods: (a) Expand 1/A + xt) as a series, (b) Integrate by parts
repeatedly.
2. Suppose that
F{x) = f e~xtf{t)dt
Jo
converges for some x = xo and / has continuous derivatives of all orders in
0 < t < a. Show that
F(x) ~ Y,f{n)(O)x-n-1
uniformly in arg x, as x ->¦ oo in |arg jc| < я/2 — e for e > 0.
3. Show that
Л
4. Suppose 6»„ is defined by
Show that
and hence
л
1!
On = M
»~i
+
л^
(I
4
j
8
J
-7 + '-
Л"
1 (yp
\ ^
as,
1 n
) ufx
n —>¦ oo.
For a discussion of this result of Ramanujan, see Bemdt [1989, pp. 181-184].
5. Prove Theorem C.2.2.

D
Euler-Maclaurin Summation Formula
D.I Introduction
Some consequences of the close connection between series and integrals are
brought out even in a first course in calculus. The integral test, for example,
states that for a decreasing continuous function on [1, oo) the series YT /(n)
and integral /j°° f(x)dx converge or diverge together. In the derivation of Stir-
Stirling's approximation for F(jc) given in Chapter 1, we saw that for finite sums
the function / need not be decreasing for the integral to provide a good ap-
approximation. The Euler-Mclaurin summation formula makes the connection be-
between the sum and the integral explicit for sufficiently smooth functions. In
this appendix we give a statement and proof of the formula and a few applica-
applications.
Start with a differentiable function defined on a set that contains the interval
[m, n], where m and n are integers. Let m < j < n, where j is an integer. As a
first approximation we consider
f(x)dx « l-(f{j) + f(j + 1)). (D.I.I)
It is possible to express the error in this approximation as an integral. Observe that
/o+d)= [J+14-\(*-j-l
-j--)f'(x)dx. (D.1.2)
j Ji
Set j = m, m + 1,..., n — 1 to get
\(f(m) + f(m + 1)) = f + f(x)dx +[ + (х-т-\ )f'{x)dx,
617

618 D Euler-Maclaurin Summation Formula
i rm+2 pm+2/ i \
-(f(m+l) + f(m+2))= f(x)dx+ [x - (m + 1) - - )f'(x)dx,
1 Jm+l Jm+l \ ?J
Ufin - 1) + /(«)) = f f(x)dx + [ (x - (л - 1) - ^]f'(x)dx.
? Jn-l Jn-l\ l)
The second integrals on the right can be added together if we note that the integrand
can be written as (x — [x] — \)f'{x). Addition gives
f (*) + \ (/(m> + /С"» = / fWdx + [ (x - W - I) f'(x)dx.
(D.1.3)
This is a particular case of the Euler-Maclaurin formula. To put it in a slightly
different form, set B\(x) = x — i, so that B\{x — [x]) = x — [x] — \. Also let
Bx = Вхф) - -1/2. Now write (D.1.3) as
J2 /(*) + Bi{f(n) - f{m)} = / /(jc)djc + /" Bi(jc - W)/'(jc)djc.
m+l ^m m
(D.1.4)
Recall that Bi (x) is the first Bernoulli polynomial and B^ the first Bernoulli number.
As an application take f(x) = logx and m = 1 in (D. 1.4) to get
n+-)logn+n = 1+ / — —-dx. (D.1.5)
Since B\(x — [x]) is periodic of period 1 and
/•l «+1
Bl(x-[x])dx= B1(x-[x])dx = 0 (D.1.6)
Jo Л
for any t > 0, we see that the limit as n —> oo in (D.1.5) exists and is equal to
This is exactly Stirling's approximation, if we can show that the expression (D. 1.7)
equals \ log 2л-. We prove this later by a method different from the one given in
Chapter 1, which uses Wallis's formula.
Formula (D.1.4) was obtained by one integration by parts. With Stirling's for-
formula in mind, it makes sense to repeat the process. This would give an ex-
expression with higher derivatives of f(x). If f(x) = log*, then f'(x) — 1/x,
f"(x) = —1/x2, and so on. These quantities get small for large x and so an
extension of (D. 1.4) would be useful.

D.2 The Euler-Maclaurin Formula 619
D.2 The Euler-Maclaurin Formula
Let B2(x) be a primitive (antiderivative) of Bi(x - [x]) = Bi(x). It follows
from (D.1.6) that B2 is periodic with period one. In particular, this implies that
B2@) = B2{\) = • • • = B2(j) = B2(j + 1) = • • •. Now, integration by parts
gives
f
•I j
[J B2(x)f"(x)dx.
j
We assume that / has continuous derivatives of as high an order as necessary.
Sum from j = m to j — n to get
Г B1(x)f'(x)dx = B2(O)(/'(n) - f'{m)) - Г B2{x)f"{x)dx. (D.2.1)
J m J m
Note that it was the periodicity of B2(x) that gave us the simple expression on the
right side of Equation (D.2.1). This suggests that we should choose the constant
of integration in B2{x) such that
/¦l rt+i
/ B2(x)dx = / B2(x)dx = 0. (D.2.2)
Jo Jt
Let \B2(x) = B2(x) for 0 < x < 1, so that B2(x) = \B2{x - [x]). From the
definition of B2(x), it follows that
B2(x)=x2 -x + -,
о
which is the second Bernoulli polynomial. We state the general Euler-Maclaurin
formula in the next theorem:
Theorem D.2.1 Suppose f has continuous derivatives up to order s. Then
J2 fix) = Г
m+\ Jm
f f +
m+\ J
/
Jm
where Bs{x) are the Bernoulli polynomials and Bi = ^@), the Bernoulli num-
numbers.
Proof. As in the derivation of (D.2.1), apply integration by parts successively
to obtain a sequence of periodic functions Bn(x) such that B'n(x) = Bn-\(x)
@ < x < 1, n > 1) and/0 Bn{x)dx = 0, n > 1. With respect to these functions,

620 D Euler-Maclaurin Summation Formula
we obtain the formula
?/(*) = Г f(x)dx+J2(-VeBd0){f(t-l)(n) - fl-l\m)}
m+\ ^т l=\
+ (-l)'-1 f Bs(x)f(s)(x)dx.
Jm
To show the relation of Bt{x) to Bernoulli polynomials, consider the generating
function
oo
G(x, t) := J2 Bn(x)tn, where B0(x) = 1, (D.2.3)
о
of the sequence {Bi(x)}. Observe that for 0 < x < 1,
„ „ OO OO 00
— = Y, Kw = Yl Ёп-лхУ = t J2 E^x">tn =tG-
„=1 n=l n=0
We then have
G(x,t) = A(t)ext. (D.2.4)
Use /0' Bn(x)dx = 0 in (D.2.3) and (D.2.4) to obtain
ё -\ t
1 = A(t) or A(t) =
e' - 1
Thus
and
G(x, t) = = V Bn(x)-, (D.2.5)
e' — 1 *-^ n\
Bn(x) = forO < x < 1.
By the periodicity of Bn (jc) we arrive at
Bn(x) = -Bn(x-[x]) for all x.
л!
Although this formal argument has not been justified, it can be done easily since
the generating function (D.2.5) is analytic in the disk \t\ < 2л-. This proves the
theorem. ¦

D.3 Applications 621
D.3 Applications
Take f{x) = x~s logx, m = 1, and n = N in (D.1.3) to get
A Nl~slogN 1 Nl~s
-5 (S -IJ E-1J
n
Let Re s > 1 and N -> oo to obtain
f
/0
(s ~ IJ
The integral on the right side of the equality converges for Re 5 > — 1 by Abel's
test. So
-' - Г
л
By Corollary 1.3.3 we have f'@) = -ilog27r. Thus the constant (D.1.7) in
Stirling's formula is \ Iog2^.
As another application, we prove the following useful theorem on the order of
t,{s) and t,'(s) for large Im s.
Theorem D.3.1 Let s = a +it. For 1 — ^r- < a < 2, where A is any positive
constant, and \t\ large enough,
= o{\Og\t\)
and
);l{s) = O{\og1\t\),
Proof, In (D.1.3), set f(x) = x~s, m = N, and let n -> oo to arrive at
¦A 1 iV1^ 1 , f00 x - [x] - \
f^n- s - 1 2 JN xs+l
The integral is 0{\t\/oNa). Note that
where the last equality holds for n < \t\. Thus, take ^V = [|г|] to get
N П\
Ы = O(log|f|).

622
D Euler-Maclaurin Summation Formula
To derive the result for f '(s), take f(x) = x s logx and do a similar calculation.
See Rademacher [1973, p. 100], if necessary. ¦
We complete this section with a proof of the following theorem stated in
Chapter 1.
Theorem D.3.2 Let z e С - (-oo, 0]. Then
2 2m Jo
B2m(x - [x])
dx.
(x + zJm
Proof Start with the following expression of the gamma function:
z—1
r^=}™U
l=\
z + I - 1
Then
logT(z) = lim
z + l-\
i=\
(D.3.2)
where the principal branch of the log function in С — (—oo, 0] is chosen. In
Theorem D.2.1, take
to get
x + z — 1
f{x) = log = log(* + г - 1) - logx
-l
i=\
= logz + / [log(x + z-D-
j^ 2jBj - 1) [(n + z - lJJ~l n2;
+ -[log(n +z - 1) - logn - logz]
1 Г
2m Jo
B2m(x-[x])
1
(z+x-lJm x
1
~2^
dx.
Here we have used the fact that Bi = -1/2 and B2j+i = 0 for j > 1. Compute
the first of the above integrals and observe that after some cancellation the terms

D.4 The Poisson Summation Formula 623
that involve n are
(n + z - 1) log(n + z — 1) — n logn + - log
2 D n
+y
B2j Г i l
2jBj -
Subtract this from (z — l)log(n + l) and let и -> oo to compute the limit in (D.3.2).
The result is
log ГЙ - (z - I) .o8Z - г + . + ± JJ^-
l r°° г l il
- — / B2m(x - [x]) — -=- - -5- \dx. (D.3.3)
2ш У! [(x + z-lJm x2mj
From (D.1.5) and (D.3.1) we know that
lim [log Г(г) - (z - 1/2) log z + z] = - log 2л-.
г—>оо 2
So let г -> oo in (D.3.3) to see that
This result combined with (D.3.3) gives the formula in Theorem D.3.2. ¦
D.4 The Poisson Summation Formula
In this section we state and prove an important and useful formula from the theory of
Fourier series. This is the Poisson summation formula. It has numerous applications
though we mention only a few. One consequence of Poisson's formula is the Euler-
Maclaurin summation formula.
Start with a result on Fourier series attributed to Jordan. Recall that a function
f(x) is said to be of bounded variation on an interval [a, b] (which may be infinite),
if there is a constant С > 0 such that for any set of points xo < x\ < ¦ ¦ ¦ < xn in
the interval
k=\
If the interval is the whole real line, then we say that / is of bounded total variation.
An important though easily proved result is that if / is of bounded variation on
[a,b], then / = g — h, where g and h are increasing functions on [a,b].

624 D Euler-Maclaurin Summation Formula
Jordan's theorem is the following: Suppose / is integrable on [0, 1] and periodic
with period one. If / is of bounded variation on [0, 1], then
f(x+)+f(x-)
f(t)e~27lintdt )e2lTinx.
2
Poisson's summation formula is contained in the next theorem.
Theorem D.4.1 Suppose g is integrable on (—oo, oo) and of bounded total vari-
variation. Suppose h is an even, positive, integrable function that is decreasing on
[0, oo) and such that \g(x)\ < h(x). Then
Г
J-
(D.4.1)
Proof. The inequality
N-l
У] \g(x+n)\<\g(x)\+ / h(x+t)dt
n=-M
implies that Yl°?<x 8(x +л) is an absolutely convergent series and defines a periodic
function f(x) with period one. Moreover, since g is of bounded total variation, /
is of bounded variation on [0, 1]. By Jordan's theorem
f(x+) + f(x-)
= lim V I'
Now
-1 oo
/ f(t)e-27lir"dt= f
Jo Jo
= Yl I S(m + t)e-2nimdt
00 rm+l
= J2 / g{u)e-2*inudu
-oo Jm
= Г
J-c
The summation in the second line can be taken outside the integral by the dominated
convergence theorem. ¦

D.4 The Poisson Summation Formula 625
Remark D.4.1 If the series on the right-hand side in Theorem D.4.1 is absolutely
convergent, then we can write
An interesting example of Poisson's formula connects the Poisson kernel for the
upper half plane given by g(x) — y/(x2 + у2), у > 0, with the kernel for the unit
disk given by A — r2)/(l — 2r cosx + r2). Observe that with g as defined here,
-oo X* + у1
The reader may verify this by contour integration or by other means. Substitution
of this in Theorem D.4.1 gives
У V4 -2ny\n\ 2ninx _ 1-е
y2 + (x + лJ ^ 1 - 2е~2жУ cos 2тгх + е~ЛлУ '
n=—oo J v ' n=—oo
The partial fraction expansions for the trigonometric functions can also be de-
derived from Theorem D.4.1. Take s > 0 and set
je~s', t>0,
8it) = [O, t< 0.
The left side of Poisson's formula becomes
e-s(n+x) + f 1/2, x integer,
I 0, otherwise.
— x
The right-hand side is given by
E
oo e2ninx
s + 2nin
n=—oo
When x = 0, we get
es + \ 1 . ^ ( 2 + 1
es — 1 5 ^ \s + 2nin s — 2nin
By analytic continuation, we can take s = 2niy. The result is
1^/1 1
Trcot ny = —(- у 1-
У ^ы^у + п y~n

626 D Euler-Maclaurin Summation Formula
When x = 1/2, we obtain
es - 1 s ^-^ \s +2nin
n=l ч
This time, s = 2niy gives
7Г CSC TCy = - + У^(- 1)" ( 1
У ^Г[ \У+п y-
An important transformation formula for theta functions follows from the
Poisson summation formula by taking g(x) = e~S71*2. The formula is
oo
V^ e-nn2/se2ninx (D 4 2)
The Euler-Maclaurin formula follows from the Poisson summation formula by
taking
\ 0, otherwise.
Here a and b are assumed to be nonnegative integers and f(x) is continuously
differentiable in [a, b]. This implies that / is of bounded variation on [a, b] and
hence g{x) is of bounded total variation. Use this g(x) in (D.4.1). The result is
f
a
/b oo rb
f(t)dt +2^ f({)c°s2nntdt. (D.4.3)
n=l ^
k=a+l ^^ \n\sN
b oo rb
Integration by parts gives
fb fb , sin2rmt
/ f(t)cos2nntdt = - fit)— dt.
Ja Ja 2Л7Г
Also,
oo ¦ „
v-^ sin 2mcx
2^Bl(x[x]),
n=l
except when x is an integer (see Exercise 44 in Chapter 1). Thus, by the dominated
convergence theorem, we can write (D.4.2) as
Y, /(«)= / f(t)dt-Bl(f(b)-f(a))+ f f'(x)Bl(x-[x])dx.
a<n<b Ja Ja

Exercises 627
If we assume that f(m) (x) is continuously differentiable, then the application of
dBn{x)
— =nBn-i(x), 0<x<\,
ax
and successive integration by parts gives Theorem D.2.1. For other applications
and extensions of the summation formulas studied here, see Berndt [1975].
Exercises
1. Prove that with the notation and conditions of Theorem D.3.1 $'(s) =
0(log2 |f |).
2. Define A/@) = /A) — /@). Show that if f(x) is a polynomial of degree m,
then
/"' m В
/@) = / f(x)dx + Y] -Z д/У-^О).
Jo j=1 J
3. Use the Euler-Maclaurin formula to obtain the relation
where 0 < в < 1 and у is Euler's constant.
4. Show that there is a constant С such that
O(
V —!— =loglogx
t^« log n
loglogx + C + O(}.
2t^,« log n \x\ogxj
5. Use the Poisson summation formula to prove that
oo . oo
Ee-s(n+xJ _ V^ c-7in2/sc2ninx
Js ^
—oo —oo
6. Use Poisson summation to obtain the following theorem of Lipschitz: Suppose
Re x > 0, 0 < a < 1, and Re s > 1. Then
°° O2nina
(n+a)= у ?
(For a different proof where or = 1, see Exercise 37 in Chapter 2.)
7. Suppose a, b, and с are integers such that ac+b is even. Use Poisson summation
to prove the reciprocity for Gaussian sums
H-l I |a|-l
eni(an2+bn)/c _ / ?
E
V a
n=0 n=0

E
Lagrange Inversion Formula
E.I Reversion of Series
There are situations in analysis in which one knows a series expansion for y(x)
but would like to obtain the series for x in terms of 3;. Newton, for example,
encountered such a problem when he had the series for sin~' x by integrating the
expansion of A - x2)~1/2 term by term and he wanted the series for sinx. Some
results of Newton on reversion of series can be found in Newton [1960, p. 147].
Suppose a series for x in powers of у is required when x = уф (х). Assume that
ф is analytic in a neighborhood of x = 0 with ф @) ф 0. Then
= ^апх\ ахф0. (E.I.I)
«=1
We shall see below that Lagrange's inversion formula gives
n=\
where
1 dn~l
This means that nbn is the coefficient of x"~l in the expansion of ф"(х) or the
coefficient of x~l in the expansion of 1/y".
More generally, suppose that (E. 1.1) holds and that f(x) is an analytic function
in a neighborhood of x = 0. Then Lagrange's formula is
fix) = /@) + > /—
n=\
d
n~l
(E.I.2)
629

630 E Lagrange Inversion Formula
Б.2 A Basic Lemma
The proof of Lagrange's formula (E. 1.2) depends on a simple lemma given below.
The treatment here follows that of Gessel and Stanton [1982], which is based on
the work of Jacobi [1830]. See also Bromwich [1926]. An extension of Lagrange
inversion to several variables is in Good [1960] and ^-analogs are discussed in
Stanton [1988]. A history of this formula can be found in W. Johnson's unpublished
manuscript, Notes on the Lagrange inversion formula.
Define the residue of a formal Laurent series f{x) = YfjL-m aixJ ^У
Res[/(x)] = a-\. The next lemma states that the residue does not change un-
under a certain change of variables.
Lemma E.2.1 Let G(x) = ?°L_m ajXJ and h(x) = ]P~i Ьх'> where bi Ф °-
Then Res[G(h(x))h'(x)] = Res[G(x)].
Proof. Since both sides of the equation are linear in G, it is sufficient to prove it
for G(x) = xm. Since Res[g'(x)] = 0 for any Laurent series g(x), it follows that,
form ф — 1,
Res[hm(x)h'(x)] = —^—Res[{hm+1(x)}'] = 0.
m + 1
For m = -\, let h(x) = bxxf(x). Then
Res f^l = Res \- + ^1 = 1 + Res[{log f(x)}'] = 1.
[h(x)j [x f(x)\
The lemma is proved. ¦
To prove (E.1.2) note that у = axx + агх1 + ¦ ¦ ¦ (x a complex variable) is
conformal in a neighborhood of 0, because a\ ф 0, and so x can be expanded in
powers of y. Since f{x) is analytic, it can be expanded in powers of y. Suppose
that
n=\
Then
fix) =
LetG(x) = T,T=incnX"~]- By Lemma E.2.1,
\Г(х)фг(х)] \f(x)] \G(y)dy/dx] \G(x)]
Res = Res = Res = Res = rcr.
L xr J L У J L У ] L ^r J

E.3 Lambert's Identity 631
But the Taylor series of /'(х)фг(х) is
and so
Thus
which proves Lagrange's formula (E.I.2).
A variant of (E.I.2) is the formula
x=0
where 3; is defined by (E. 1.1). To prove (E.2.1), note that since x — уф(х) = 0,
we have
(E.2.2)
or
dy = 1 - уф'(х)
dx ф(х)
Take the derivative of (E. 1.2) with respect to x and use (E.2.2) to get
Set F(x) = /'(х)ф(х) to obtain (E.2.1).
E.3 Lambert's Identity
Lagrange's inversion formula can be used to obtain a number of interesting series
expansions for specific functions. A few examples are given in the exercises. Here
we derive an identity of Lambert. This is given by
(ЕЗЛ)
where 3; = x{\ +x)~P. Take f(x) = A + x)a апйф(х) = A + xf in (E.I.2).
Then
v" Г dn~x
n=l
x=0

632 E Lagrange Inversion Formula
The expression in square brackets equals
a(nP + a - 1) ¦ ¦ ¦ (и/3 + a - n + 1) = a(n - 1)!,
n — 1
and this gives (E.3.1). The identity corresponding to (E.2.1) is
An interesting binomial identity comes from (E.3.1) and (E.3.2). Equate the co-
coefficient of y" on both sides of
(
The result is
^ у {у+рк\{а + Р(пк)\ fa + y+Pn\
A ){ ) = { n У (R33)
E.4 Whipple's Transformation
Gessel and Stanton showed how to derive Whipple's transformation, which we
obtained in different ways in Chapters 3 and 7, from Lemma E.2.1.
Theorem E.4.1 Suppose A(x), B{x), C(x), and D(x) are power series whose
coefficients ofxk are At, Дь С*, and D^ respectively. Suppose
= B(x),
- x)-yC(x/(l - x)P+l) = D(x). (E.4.2)
Ifn(P + l)= 1 -a-y, then
k=0 k=0
Proof. First note that
Res , = coefficient of x" in B(x)D(x) = V^ BkDn^k.
L x J
L x J k=o
Now write h(x) = x(l - x)~^1 so that h'(x) = A + /?*)A - x)"^ and
B(x)D(x) A +/?лс)A - jc)-
1 A(h(x))C(h(x))
A{h{x))C{h{x))h'{x)

E.4 Whipple's Transformation 633
By Lemma E.2.1,
-B(x)D(x)] \A(x)C(x)
Res — — Res'
Kn+x j у xn+i
By the remark at the beginning of the proof, this implies that
*=0 k=0
This proves the theorem. ¦
To derive Whipple's transformation for a jF^, recall Whipple's quadratic trans-
transformation noted in Chapter 2, namely,
'a/2,(l+a)/2,l+a-b-c 4x
(I — X) з/-2
l+a-b,l+a-c ' A-хJ,
h r \
;x). (E.4.3)
l +a — b, \ +a — с J
This corresponds to (E.4.1) with a = a and ft = 1. The transformation corres-
corresponding to (E.4.3) is a result of Bailey [1929]:
+ l-b-c4x
a, I + a/2, b, с
~4 3[a/2,l+a-b,l+a-c'X
l+a-b,l+a-c A-хJ
(E.4.4)
This can be obtained from (E.4.3) by differentiation or directly by equating the
coefficients of x". To apply Theorem E.4.1, note that from (E.4.3),
Ak = (fl/2)t((l + fl)/2)t(l +a-b- с)к(-4)к/[к\A +а- b)k(l +a- c)k],
Bk = (а)к(Ь)к(с)к/[к\(\ + a - b)k(l + a - c)k],
and from (E.4.4) after renaming the parameters,
Ck = (A + d)/2)k(B + d)/2)k(l +d-e- /M-4)V
[k\(l+d-e)k(l+d-f)k],
Dk = (d)k(l + d/2)k(e)k(f)k/[k\ (d/2)k(l +d- e)k(\ +d- f)k].
So
+d - e -
•E:
n\(l + d - e)n(l + D - f)n
(a/2)k((\ + a)/2)k(l +a-b- c)k(-n)k(e -d- n)k(f - d - n)k
k\(\ +a- b)k(\ + a ~ c)*((l - d)/2 - n)k(-n - d/2)k(e + f - n - d\

634 E Lagrange Inversion Formula
Theorem E.4.1 is applicable when 2n = — a - d. This simplifies the expression in
the sum. For example, — n — d/2 = a/2, and the terms involving these expressions
cancel. After simplification the sum becomes
/ — n, 1 + a — b — c, e + a + n, f + a + n
43V l+a-b, l+a-c, e +f + a+n '
The sum Y^k=o ^k^n-k leads to a iF^ and the equality
n n
fc=0 k=0
results in Whipple's theorem
/ a, l + a/2, b, c, e + a + n, f + a+n, -n
1 6\a/2,l+a-b, l+a-c, 1-е- n, I- f-n, 1+a+n'
_ (fl+ \)n(a + с + f + n)n
/l+a-b-c,e + a + n,f + a+n,-n\
X4 3\ l+a-b, l+a-c, e +f + a + n ' )'
Exercises
1. Show that
n=0
and that
00 „л-1
n=\
2. Show formally that
n
Give sufficient conditions for this to be correct.
3. Show that
{хкЪ)
k\
k=0
4. Show that if f~l(x) is the inverse of /(x) and /@) = 0, then assuming the
necessary analyticity of the functions,

Exercises 635
5. Show that
when у = x(l + x)~P.
6. Suppose that xm+1 + ax - b = 0. Show that
2m + 2 fr2m+1 Cwi + 2№m + 3) b3m+l
X ~ a am+2 + 2! a2m+3 3! a3m+4 + '
Use this formula to find a solution of x5 + 4x + 2 = 0 to four decimal places of
accuracy. When m = 0, this series reduces to the geometric series. Write this
sum as a hypergeometric series.
7. Use the Lagrange inversion formula to derive the generating function for
Laguerre polynomials.

Series Solutions of Differential Equations
F.I Ordinary Points
For readers not familiar with series solutions of differential equations, we give
a few basic definitions so that the discussions of the hypergeometric equation in
Chapter 2 and of the confluent and Bessel's equations in Chapter 4 are intelligible.
Consider the differential equation
^ + c(x)yO (F.I.I)
dx
with a(x), b{x), and c(x) analytic in the neighborhood of x = x0. We take x to be a
complex variable in this discussion. The simplest situation occurs when a (x0) ф 0.
In this case, x0 is called an ordinary point of the Equation (F.I.I).
It is not difficult to show that if xo is an ordinary point of (F.I.I), then (F.I.I)
has a unique solution /(x) analytic in a neighborhood of x0 with prescribed values
/(x0) = /o and /'(xo) = f\. This implies that there are exactly two linearly inde-
independent solutions in a neighborhood of x0. To prove this result, it is convenient to
divide (F.I.I) by a(x) and rewrite the equation as
y" = B{x)y' + C(x)y. (F.I.2)
Again for convenience, take x0 = 0. Then
oo
and С(х) = ^с„х".
л=0 п=0
Suppose that (F.1.2) has an analytic solution /(x) with
Then
л=0
637

638 F Series Solutions of Differential Equations
Substitute this series and the series for B(x) and C(x) in (F.I.2); equate the coeffi-
coefficients of x". The result is
(и + 2) (и + 1)/я+2 = 5> + 1 - *)/»+1-*** + ? ^«-*С*' (RL4)
<fc=0 *=0
и = 0, 1, 2, This shows that /0 and /i uniquely determine /„ for n > 2. It is
now enough to prove that the series (F.1.3) has a positive radius of convergence.
Since the series for B(x) and C(x) have a positive radius of convergence, there
exist constants M and R such that
|&Л|<М/Л" and \cn\<M/Rn.
We show by induction that there exist suitable positive numbers M\ and r such
that
\fn\<Ml/rn. (F.1.5)
This implies the needed result. Take M\ such that \f\\ < Mi and choose r such
that r < /?, |/i| < Mi/r, and M(r/2 + r2) < 1. Suppose n > 2 and assume
(F.1.5) true up to n — 1. By (F.1.4), the inequality (F.1.5) follows after a small
calculation.
F.2 Singular Points
Whena(xo) = 0 and b(x0) and/or c(x0) is not zero, then x = x0 is called a singular
point of (F.I.1). Divide (F.I.I) by a(x) and write it as
y" + d(x)y' + e(x)y = 0. (F.2.1)
Consider first the simplest case where xo is a singular point and a(x) has a simple
zero at xo. Then d(x) and e(x) have at most simple poles. Take xo = 0. Then
oo oo
d(x) = J2 dnx" and e(x) = ^ enx". (F.2.2)
л=-1 л=-1
Substitute the series (F. 1.3) in (F.2.1) and equate coefficients as before. The reader
may check that in this case we get only one solution, since the value of /0 determines
/„ for n > 1. Moreover, if d-\ is a nonnegative integer, then this method may fail
to produce a solution.
Now consider the case where d{x) has at most a simple pole and e(x) at most
a double pole. The simplest special case of (F.I. 1) that leads to this is when
a(x) = a2(x - x0J, b{x) = bi(x - x0), c(x) = c0. (F.2.3)

F.3 Regular Singular Points 639
Two linearly independent solutions exist, at least one of the form у = (x — Хо)д.
To determine д, substitute this expression for у in (F.I.I) to get
а21лAл-1) + Ь11л + с0 = 0. (F.2.4)
If this quadratic has two unequal roots, these roots give two independent solutions
of (F.I.I). If the two roots equal Д] (say), then we have the solution (x — хо)Д|.
To find the other independent solution, set у = (x — xq)^w. The differential
equation for w has log(x — x0) as a solution. Thus the second independent solution
is (x - х0У log(x - x0).
F.3 Regular Singular Points
A point x = xo is called a regular singular point of (F.I. 1) if
(x - xo)b(x) (x - xoJc(x)
lim and hm
x^x0 a(x) x-*xo a(x)
both exist. If one of these limits does not exist, the singular point is irregular.
To see the difference between a regular and an irregular singular point, the
easiest case to consider is the first-order analog of (F. 1.1). A regular singular point
occurs when
b(x)y' + c(x)y = 0 (F.3.1)
with b(x) and c(x) analytic in a neighborhood of x = x0, and
lim (x - xo)c(x)/b(x)
x^x0
exists.
When b(x) = (x — x0J and c(x) = c0, then
y(x) = Ae-^*-^
is the solution, and it has an essential singularity at x = xo.
The simple case considered in the previous section, where a (x), b (x), с (х) are as
in (F.2.3), shows that, at a regular singular point, solutions may involve noninteger
powers of x —x0. Moreover, the second solution may have a logarithmic singularity.
Now suppose that Xo is a regular singular point of (F.I. 1) such that a(x) has a
double zero at x0. Then
п{х-х^)п, and с(х)
л=2 л=1 л=0

640 F Series Solutions of Differential Equations
Suppose у = (x — хо)д/(х) is a solution of (F.I.1). The equation for /(x) can be
seen to be
a(x)f" +
V n=l
oo
" - Xo)" + H(JI ~ 1)
n=0
Now if [L is chosen so that
ц(ц - \)a2 + fj-bi + c0 = 0, (F.3.3)
then (F.3.1) is an equation of the same type as the one considered at the beginning
of Section F.2. In that case a solution of the form /(x) = YlT=o fnix ~ xo)" is
always possible provided 2д + Ь\/а2 is not a nonnegative integer.
Note that the quadratic (F.3.2) has two solutions Д] and д2- Thus it may be
possible to obtain two solutions of the form (x — хо)д/(х). It follows from (F.3.2)
that
Д! -Д2 = 2/Л -(Д! +Д2)
= 2m-(l-bi/a2)
The remarks in the previous paragraph now imply that if the difference of the two
roots of (F.3.2) is not an integer, then (F. 1.1) has two independent solutions of the
form Y^=o fn(x — -Хо)"+Д- However, if /л\ — \i2 > 0 is an integer, then a solution
of this form with \i = \i\ exists. The other solution may involve a logarithmic
singularity.
Equation (F.3.2) is called the indicial equation. In practice it is obtained by
substituting the series Е/„(х — хо)"+д in (F.I. 1) and equating the coefficients of
the lowest power of x - x0.
For the hypergeometric equation B.3.5), there are three regular singular points
0, 1, and infinity. For the confluent equation D.1.3) and the Bessel equation D.5.1),
0 is a regular singular point, and infinity is an irregular singular point.

Bibliography
Abel, N. H. [1826]. Auflosung einer mechanischen Aufgabe, J. Reine Ang. Math., 1,
153-157. English translation in Source Book in Mathematics, D. E. Smith, ed.,
McGraw-Hill, New York, 656-662.
Adams, J. С [1877]. On the expression of the product of any two Legendre coefficients by
means of a series of Legendre's coefficients, Proc. R. Soc. London, 27, 63-71.
Agarwal, R. P. [1953]. On the partial sums of series of hypergeometric type, Proc.
Cambridge Phil. Soc, 49, 441^45.
Ahern, P. and Rudin, W. [1996]. Geometric properties of the gamma function, Am. Math.
Monthly, 103,678-681.
Al-Salam, W. A. and Ismail, M. E. H. [1988]. g-Beta integrals and the g-Hermite
polynomials, Рас. J. Math., 209-221.
Amend, B. [1996]. Fox Trot, Comic strip, June 2.
Anastassiadis, J. [1964]. Definition des Functions Euliriennes par des Equations
Functionnelles, Gauthier-Villars, Paris.
Anderson, G. W. [1990]. The evaluation of Selberg sums, Сотр. Rend. Acad. Sci., Paris,
311, Serie 1, 469^72.
Anderson, G. W. [1991]. A short proof of Selberg's generalized beta formula, Forum
Math., 3,415-417.
Andrews, G. E. [1976]. The Theory of Partitions, Addison-Wesley, Reading, MA.
Andrews, G. E. [1986]. q-Series: Their Development and Application in Analysis, Number
Theory, Combinatorics, Physics, and Computer Algebra, Amer. Math. Soc,
Providence, RI.
Andrews, G. E. [1994]. The death of proof? Semi-rigorous mathematics? You've got to
be kidding!, Math. Intelligencer, 16 D), 16-18.
Andrews, G. E. and Askey, R. [1977]. Enumeration of partitions: The role of Eulerian
series and ^-orthogonal polynomials, in Higher Combinatorics, M. Aigner, ed.,
Reidel, Dordrecht, 3-26.
Andrews, G. E. and Burge, W. H. [1993]. Determinant identities, Рас. J. Math., 158, 1-14.
Andrews, G. E., Dyson, F. J., and Hickerson, D. [1988]. Partitions and indefinite quadratic
forms, Invent. Math., 91, 391^07.
Anno, M. and Mori, T. [1986]. Socrates and the Three Little Pigs, Philomel Books, New
York.
Aomoto, K. [1987]. Jacobi polynomials associated with Selberg integrals, SIAMJ. Math.
Anal, 18,545-549.
Artin, E. [1964]. The Gamma Function, Holt, Rinehart and Winston, New York.
641

642 Bibliography
Askey, R. [1970]. Orthogonal polynomials and positivity, Studies in Applied Mathematics
6, in Wave Propagation and Special Functions, D. Ludwig and F. W. J. Olver, eds.,
SIAM, Philadelphia, 68-85.
Askey, R. [1975]. Orthogonal Polynomials and Special Functions, SIAM, Philadelphia.
Askey, R. [1987]. Ramanujan's i i^i and formal Laurent series, Indian J. Math., 19,
101-105.
Askey, R. [1988]. Beta integrals and ^-extensions, Papers of the Ramanujan Centennial
International Conference, Ramanujan Math. Soc, 1987, 85-102.
Askey, R. [1994]. A look at the Bateman project, Contemporary Math., 169, 29-43.
Askey, R. and Gasper, G. [1971]. Jacobi polynomial expansions of Jacobi polynomials
with non-negative coefficients, Proc. Cambridge Phil. Soc, 70, 243-255.
Askey, R. and Gasper, G. [1977]. Convolution structures for Laguerre polynomials,
J. Analyse Math., 31, 48-68.
Askey, R., Gasper, G., and Ismail, M. [1975]. A positive sum from summability theory, J.
Approx. Theory, 13, 413^120.
Askey, R., Koornwinder, Т., and Rahman, M. [1986]. An integral of products
of ultraspherical functions and a g-extension, J. London Math. Soc, 33 B), 133-148.
Askey, R. and Steinig, J. [1974]. Some positive trigonometric sums, Trans. Am. Math.
Soc, 187, 295-307.
Askey, R. and Wilson, J. A. [1984]. A recurrence relation generalizing those of Apery, J.
Austral. Math. Soc. (Series A) 36, 267-278.
Askey, R. and Wilson, J. A. [1985]. Some Basic Hypergeometric Orthogonal Polynomials
that Generalize Jacobi Polynomials, Am. Math. Soc., Providence, RI.
Auslander, L. and Tolimieri, R. [1979]. Is computing with finite Fourier transform pure or
applied mathematics?, Bull. Am. Math. Soc. (New Series), 11, 847-897.
Azor, R., Gillis, J., and Victor, J. D. [1982]. Combinatorial applications of Hermite
polynomials, SIAM J. Math. Anal, 13, 879-890.
Baernstein II, A., Drasin, D., Duren, P., and Marden, A. [1986]. The Bieberbach
Conjecture, Amer. Math. Soc. Surveys and Monographs No. 21, Providence, RI.
Bailey, W. N. [1929]. Some identities involving generalized hypergeometric series, Proc.
London Math. Soc, 26 B), 503-516.
Bailey, W. N. [1931]. The partial sum of the coefficients of the hypergeometric series, J.
London Math. Soc, 6, 40-41.
Bailey, W. N. [1932]. On one of Ramanujan's theorems, J. London Math. Soc, 7, 34-36.
Bailey, W. N. [1933]. On the product of two Legendre polynomials, Proc. Cambridge
Phil. Soc, 29, 173-177.
Bailey, W. N. [1935]. Generalized Hypergeometric Series, Cambridge University Press,
Reprinted by Hafner Pub. Co., New York, 1972.
Bailey, W. N. [1938]. The generating function of Jacobi polynomials, J. London Math.
Soc, 13, 8-11.
Bailey, W. N. [1941]. A note on certain ^-identities, Quart. J. Math. (Oxford), 18,
157-166.
Bailey, W. N. [1949]. Identities of the Rogers-Ramanujan type, Proc. London Math. Soc,
50B), 1-10.
Bailey, W. N. [1954]. Contiguous hypergeometric functions of the type 3^A), Proc.
Glasgow Math. Assoc, 2, 62-65.
Bak, J. and Newman, D. J. [1982]. Complex Analysis, Springer-Verlag, New York.
Barnes, E. W. [1908]. A new development of the theory of the hypergeometric functions,
Proc. London Math. Soc, 6 B), 141-177.
Barnes, E. W. [1910]. A transformation of generalized hypergeometric series, Quart. J.
Math., 41, 136-140.
Bateman, H. [ 1909]. The solution of linear differential equations by means of definite
integrals, Trans. Cambridge Phil. Soc, 21, 171-196.

Bibliography 643
Bateman, H. [1932]. Partial Differential Equations of Mathematical Physics, Cambridge
University Press, Cambridge, UK.
Bellman, R. [1961]. A Brief Introduction to Theta Functions, Holt, Rinehart, and Winston,
Inc., New York.
Berggren, L., Borwein, J., and Borwein, P. [1997]. Pi: A Source Book, Springer-Verlag,
New York.
Berndt, B. [1975]. Periodic Bernoulli numbers, summation formulas and applications, in
Theory and Application of Special Functions, R. Askey, ed., Academic Press, New
York.
Berndt, B. [1985]. Ramanujan'sNotebooks, Parti, Springer-Verlag, New York.
Berndt, B. [1989]. Ramanujan's Notebooks, Part II, Springer-Verlag, New York.
Beukers, F. [1979]. A note on the irrationality of f B) and f C), Bull. London Math. Soc,
11, 2268-2272.
Boas, R. P. [1954]. Entire Functions, Academic Press, New York.
Bochner, S. [1952]. Remarks on Gaussian sums and Tauberian theorems, J. Indian Math.
Soc, 15, 99-106.
Bochner, S. [1955]. Harmonic Analysis and the Theory of Probability, University of
California Press, Berkeley.
Bochner, S. [1979]. Review of Gesammelte Schriften by Gustav Herglotz, Bull. Am.
Math. Soc. (New Series), 1, 1020-1022.
Bohr, H. and Mollerup, J. [1922]. Laerebog i Matematisk Analyse, Vol. Ill, J. Gjellerup,
Kopenhagen.
Borwein, J. and Borwein, P. [1987]. Pi and the AGM, Wiley, New York.
Boyarsky, M. [1980]. p-adic gamma functions and Dwork cohomology, Trans. Am. Math.
Soc, 257, 359-369.
Brafman, F. [1951]. Generating functions of Jacobi and related polynomials, Proc Am.
Math. Soc, 2, 924-949.
Brent, R. P. [1976]. Fast multiple-precision evaluation of elementary functions, J. Assoc.
Сотр. Mech., 23, 242-251.
Bressoud, D. M. [1987]. Almost poised basic hypergeometric series, Proc. Indian Acad.
Sci. (Math./Sci.), 97, 61-66.
Bromwich, T. J. ГА. [1926]. An Introduction to the Theory of Infinite Series, 2nd ed.,
Macmillan, London.
Brown, G. and Hewitt, E. [1984]. A class of positive trigonometric sums, Math. Ann., 268,
91-122.
Brown, M. and Ismail, M. E. H. [1995]. A right inverse of the Askey-Wilson operator,
Proc Amer. Math. Soc, 123, ^071-2079.
Brualdi, R. A. and Ryser, H. J. [1991]. Combinatorial Matrix Theory, Cambridge
University Press, New York.
Bustoz, J. [1974]. Note on "Positive Cesaro means of numerical series," Proc. Amer.
Math. Soc. ,45, 69.
Caratheodory, C. [1954]. Funktionentheorie, English translation, Theory of Functions,
Vols. I and II, [1958, 1960], Chelsea, New York.
Cartier, P. and Foata, D. [1969]. Problimes Combinatoires de Commutation et
Rearrangements, Lecture Notes in Math., No. 85, Springer-Verlag, Berlin.
Cauchy, A.-L. [1843]. Memoire surles fonctions dont plusieurs valeurs..., C.R. Acad.
Sci. Paris, 17, 523, reprinted in Oeuvres de Cauchy, [1893], Ser. 1, 8, 42-50.
Cauchy, A.-L. [1843a]. Deuxieme Memoir sur les fonctions dont plusieurs valeurs...,
C.R. Acad. Sci. Paris, reprinted in Oeuvres de Cauchy, [1893], 8, 50-55.
Chebyshev, P. L. [1870]. Sur les fonctions analogues a celles de Legendre, in Oeuvres de
P. L Tchebychef, Vol. 2, A. Markoff and N. Sonin, eds., St. Petersburg [1907],
61-68, reprinted Chelsea, New York [1961].

644 Bibliography
Chihara, J. S. [1978]. An Introduction to Orthogonal Polynomials, Gordon & Breach,
New York.
Clausen, T. [1828]. Ueber die Falle, wenn die Reihe von der Form у = 1 + f • &x + 2^±i
¦^ц\х2+ etc. ein quadrat von der Form z=\ + j'^p'7x + " °2+' ' ^v+T
¦jr^x2+ etc. hat, J. Reine Ang. Math., 3, 89-91.
Cooley, J. W. and Tukey, J. W. [1965]. An algorithm for the machine calculation of
complex Fourier series, Math. Сотр., 19, 297-301.
Copson, E. T. [1935]. Theory of Functions of a Complex Variable, Oxford Univ. Press,
London.
Cox, D. [1984]. The arithmetic-geometric mean of Gauss, Enseign. Math., 30, 270-330.
Davenport, H. and Hasse, H. [1934]. Die Nullstellen der Kongruenzzetafunktion in
gewissen zyklischen Fallen, J. Reine Ang. Math., 172, 151-182.
de Boor, С [1980]. FFT as nested multiplication, with a twist, SIAM J. Sci. Stat. Сотр., 1,
173-178.
de Branges, L. [1972]. Gauss spaces of entire functions, J. Math. Anal. Appl., 37, 1—41.
de Branges, L. [1972a]. Tensor product spaces, J. Math. Anal. App., 38, 109-148.
de Branges, L. [1985]. A proof of the Bieberbach conjecture, Ada Math., 154, 137-152.
de Bruijn, N. G. [1967]. Uncertainty principles in Fourier analysis, in Inequalities, O.
Shisha, ed., Academic Press, New York,
de Bruijn, N. G., Saff, E. В., and Varga, R. S. [1981]. On the zeros of generalized Bessel
polynomials, Proc. K. Ned. Akad. Wet., Series A, 84 A), 1-25.
de Moivre, A. [1730]. Miscellanea Analytica de Seriebus et Quadraturis, Tonson and
Watts, London.
Dedekind, R. [1853]. Uber ein Eulersches Integral, J. Reine Ang. Math.
DeSainte-Catherine, M. and Viennot, G. [1983]. Combinatorial interpretation of integrals
of products of Hermite, Laguerre and Tchebycheff polynomials, Polynomes
Orthogonaux et Applications, С Brezinski et al., eds., Lecture Notes in Math. 1171,
Springer-Verlag, 120-128.
Din, A. M. [1981]. Lett. Math. Phys., 5, 207.
Dixon, A. C. [1903]. Summation of a certain series, Proc. London Math. Soc, 35 A),
285-289.
Dougall, J. [1907]. On Vandermonde's theorem and some more general expansions, Proc.
Edinburgh Math. Soc, 25, 114-132.
Dougall, J. [1919]. A theorem of Sonine in Bessel functions, with two extensions to
spherical harmonics, Proc. Edinburgh Math. Soc, 37, 33^7.
Edwards, С. Н. [1979]. The Historical Development of Calculus, Springer-Verlag, New
York.
Edwards, A. W F. [1987]. Pascal's Arithmetical Triangle, Charles Griffin and Co.,
London.
Eisenstein, G. [1847]. Genaue Untersuchung der unendlichen Doppelproducte, J. Reine
Ang. Math., 35, 153-274.
Elliot, E. B. [1904]. A formula including Legendre's EK' + KE' - KK' = \it,
Messenger of Math., 33, 31^Ю.
Erdelyi, A. [1937]. Der Zusammenhang zwischen verschiedenen Integraldarstellungen
hypergeometrischer Funktionen, Quart. J. Math. (Oxford), 8, 200-213.
Erdelyi, A. [1938]. Note on the transformation of Eulerian hypergeometric functions,
Quart. J. Math. (Oxford), 9, 129-134.
Erdelyi, A. [1939]. Transformation of hypergeometric integrals by means of fractional
integration by parts, Quart. J. Math. (Oxford), 9, 176-189.
Erdelyi, A. [1953]. Higher Transcendental Functions, Vols. I, II, III, A. Erdelyi, ed.,
McGraw-Hill. Reprinted by Krieger Publishing Co., Malabar, FL [1981].
Erdelyi, A. [1956]. Asymptotic Expansions, Dover, New York.

Bibliography 645
Erdelyi, A. [1960]. Asymptotic solutions of differential equations with transition points or
singularities, J. Mathematical Phys., 1, 16-26.
Euler, L. [1730]. De progressionibus transcendentibus sen quaroum termini generales
algebrare dari nequeunt, Comm. Acad. Sci. Petropolitanae, 5, 36-57.
Euler, L. [1739]. De productis ex infinitis factoribus ortis, Comm. Acad. Sci.
Petropolitanae, 11, 3-31. Reprinted in Opera Omnia, 14 [1924], 260-290.
Euler, L. [1748]. Introductio in Analysin Infinitorum, Marcum-Michaelem Bousquet,
Lausannae. English translation published by Springer-Verlag, [1988].
Euler, L. [1769]. Institutiones Calculi Integralis, II, Opera Omnia, Ser. 1, Vols. 11-13.
Euler, L. [1794]. Specimen transformationis singularis serierum, Nova Ada Acad. Sci.
Petropolitanae, 12, 58-70. Reprinted in Opera Omnia, Ser. 1, Vol. 16, Part 2, 41-55.
Evans, R. [1981]. Identities for Gauss sums over finite fields, Enseign. Math., 27 B),
197-209.
Evans, R. [1991]. The evaluation of Selberg character sums, Enseign. Math., 37 B),
235-248.
Fejer, L. [1925]. Abschatzungen fur die Legendreschen und verwandte Polynome,
Math. Zeit., 24, 267-284.
Feldheim, E. [1941]. Contribution a la theorie des polynomes de Jacobi (Hungarian,
French summary), Mat. Fiz. Lapik, 48, 453-504.
Feldheim, E. [1941a]. Sur les polynomes generalises de Legendre, Izv. Akad. Nauk. SSSR
Ser. Math., 5,241-248.
Feldheim, E. [1943]. Contributi alia teoria della funzioni ipergeometriche di piu variabili,
Annali della Scuola Norm. Super, di Pisa, Ser. II, 12, 17-60.
Feldheim, E. [1963]. Appendix by G. Szego. On the positivity of certain sums of
ultraspherical polynomials, J. Anal. Math., 11, 275-284.
Ferrers, N. M. [1877]. An Elementary Treatise on Spherical Harmonics and Subjects
Connected with Them, Macmillan, London.
Fields, J. L. [1964]. A note on the asymptotic expansion of a ratio of gamma functions,
Proc. Edinburgh Math. Soc, 15, 43^45.
Fine, N. J. [1988]. Basic Hypergeometric Series and Applications, Amer. Math. Soc,
Providence, RI.
Foata, D. [1965]. Etude algebrique de certains problemes d'analyse combinatoire et du
calcul des probabilites, Publ. Inst. Stat. Univ. Paris, 14, 81-241.
Forrester, P. J. and Rogers, J. B. [ 1986]. Electrostatics and the zeros of the classical
polynomials, SIAM J. Math. Anal, 17, 461^68.
Frobenius, F. G. [1871]. Uber die Entwicklung analytischer Functionen in Reihen, die
nach gegebenen Functionen fortschreiten, J. Reine Ang. Math., 73, 1-30. Reprinted
in Gesammelte Abhandlungen, Band I, Springer-Verlag, [1968], 35-64.
Funk, P. [1916]. Beitrage zur Theorie der Kugelfunktionen, Math. Ann., 77, 136-152.
Fuss, N. [1843]. Correspondance Mathematique et Physique de Quelques Cilebres
Geometres du XVIIF™ siecle, 1, Saint-Petersbourg.
Gasper, G. [1975]. Positive integrals of Bessel functions, SIAMJ. Math. Anal, 6, 868-881.
Gasper, G. [1977]. Positive sums of the classical orthogonal polynomials, SIAMJ. Math.
Anal, (8), 423^47.
Gasper, G. [1985]. Rogers' linearization formula for the continuous g-ultraspherical
polynomials and quadratic transformation formulas, SIAMJ. Math. Anal, 16,
1061-1071.
Gasper, G. and Rahman, M. [1990]. Basic Hypergeometric Series, Cambridge University
Press, Cambridge, UK.
Gauss, C. F. [1808]. Summatio quarumdam serierum singularium, Commen. Soc. Reg. Sci.
Getting. Rec, vol. I. German translation in Arithmetische Untersuchungen, Chelsea,
New York [1981].

646 Bibliography
Gauss, С. F. [1812]. Disquisitiones generates circa seriem infinitam, Comm. Soc. Reg
Gott. II, Werke, 3, 123-162.
Gauss, С F. [1866]. Zur Theorie der neuen Transscendenten, II, Werke, 3, 436-445.
Gauss, C. F. [1866a]. Hundert Theoreme tiber die neuen Transscendenten, Werke, 3,
461^69.
Gauss, С F. [1866b]. Arithmetisch Geometrisches Mittel, Werke, vol. 3, 361^03.
Gautschi, W. [1967]. Computational aspects of three-term recurrence relations, SIAM
Review, 9, 24-82.
Gegenbauer, L. [1875]. Ueber einige Bestimmte Integrale, Sitz- Math. Natur. Klasse Akad.
Wiss. Wien, 70, 433-443.
Gessel, I. and Stanton, D. [1982]. Strange evaluations of hypergeometric series, SIAM J.
Math. Anal, 11, 295-308.
Godsil, С D. [1981]. Hermite polynomials and a duality relation for the matching
polynomial, Combinatorica, 1, 257-262.
Godsil, С D. [1993]. Algebraic Combinatorics, Chapman and Hall, New York.
Good, I. J. [1960]. Generalization to severable variables of Lagrange's expansion, with
applications to stochastic processes, Proc. Cambridge Phil. Soc, 56, 367-380.
Good, I. J. [1962]. A short proof of MacMahon's "master theorem," Proc. Cambridge
Phil. Soc, 58, 160.
Good, I. J. [1970]. Short proof of a conjecture of Dyson, J. Math. Phys., 11, 1884.
Gosper, R. W., Jr. [1978]. Decision procedure for indefinite hypergeometric summation,
Proc. Natl. Acad. Sci. USA, 75, 40-42.
Gould, H. W. and Hsu, L. С [1973]. Some new inverse series relations, Duke Math. J., 40,
885-891.
Gray, J. [1986]. Linear Differential Equations and Group Theory from Riemann to
Poincare, Birkhauser, Boston.
Gronwall, T. H. [1912]. Ueber die Gibbssche Erscheinung und die trigonometrischen
Summen sinx + 1 /2 sin 2x + 1 /3 sin 3x-\ h 1 /n sinnx, Math. Ann., 72, 228-243.
Gross, B. and Koblitz, N. [1979]. Gauss sums and the />-adic gamma function, Ann.
Mart., 109, 569-581.
Halmos, P. [1950]. Measure Theory, Van Nostrand, Princeton.
Hamburger, H. [1922]. Ueber einige Beziehungen, die mit der Funktionalgleichung der
Riemannschen f -Funktion aequivalent sind, Math. Ann., 85, 129-140.
Hankel, H. [1875]. Bestimmte Integrale mit Cylinderfunctionen, Math. Ann., 8, 453-470.
Hardy, G. H. [1922]. A new proof of the functional equation for the zeta function, Mat.
Tidsskrift,B,l\-73.
Hardy, G. H. [1933]. A theorem concerning Fourier transforms, J. London Math. Soc, 8,
227-231.
Hardy, G. H. [1940]. Ramanujan, Cambridge University Press, Cambridge, UK.
Hardy, G. H. [1949]. Divergent Series, Cambridge University Press, Cambridge, UK.
Hecke, E. [1918]. Ueber orthogonal-invariante Integralgleichungen, Math. Ann., 78,
398^04.
Heckman, G. and Schlicktkrull, H. [1994]. Harmonic Analysis and Special Functions on
Symmetric Spaces, Academic Press, San Diego.
Heine, E. [1847]. Untersuchungen iiber die Reihe..., J. Reine Ang. Math., 34, 285-328.
Helversen-Pasotto, A. [1978]. L'identite de Barnes pour les corps finis, C. R. Acad. Sci.
Paris, Ser. A-B, 286, A297-A300.
Helversen-Pasotto, A. and Sole, P. [1993]. Barnes' first lemma and its finite analogue,
Canadian Math. Bull, 36, 273-282.
Hermite, С [1890]. Sur les Polynomes de Legendre, Rend. Circ. Mat. Palermo, IV,
146-152. Reprinted in Oeuvres, 4, 314-320.

Bibliography 647
Hewitt, E. [1954]. Remark on orthonormal sets in Ьг(а, b), Amer. Math. Monthly, 61,
249-250.
Hill, M. J. M. [1908]. On a formula for the sum of a finite number of terms of the
hypergeometric series when the fourth element is equal to unity, Proc. London Math.
Soc, B), 6, 339-348.
Hille, E. [1962]. Analytic Function Theory, Vol. II, Ginn and Co., Republished, Chelsea,
New York [1977].
Holder, O. [1889]. Uber einen Mittelwertsatz, Goettinger Nach., 38^7.
Hsu, H.-Y. [1938]. Certain integrals and infinite series involving ultraspherical
polynomials and Bessel functions, Duke Math. J., 4, 374-383.
Hylleraas, E. [1962]. Linearization of products of Jacobi polynomials, Math. Scand., 10,
189-200.
Ingham, A. E. [1932]. The Distribution of Prime Numbers, Cambridge University Press,
London.
Ireland, K. and Rosen, M. [1991]. A Classical Introduction to Modern Number Theory,
2nd Ed., Springer-Verlag, New York.
Ismail, M. E. H. [1977]. A simple proof of Ramanujan's 1i/f1 sum, Proc. Amer. Math. Soc,
63, 185-186.
Ismail, M. E. H. and Stanton, D. [1988], On the Askey-Wilson and Rogers polynomials,
Canadian J. Math., 40, 1025-1045.
Ismail, M. E. H. and Tamhankar, M. V. [1979]. A combinatorial approach to some
positivity problems, SIAM J. Math. Anal, 10, 478^85.
Jackson, D. [1911]. Ueber eine trigonometrische Summe, Rend. Circ. Mat. Palermo, 32,
257-262.
Jackson, F. H. [1910]. On ^-definite integrals, Quart. J. Pure Appl. Math., 41, 193-203.
Jackson, F. H. [1921]. Summation of a g-hypergeometric series, Messenger of Math., 50,
101-112.
Jacobi, С G. [1826]. Uber Gauss' neue Methode, die Werthe der Integrate
naherungsweise zu finden, J. Reine Ang. Math., 1, 301-308.
Jacobi, С G. [1829]. Fundamenta Nova, Regiomontis, fratrum Borntraeger. Reprinted in
Werke, Vol. 1, Chelsea, New York [1969], 49-239.
Jacobi, С G. [1830]. De resolutione aequationum per series infinitam, J. Reine Ang.
Math., 6, 257-286.
Jacobi, С G. [1834]. Demonstratio Formulae..., /. Reine Ang. Math., 11, 307.
Jacobi, C. G. [1859]. Untersuchung iiber die Differentialgleichung der
hypergeometrischen Reihe, / Reine Ang. Math., 56, 149-165.
Jensen, J. L. W. V. [1915-1916]. An elementary exposition of the theory of the gamma
function, Ann. Math., 17, 124—166.
John, F. [1938]. Special solutions of certain difference equations, Ada Math., 71,
175-189.
Karlin, S. and McGregor, J. L. [1957]. The differential equations of birth-and-death
processes, Trans. Amer. Math. Soc, 85, 489-546.
Kirillov, A. N. [1994]. Dilogarithm Identities, Preprint Series, Dept. of Math. Sci.,
University of Tokyo.
Klein, F. [1894]. Vorlesungen iiber die Hypergeometrische Funktion, Gottingen.
Knopp, M. [1971]. Modular Forms and Analytic Number Theory, Markham, New York.
Koblitz, N. [1977]. P-Adic Numbers, p-adic Analysis, andZeta-Functions,
Springer-Verlag, New York.
Koblitz, N. [1984]. Introduction to Elliptic Curves and Modular Forms, Springer-Verlag,
New York.

648 Bibliography
Koekoek, R. and Swarttouw, R. F. [1994]. The Askey-scheme of hypergeometric
orthogonal polynomials and its ^-analogue, Reports of the Faculty of Technical
Mathematics and Informatics, no. 98-117, Delft.
Kogbetliantz, E. [1924]. Recherches sur la sommabilite des series ultraspheriques par la
methode des moyennes arithmetiques, J. Math. Pures Appl, (9), 3, 107-187.
Kolberg, O. [1957]. Some identities involving the partition functions, Math. Scand., 5,
77-92.
Koornwinder, Т. Н. [1972]. The addition formula for Jacobi polynomials, I, Summary of
results, Indag. Math., 34, 188-191.
Koornwinder, Т. Н. [1974]. Jacobi polynomials, II. An analytic proof of the product
formula, SIAMJ. Math. Anal, 5, 125-137.
Koornwinder, Т. Н. [1975]. Jacobi polynomials, III. An analytic proof of the addition
formula, SIAMJ. Math. Anal, 6, 533-543.
Koornwinder, Т. Н. [1978]. Positivity proofs for linearization and connection coefficients
of orthogonal polynomials satisfying an addition formula, / London Math. Soc. B),
18, 101-114.
Koornwinder, T. H. [1990]. Jacobi functions as limit cases of g-ultraspherical
polynomials, / Math. Anal. Appl, 148, 44-54.
Kummer, E. [1836]. Ueber die hypergeometrische Reihe, J. Reine Ang. Math., 15, 39-83,
127-172.
Kummer, E. [1847]. Beitrag zur Theorie der Function Г(х), J. Reine Ang. Math., 35, 1-4.
Lang, S. [1980]. Cyclotomic Fields II, Springer-Verlag, New York.
Lanzewizky, I. L. [1941]. Ueber die Orthogonalitat der Fejer-Szegoschen Polynome, С R.
(Dokl)Acad. Sci. USSR, 31, 199-200.
Lewin, L. [1981]. Poly logarithms and Associated Functions, North-Holland, New York.
Lewin, L. [1981a]. Structural Properties of Polylogarithms, Amer. Math. Soc,
Providence, RI.
Liouville, J. [1839]. Sur quelques integrates definies, / Math. Pures Appl, Ser. 1, 4,
229-235.
Liouville, J. [1855]. Sur un theoreme relatif а Г integrate eulerienne de seconde espece, /
Math. Pure Appl, Ser. 1, 20, 157-160.
Littlewood, J. E. [1986]. Littlewood's Miscellany (B. Bollobas, ed.), Cambridge
University Press, Cambridge, UK.
Lorch, L., Muldoon, M. E., and Szego, P. [1970]. Higher monotonicity properties of
certain Sturm-Liouville functions, III, Canadian J. Math., 22, 1238-1265.
Lorch, L., Muldoon, M. E., and Szego, P. [1972]. Higher monotonicity properties of
certain Sturm-Liouville functions, IV, Canadian J. Math., 2Д, 349-368.
Lorch, L. and Szego, P. [1963]. Higher monotonicity properties of certain
Sturm-Liouville functions, Ada Math., 109, 55-73.
Lorentz, G. G. and Zeller, K. [1964]. Abschnittlimitierbarkeit und der Satz von
Hardy-Bohr, Arch. Math. (Basel), 15, 208-213.
Lorentzen, L. and Waadeland, H. [1992]. Continued Fractions with Applications,
North-Holland, Amsterdam.
Macdonald, I. G. [1995]. Symmetric Functions and Hall Polynomials, 2nd ed., Oxford
University Press, Oxford.
MacMahon, PA. [1917-1918]. Combinatory Analysis, Cambridge University Press,
Cambridge, UK. Reprinted by Chelsea, New York [1984].
Magnus, A. [1988]. Associated Askey-Wilson polynomials as Laguerre-Hahn orthogonal
polynomials, in Orthogonal Polynomials and Their Applications (Segovia, 1986),
M. Alfaro et al, eds., Lecture Notes in Math. no. 1329, Springer-Verlag, New York,
261-378.
Makai, E. [1974]. An integral inequality satisfied by Bessel functions, Ada Math. Acad.
Sci. Hungaricae, 25, 387-390.

Bibliography 649
Mehta, M. L. [1991]. Random Matrices, 2nd Ed., Academic Press, Boston.
Miller, W., Jr. [1972]. Symmetry Groups and Their Applications, Academic Press, New
York.
Milne, S. С [1988]. Multiple ^-series and U(n) generalizations of Ramanujan's 1i/r) sum,
in Ramanujan Revisited, G. Andrews et al., eds., Academic Press, New York,
473-524.
Milne, S. С and Lilly, G. M. [1995]. Consequences of the At and Ce Bailey transform and
Bailey lemma, Discrete Math., 139, 319-346.
Moak, D. S. [1980]. The g-gamma function for q > 1, Aequat. Math., 20, 278-285.
Monsky, P. [1994]. Simplifying the proof of Dirichlet's theorem, Amer. Math. Monthly,
100, 861-862.
Morita, Y [1975]. A p-adic analog of the Г-function, J. Fac. Sci. Tokyo, Sec. 1A, 22,
255-266.
Morris, W. G. [1984]. Constant term identities for finite and affine root systems:
Conjectures and theorems, Ph.D. Thesis, University of Wisconsin-Madison.
Miiller, С [1966]. Spherical Harmonics, Lecture Notes in Mathematics, 7,
Springer-Verlag, New York.
Murphy, R. [1835]. Second memoir on the inverse method of definite integrals, Trans.
Cambridge Phil. Soc, 5, 113-148.
Nassarallah, B. and Rahman, M. [1985]. Projection formulas, a reproducing kernel and a
generating function for ^-Wilson polynomials, SIAM J. Math. Anal., 16, 186-197.
Natanson, I. P. [1965]. Constructive Function Theory, Vol. Ill, Frederick Ungar
Publishing Co., New York.
Nemes, I., Petkovsek, M., Wilf, H., and Zeilberger, D. [1997]. How to do Monthly
Problems with your computer, Amer. Math. Monthly, 104, 505-519.
Nevai, P. [1979]. Orthogonal Polynomials, Amer. Math. Soc, Providence, RI.
Nevai, P. [1990]. Orthogonal Polynomials: Theory and Practice, P. Nevai, ed., Kluwer
Academic Publishers, Dordrecht.
Newton, I. [1960]. The Correspondence of Isaac Newton, 2, 1676-1687, Cambridge
University Press, Cambridge, UK.
Nielsen, N. [1906]. Handbuch der Theorie der Gamma Funktion, В. В. Teubner, Leipzig.
Olver, F. W. J. [1974]. Asymptotics and Special Functions, Academic Press, New York.
Papperitz, E. [1889]. Ueber die Darstellung der hypergeometrischen Transcendenten
durch eindeutige Functionen, Math. Ann., 34, 247-296.
Petkovsek, M., Wilf, H., and Zeilberger, D. [1996]. A = B, A. K. Peters, Wellesley, MA.
Pfaff, J. F. [1797]. Disquisitiones Analydcae, I, Helmstadt.
Pfaff, J. F. [1797a]. Observations analyticae ad L. Euleri Institutiones Calculi Integralis.
Vol. IV, Supplem. II et IV, Historie de 1793, Nova Acata Acad. Scie. Petropolitanae.
XI, 38-57. (Note: The history section is paged separately from the scientific section
of this journal.)
Poincare, H. [ 1886]. Sur les integrales irregulieres des equations lineares, Acta Math.. 8.
295-344.
Poisson, S. D. [1823]. Suite du Memoire sur les integrales definies et sur la sommation de*
series, Paris Jour, de I'Ecole Poly technique, 19, 404—509, especially, pp. 477—4~S.
Polya, G. [1984]. Collected Papers, Vols. I-IV, MIT Press, Cambridge, MA.
Polya, G. and Szego, G. [1972]. Problems and Theorems in Analysis, Vols. I and II.
Springer-Verlag, New York.
Rademacher, H. [1955]. On the transformation of log r](r), J. Indian Math. Soc. 19.
25-30.
Rademacher, H. [1973]. Topics in Analytic Number Theory, Springer-Verlag. New YorV
Rahman, M. [1981]. A non-negative representation of the linearization coefficient- of t.v
product of Jacobi polynomials, Canadian J. Math., 33, 915-928.

650 Bibliography
Rahman, M. [1986]. ^-Wilson functions of the second kind, SIAM J. Math. Anal, 17,
1280-1286.
Ramanujan, S. [1927]. Collected Papers, Cambridge University Press. Reprinted by
Chelsea, New York [1962].
Ramanujan, S. [1988]. The Lost Notebook, Narosa Publishing House, New Delhi.
Raynal, J. [1979]. On the definition and properties of generalized 6-j symbols, /. Math.
Phys., 20, 2398-2415.
Riemann, B. [1857]. Beitrage zur Theorie der durch Gauss'sche Reihe F(a, /б, у, x)
darstellbaren Functionen, K. Gess. Wiss. Gottingen, 7, 1-24.
Riemann, B. [1859]. Uber die Anzahl der Primzahlen unter einer gegebene Grosse,
Monatsb. Berliner Akad., 671-680. Reprinted in Gesammelte Mathematische Werke,
paper 7, Springer-Verlag, [1990].
Rogers, L. J. [1888]. An extension of a certain theorem in inequalities, Messenger of
Math., 17, 145-150.
Rogers, L. J. [1894]. Second memoir on the expansion of certain infinite products, Proc.
London Math. Soc, 25, 318-343.
Rogers, L. J. [1895]. Third memoir on the expansion of certain infinite products, Proc.
London Math. Soc, 26, 15-32.
Rogers, L. J. [1907]. On function sum theorems connected with the series ?^1, x"/n2,
Proc. London Math. Soc, 4, 169-189.
Rogers, L. J. [1917]. On two theorems of combinatory analysis and allied identities, Proc.
London Math. Soc, B), 16, 315-336.
Roosenraad, С. Т. [1969]. Inequalities with orthogonal polynomials, Ph.D. Thesis,
University of Wisconsin-Madison.
Rothe, H. A. [1811]. Systematisches Lehrbuch der Arithmetic, Leipzig.
Roy, R. [1990]. The discovery of the series formula for n by Leibniz, Gregory and
Nilakantha, Math. Mag., 63, 291-306.
Roy, R. [1993]. The work of Chebyshev on orthogonal polynomials, in Topics in
Polynomials of One and Several Variables and Their Applications, Th. Rassias,
H. M. Srivastava, A. Yanushauskas, eds., World Scientific, Singapore, 495-512.
Rudin, W [1976]. Principles of Mathmatical Analysis, 3rd ed., McGraw-Hill, New York.
Saalschutz, L. [1890]. Eine Summationsformel, Zeitschrift Math. Phys., 35, 186-188.
Saff, E. B. and Varga, R. S. [1976]. Zero-free parabolic regions for sequences of
polynomials, S/АМУ. Math. Anal, 1, 344-357.
Salamin, E. [1976J. Computation of n using arithmetic-geometric mean, Math. Comput.,
30, 565-570.
Sapiro, R. L. [1968]. Special functions related to representations of the group SU(n), of
class I with respect to SU(n — 1) (и > 3), Izv. Vyssh. Uchebn. Zaved. Matematika,
71 D), 97-107 (in Russian).
Sarmanov, I. O. [1968]. A generalized symmetric gamma correlation, Dokl Akad. Nauk
SSSR, 179, 1276-1278; Soviet Math. Dokl, 9, 547-550.
Scharlau, W and Opolka, H. [1985]. From Fermat to Minkowski, Springer-Verlag, New
York.
Schiff, L. I. [1947]. Quantum Mechanics, Addison-Wesley, New York.
Schur, I. [1918]. Uber die Verteilung der Wurzeln bei gewissen algebraischen
Gleichungen mit ganzzahligen Koeffizienten, Math. Zeit., 1, 377^02.
Schur, I. [1921]. Uber die Gaussschen Summen, Nach. Gessel Gottingen, Math-Phys.
Klasse, 147-153.
Schiitzenberger, M.-P. [1953]. Une interpretation de certaines solutions de l'equation
fonctionelle: F(x + y) = F(x)F(y), С R. Acad. Sci. Paris, 236, 352-353.

Bibliography 651
Sears, D. B. [1951]. Transformations of basic hypergeometric functions of special type,
Proc. London Math. Soc, B), 53, 138-157.
Selberg, A. [1944]. Bemerkninger om et multipelt integral, Norske Mat. Tidsskr., 26,
71-78.
Sheppard, W. F. [1912]. Summation of the coefficients of some terminating
hypergeometric series, Proc. London Math. Soc, B), 10, 469-478.
Siegel, C. L. [1945]. The trace of totally positive and real algebraic integers, Ann. Math.,
46,302-313.
Siegel, С L. [1954]. A simple proof of ??(-1/т) = ti(t)</tJ2, Mathematica, 1, 4.
Simpson, T. [1759]. The invention of a general method for determining the sum of every
second, third, fourth, or fifth, etc. term of a series, taken in order; the sum of the
whole series being known, Phil. Trans. Royal Soc. London, 50, 757-769.
Smith, H. J. S. [1859-1865]. Report on the Theory of Numbers, Reprinted by Chelsea,
New York [1965].
Stanton, D. [1988]. Recent results for the g-Lagrange inversion formula, in Ramanujan
Revisited, G. E. Andrews, R. Askey, B. Berndt, K. G. Ramanathan, and R. A.
Rankin, eds., Academic Press, San Diego, 525-536.
Stieltjes, T. J. [1993]. Collected Papers, Vols. I and II, Springer-Verlag, New York.
Stirling, J. [1730]. Methodus Differential, London.
Stone, M. H. [1962]. A generalized Weierstrass approximation theorem, in Studies in
Modern Analysis, R. С Buck, ed., Math. Assoc. America,
Sylvester, J. J. [1853]. On Mr. Cayley's impromptu demonstration of the rule for
determining at sight the degree of any symmetrical function of the roots of an
equation expressed in terms of the coefficients, Phil. Mag., 5, 199-202. Collected
Papers, Vol. 1, 594-598.
Sylvester, J. J. [1883]. Note on the graphical method in partitions of n, Proc. Cambridge
Phil. Soc, 19,207-210.
Sylvester, J. J. [1886]. Lectures on the theory of reciprocants, Amer. J. Math., 8, 196-260;
9, 1-37, 113-11661, 297-352; 10, 1-116.
Szasz, O. [1950]. On the relative extrema of ultraspherical polynomials, Bollettino della
Unione Matematica Italiana, 5 C), 125-127.
Szasz, O. [1951]. On the relative extrema of the Hermite orthogonal functions, J. Indian
Math. Soc, 25, 1340-1345.
Szego, G. [1933]. Ueber gewisse Potenzreihen mit lauter positiven Koeffizienten, Math.
Zeit., 37, 674-688.
Szego, G. [1936]. Inequalities for the zeros of Legendre polynomials and related
functions, Trans. Amer. Math. Soc, 39, 1-17.
Szego, G. [1948]. On an inequality of Turan concerning Legendre polynomials, Bull.
Amer. Math. Soc, 54, 401^405.
Szego, G. [1950]. On the relative extrema of Legendre polynomials, Bollettino della
Unione Matematica Italiana, 5 D), 120-121.
Szego, G. [1975]. Orthogonal Polynomials, Amer. Math. Soc, Providence, RI.
Szwarc, R. [1992]. Orthogonal polynomials and a discrete boundary value problem II,
SIAMJ. Math. Anal, 23, 965-969.
Takacs, L. [1973]. On an identity of Shih-Chieh Chu, Ada Sci. Math., (Szeged), 34,
383-391.
Thomae, J. [1869]. Beitrage zur Theorie der durch die Heinesche Reihe..., J. Reine Ang.
Math., 70, 258-281.
Thomae, J. [1879]. Uber die Funktionen welche durch Reihen von der Form dargestellt
werden: 1 + ^jf + ¦¦¦, Journal Math., 87, 26-73.

652 Bibliography
Titchmarsh, E. С [1937]. Introduction to the Theory of Fourier Integrals, Oxford
University Press, London.
Todd, J. [1950]. On the relative extrema of Laguerre orthogonal functions, Bollettino della
Unione Matematica Italiana, 5 C), 120-125.
Tricomi, F. G. and Erdely i, A. [ 1951 ]. The asymptotic expansion of a ratio of gamma
functions, Рас. J. Math., 1, 133-142.
Turan, P. [1952]. On a trigonometrical sum, Ann. Soc. Polonaise Math., 25, 155-161.
Tweddle, I. [1988]. James Stirling, Scottish Academic Press, Edinburgh,
van der Poorten, A. [1979]. A proof that Euler missed.. .Apery's proof of the irrationality
of f C), Math. Intelligencer, 2, 195-203.
Van Loan, C. [1992]. Computational Frameworks for the Fast Fourier Transform, SIAM,
Philadelphia.
Viennot, G. [1983]. Une Theorie Combinatoire des Polynomes Orthogonaux Generaux,
Lecture Notes, UQAM.
Vietoris, L. [1958]. Ueber das Vorzeichen gewisser trigonometrischer Summen,
Sitzungsber. Oest. Akad. Wiss., 167, 125-135.
Vietoris, L. [1959]. Ueber das Vorzeichen gewisser trigonometrischer Summen, II,
Sitzungsber. Oest. Akad. Wiss., 167, 192-193.
Vilenkin, N. J. [1958]. Some relations for Gegenbauer functions, Uspekhi Matem. Nauk
(N.S.), 13 C), 167-172.
Vilenkin, N. J. [1968]. Special Functions and the Theory of Group Representations,
Translations of Math. Monographs 22, Amer. Math. Soc, Providence, RI.
Vilenkin, N. J. and Klimyk, A. [1992]. Representation of Lie Groups, Special Functions
and Integral Transforms, Kluwer Academic, Amsterdam.
Wallis, J. [1656]. Arithmetica Infinitorum, Oxford.
Wang, Z. X. and Guo, D. R. [1989]. Special Functions, World Scientific, Singapore.
Watson, G. N. [1918]. Asymptotic expansions of Hypergeometric functions, Trans.
Cambridge Phil. Soc, 22, 277-308.
Watson, G. N. [1925]. A note on generalized hypergeometric series, Proc. London Math.
Soc, B), 23, xiii-xv (Records for 8 Nov. 1923).
Watson, G. N. [1929]. A new proof of the Rogers-Ramanujan identities, J. London Math.
Soc, 4, 4-9.
Watson, G. N. [1944]. A Treatise on the Theory ofBessel Functions, 2nd Ed., Cambridge
University Press, Cambridge, UK.
Weil, A. [1949]. Number of solutions of equations in a finite field, Bull. Amer. Math. Soc,
55, 497-508.
Weil, A. [1968]. Sur une formule classique, /. Math. Soc. Japan, 20, 400-402.
Weil, A. [1974]. Two lectures on number theory, past and present, Collected Papers, Vol.
Ill, Springer-Verlag, New York.
Weil, A. [1976]. Elliptic Functions According to Eisenstein and Kronecker,
Springer-Verlag, New York.
Weil, A. [1983]. Number Theory, from Hammurabi to Legendre, Birkhauser, Boston.
Whipple, F. J. W [1926]. On well-poised series, generalized hypergeometric series having
parameters in pairs each pair with the same sum, Proc. London Math. Soc, B), 24,
247-263.
Whittaker, E. T. [1904]. An expression of certain known functions as generalized
hypergeometric series, Bull. Amer. Math. Soc, 10, 125-134.
Whittaker, E. T. and Watson, G. N. [1940]. A Course of Modern Analysis, 4th Ed.,
Cambridge University Press, London.
Wiener, N. [1933]. The Fourier Integral and Certain of Its Applications, Cambridge.
Reprinted by Dover, New York [1958].

Bibliography 653
Wilkins, J. E. [1948]. Nicholson's integral for У„2(г) + Уи2(г), Bull. Amer. Math. Soc, 54,
232-234.
Wilson, J. A. [1977]. Three-term contiguous relations and some new orthogonal
polynomials, Pade and Rational Approximations, E. B. Saff and R. S. Varga, eds.,
Academic Press, New York.
Wilson, J. A. [1978]. Hypergeometric series, recurrence relations and some new
orthogonal polynomials, Ph.D. Thesis, University of Wisconsin, Madison.
Wilson, J. A. [1991]. Orthogonal functions for Gram determinants, SIAM J. Math. Anal,
22, 1147-1155.
Wong, R. [1989]. Asymptotic Approximations of Integrals, Academic Press, New York.
Zagier, D. [1989]. The dilogarithm in geometry and number theory, in Number Theory and
Related Topics, R. Askey et al., eds., Oxford University Press, Oxford, UK, 231-249.
Zeilberger, D. [1982]. Sister Celine's technique and its generalizations, / Math. Anal.
Appl, 85, 114-145.
Zeilberger, D. [1994]. Theorems for a price: Tomorrow's semi-rigorous mathematical
culture, The Math. Intelligencer, 16 D), 11-14.

Index
Abel, N. H., 14, 106, 600, 605
Adams, J. C, 320
Agarwal, R. P., 586
Ahern, P., 36
Airy, G. В., 201
Al-Salam, W. A.,532
Amend, В., 61
Anastassiadis, J., 36
Anderson, G. W., 401, 402, 411, 430
Andrews, G. E., 171, 175, 574, 590, 592
Anno, M., 450
Aomoto, K., 401, 402, 406, 407, 428, 440
Apery, R., 355
Archimedes, 485
Artin, E., 34, 36, 53
Askey, R. A., 67, 185, 314, 322, 363, 369, 391,
394, 505, 519, 575, 592
Auslander, L., 43
Azor, R., 328
Baernstein, A., 355
Bailey, W. N., 93, 124, 143, 144, 147, 150, 152,
154, 170, 173, 174, 181, 182, 184, 185, 277,
319,386,398,400
Bak, J., 306
Barnes, E. W., 61
Bateman, H., 277, 313, 314, 475
Baxter, R. J., xv
Bellman, R., 535, 542
Berggren, L., 139
Berndt, В., 532, 627
Bernoulli, D., 2
Bernoulli, J., 605, 615, 619
Bessel, F. W., 187, 200, 202, 203, 222, 225, 228,
230,231,288,321,464,475
Beukers, E, 355
Bieberbach, L., 355, 362, 382
Binet, J. P. M., 28, 29
Boas, R. P., 410
Bochner, S., 464, 542
Bohr, H., 363, 494
Borel, E., 608
Borwein, J., 139
Borwein, P., 139
Brafman, E, 400
Brent, R. P., 138
Bressoud, D. M., 401, 429, 590, 591
Bromwich, T. J., 630
Brown, G., 376
Brown, M., 552
Brualdi, R. A., 371
Burge, W. H., 171
Bustoz, J., 608
Butlewski, Z., 238
Caratheodory, C, 49
Carlson, F., 109
Cartier, P., 371
Cauchy, A. L., 263, 442, 491, 497, 523, 541,
542, 548
Cayley, A., 268
Cesaro, E., 355, 599, 604, 605
Charlier, С V. L., 349
Chebyshev, P. L., 117
Christoffel, E. В., 240, 261, 272, 373
Chu, S.-C, 67
Clausen, Т., 116
Connor, W. G., 576
Cooley, J. W., 43
Cooper, S., 409
Copson, E. Т., 463
Cox, D., 134
Darboux, J. G., 240, 243, 271, 272
Davenport, H., 430, 432
de Boor, C, 43
de Branges, L., 152, 355, 382, 520
de Bruin, N. G., 234, 282
deMoivre, A., 18
Dedekind, R., 1
655

656
Index
DeSainte-Cathenne, M., 330
Din, A. M., 322
Dirichlet, P. L., 1, 2, 401, 413, 434, 463,
532, 601
Dixon, A. C, 72, 239
Dougall, J., 71, 143, 147, 182, 183
Durfee, W. P., 567
Dwork, В., 46
Dyson, F. J., 410, 426, 427, 590
Edwards, A. W. E, 243, 485
Edwards, С. Н., 485
Eisenstein, E G. M., 12, 42
Elliot, E. В., 138
Erdelyi, A., 68, 111,615
Euler, L., 1, 2, 4, 469, 471, 490, 493, 497, 501,
508, 521, 523, 524, 553, 564, 627
Evans, R., 415
Favard, J., 245, 248
Fejer, L., 269, 272, 313, 334, 355, 371, 372,
388, 602
Feldheim, E., 278, 291, 315, 382, 390
Fermat, P., 1, 485, 486
Ferrers, N. M., 320, 565, 566
Fields, J. L., 615
Fine, N. J., 506
Foata, D., 371
Forrester, P. J., 426, 441
Fourier, J., 445, 464, 466, 540, 600, 602, 623
Friedrichs, K., 365
Funk, P., 445, 458
Fuss, N., 2
Gasper, G., 355, 363, 369, 382, 390, 523,
530, 545
Gauss, С F, 1, 61, 63, 67, 78, 94, 95, 240, 254,
262, 355, 411, 431, 491, 496, 497, 499, 500,
508,521,523
Gautschi, W., 212
Gegenbauer, L., 187, 204, 302, 316, 318, 334,
464, 475, 478
Gelfond, A., 410
Gessel, I., 630
Gillis, J., 328
Godsil, С D., 266, 277, 328
Goldbach, C, 2
Good, I. J., 371,427, 630
Gordon, В., 576
Gosper.R. W., 116
Gould, H. W., 186
Graf, J. H., 239, 475
Gram, J.-P, 277
Gray, J., 84
Gronwall, Т. Н., 372, 380
Gross, В., 45
Guo, D. R., 29
Hahn.W., 331,332
Halmos, P., 468
Hamburger, H., 536
Hamilton, W. R., 268
Hankel.H., 121,289-291
Hardy, G. H., 363, 410, 532, 572,
578, 608
Hasse, H., 430, 432, 436
Hecke, E., 445, 458
Heine, E., 274, 491, 520-523
Heisenberg, W., 282
Helversen-Pasotto, A., 93
Herglotz, G., 1
Hermite, C, 100, 198, 266, 269, 271, 273,
299,531
Hewitt, E., 306, 376
Hickerson, D., 590
Hill, M. J. M., 63
Hille, E., 226, 496
Holder, O., 647
Hsu, H.-Y., 321
Hsu,L. C, 186
Hurwitz, A., 1,226
Hylleraas, E., 322
Ingham, A. E., 394
Ireland, K., 432
Ismail, M. E. H., 363, 370, 504, 532, 552
Ivory, J., 99
Jackson, D., 372, 375, 380
Jackson, F. H., 485, 486, 588
Jacobi, С G., 240, 277, 297, 298, 401, 411, 485,
496,500,501,508-510,548,591, 592
Jensen, J. L. W. V., 14
Jimbo, M., xv
John, F, 544
Johnson, W., 542, 630
Jordan, C, 623
Karamata, J., 609
Karlin, S., 396
Kenko, Т., 183
Kirillov, A. N., 102
Klein, F, 65
Klimyk, A., xiii
Knopp, M., 573
Koblitz, N., 540
Koekoek,R.,331
Kogbetliantz, E., 388
Kolberg, O., 573
Koornwinder, Т. Н., 322, 369, 371, 475, 491
Kummer, E., 1, 63, 73, 124, 144, 154, 191,
522, 526
Laguerre, E. N., 266, 273, 282, 290, 292, 306,
330,363,368,370,418,635
Lambert, J. H., 608

Index
657
Landen,J., ЮЗ
Landsberg, M., 542
Lang, S., 46
Lanzewizky, I. L., 278
Laplace, P. S., 100
Laurent, P. A., 426, 496, 497
Lebesgue, H., 6, 69, 311
Legendre, A. M., 132, 136, 162, 199, 240,
252,277,311,334,496,535
Leibniz, G. W., 283, 485, 605
Lerch, M., 17
Lewin, L., 102
Lewy, H., 365
Lilly, G. M., xv
Lindelof, E., 539
Liouville, J., 82, 99, 307
Lipschitz, R. O., 121
Littlewood, J. E., xvi, 648
Lorch,L., 188,229,231
Lorentz, G. G., 363
Lorentzen, L., 98
Maclaurin, C, 618, 626
MacMohan, P. A., 369, 553, 555, 565,
569, 572
Madhava, 59
Magnus, A., 552
Makai, E., 391
Markov, A., 240, 254, 255, 259, 275
Maxwell, J. C, 479
McGregor, J. L., 396
Mehler, F. G., 313
Mehta, M. L., 410, 411,425
Meixner, J., 349
Mellin, R. H., 61
Miller Jr., W., 445
Milne, S. C, xv, 649
Moak, D. S., 544
Mollerup, J., 494
Monsky, P., 608
Mori, Т., 450
Morita, Y., 45
Morris, W. G., 428, 442
Muldoon, M. E., 231
Muller, C, 445
Murphy, R., 240
Nassarallah, В., 519
Natanson, I. P., 252, 269
Nemes, I., 166
Neumann, C. G., 235
Nevai, P., 277, 312, 334
Newman, D. J., 306
Newman, E W., 4
Newton, I., 248, 534
Nicholson, J. W., 187,224
Nielsen, N., 4
Olver, F. W.J., 312, 615
Opolka, H., 55
Orr, M. McE, 184
PopperitzE., 73
Parseval, M. A., 265
Pascal, В., 482, 485
Petkovsek, M., 166
Pfaff, J. E, 63
Phragmen, E., 539
Plana, G. A., 55
Poincare, H., 649
Poisson, S. D., 401, 446, 531, 601, 602, 623,
625, 627
Pollaczek, E, 349
Rademacher, H., 622
Rahman, M., xvi, 322, 519, 523, 545
Ramanujan, S., 182, 485, 501, 505, 506, 509,
532,550,563,569,577, 589
Raynal, J., 124
Riemann, В., 1, 63, 65, 73, 160, 497,
532, 536
Rodrigues, O., 99
Rogers, J. В., 426, 441
Rogers, L. J., 72, 319, 530, 575, 577,
579, 582
Rosen, M., 432
Rothe, H. A., 490, 491, 497
Roy, R., 59, 100,183
Rudin, W., 10,61
Ryser,H.J.,371
Saff.E. В., 188,231
Salamin, E., 138
Sapiro, R. L., 475
Sarmanov, I. O., 368
Schutzenberger, M.-R, 485
Schafheitlin, P., 236
Scharlau, W., 55
Schiff, L. I., 199
Schlomilch, O., 4
Schur,I.,401,423, 578
Sears, D. В., 524
Selberg, A., 110, 401, 402, 410, 417, 428
Sheppard, W. E, 141
Siegel, С L., 401,419, 423
Simpson, Т., 14
Slater, D., 590
Smith, H. J. S., 542
Sole, P., 93
Sonine, N. J., 236, 291, 292, 466
Spence, W., 106
Stanton, D., 630
Steinig, J., 381
Stieltjes, T. J., 240, 254, 259, 401, 416, 419,
425, 496, 594
Stirling, J., 356, 496, 617, 618

658
Index
Stokes, G., 223
Stone, M. H., 265
Subbarow, M. V., 574
Swarttouw, R. R, 331
Sylvester, J. J., 566
Szasz, O., 306
Szego, G., 238, 245, 274, 304, 312, 313, 342,
365,368,380, 390,419
Szego, P., 188,229,231
Szwarc, R., 323
Takacs, L., 70
Tamhankar, M. V., 370
Tannery, J., 72
Thiruvenkatachar, V. R., 569
Thomae.J., 142, 485,521
Titchmarsh, E. C, 89
Todd, J., 304
Tolimieri, R., 43
Tricomi, E G., 615
Tukey, J. W., 43
Turan, P., 342, 343, 384
Tweddle, I., 18, 652
van der Poorten, A., 394
Van Loan, C, 43
Vandermonde, A., 70
Varga,R. S., 188,231
Venkatachaliengar, K., 569
Victor, J. D., 328
Viennot, G., 277, 330
Viete, E, 243
Vietoris, L., 355, 375, 377, 380, 398
Vilenkin, N. J., 315,382, 445
Vitali, G., 496
von Staudt, C, 57
Waadelund, H., 98
Wallis, J., 4
Wang, Z. X., 29
Watson, G. N., 149, 179, 227, 237, 385, 512,
547,548,587,614
Weierstrass, C, 4, 463
Weil, A., 501, 538
Whipple, E J. W., 124, 130, 141, 143, 144, 146,
181,355,362,632-634
Whittaker, E. Т., 187, 195, 198, 512
Wiener, N., 289
Wiles, A., 509
Wilf.H., 125
Wilkins, J. E., 224
Wilson, J. A., 124, 152, 154, 157, 184, 277,
293,331
Wong, R., 614
Zagier, D., 102
Zeilberger, D., 125
Zeller, K., 363

Subject Index
Abel summability, 599
absolutely monotonic series, 389
adjacency matrix, 266
arithmetic-geometric mean, 134
associated Legendre function, 456
asymptotic expansion, 611
Bailey chain, 586
Bailey pair, 586
Bailey's 9Fg transformation, 182
Bailey's lemma, 584
Barnes's beta integral, 90
basic hypergeometric series, 523
it-balanced, 524
well-poised, 524
Bernoulli numbers, 12
Bernoulli polynomials, 20, 615
generalized, 615
Bessel functions, 200, 204-206, 212
Bessel's formula, 212
Gegenbauer's formula, 205
generating function, 212
Hankel's formula, 206
Poisson integral, 204
second kind, 200
Bessel's equation, 200
Bessel's inequality, 264
beta integral, 4
Carlson's theorem, 110
Cesaro summability, 599
characters, 38, 57
additive, 38
even, 58
multiplicative, 38
odd, 58
primitive, 79
Charlier polynomials, 347
Chebyshev polynomials, 101, 102
first kind, 101
fourth kind, 102
second kind, 101
third kind, 102
Christoffel-Darboux formula, 246
Chu-Vandermonde identity, 67
Clausen's identity, 116
confluent hypergeometric equation, 188,
190, 194
asymptotic solutions, 190
recessive solutions, 194
contiguous 4 F3, 155
cosine integral, 235
Coulomb wave function, 199
Darboux's method, 310
Dedekind ^-function, 538
digamma function, 13
dilogarithm, 102
Dirichlet /.-function, 58, 59
functional equation, 60
Dirichlet problem, 463
discriminant, 418
Dougall's 7F6 formula, 147, 152
Bailey's 7F6 integral analog, 152
Dougall's bilateral sum, 110
Dougall's identity, 71
Dyson's integral, 426
elliptic functions, 508
elliptic integrals, 132, 136
first kind, 132
second kind, 136
Erdelyi's formula, 113
error function, 196
Euler's angles, 469
Euler's reflection formula, 9
Euler's transformation, 69
Euler-Maclaurin summation formula, 18
Eulerian integral, 6
first kind, 6
second kind, 6
659

660
Subject Index
Fermat measure, 486
finite-dimensional representation of a group,
466, 467, 469
irreducible representation, 469
isomorphism of, 467
subrepresentation, 469
unitary representation, 467
fractional integral, 605
fractional integration, 111
Fresnel integrals, 235
Funk-Hecke formula, 459
gamma function, 3, 6, 23, 44
Gauss's multiplication formula, 23
integral representation, 6
p-adic, 44
product representation, 3
Gauss quardrature formula, 250
Gauss sums, 38, 627
reciprocity, 627
Gauss's summation formula, 67, 90
integral analog, 90
Gegenbauer polynomials, 302
Gegenbauer's addition formula, 215
Gegenbauer's product formula, 478
Gibbs phenomenon, 372
Gosper's algorithm, 163
Graf's formula, 215
Harr measure, 468
Hahn polynomials, 331, 345, 346
continuous, 331
continuous dual, 331
dual Hahn, 346
Hankel functions, 208
Hankel pair, 216, 290
Hankel transforms, 216, 289
harmonic polynomial, 446
Hasse-Davenport relation, 432
Heine's transformation formula, 520
Hermite polynomials, 278-280, 318,
419
discriminant, 419
generating function, 279
linearization formula, 318
orthogonality, 278
Poisson kernel, 280
Hurwitz zeta function, 15
hypergeometric differential equation, 75
hypergeometric functions, 64, 65, 87, 88, 94, 97,
125, 130
asymptotic expansion, 88
Barnes's integral, 87
contiguous relations, 94
continued fraction, 97
cubic transformations, 130
integral representation, 65
quadratic transformations, 125
hypergeometric series, 61, 70, 140,
146
balanced, 70
it-balanced, 140
nearly poised, 140
very well poised, 146
well-poised, 140
hypergeometric term, 163
incomplete gamma function, 197
Jackson's formula, 587
Jacobi elliptic functions, 510
Jacobi polynomials, 99, 298, 475, 476
generating function, 298
Koornwinder's product formula, 476
Laplace-type integral, 475
Poisson kernel, 385
Jacobi sums, 39
kernel polynomial, 260
Krawtchouk polynomials, 347
Kummer's first transformation, 191
Kummer's second transformation, 191
/.-function, 434
Lagrange interpolation, 249
Lagrange inversion formula, 629
Laguerre polynomials, 283, 288, 418
discriminant, 418
generating-function, 283
Poisson kernel, 288
Lambert summable, 608
Landen's transformation, 103
Laplace equation, 198
Laplace transforms, 536
Legendre polynomials, 252
Legendre's differential equation, 162
Legendre's duplication formula, 22
Legendre-Fejer polynomials, 334
Lerch's theorem, 18
logarithmic convexity, 34
logarithmic integral, 197
MacMahon's Master Theorem, 369
Markov-Stieltjes inequalities, 254
matching polynomials, 324
Meixner polynomials, 346
Meixner-Pollaczek polynomials, 348
Mellin transforms, 85, 534
Miller's algorithm, 211
modified Bessel functions, 222
modular forms, 540
modular group, 540
Nicholson's formula, 224
noncommutative binomial theorem, 485

Subject Index
661
orthogonal matrix, 452
orthogonal polynomials, 452
parabolic cylinder functions, 198
Parseval's formula, 264
partition analysis, 555
partitions, 553, 559, 566-569
congruences, 569
conjugate, 567
Durfee square, 567
Ferrers graph, 566
Frobenius symbol, 568
generating functions, 559
pentagonal number
theorem, 501
Pfaff's transformation, 69
Pfaff-Saalschiitz identity, 69, 91, 92
integral analog, 91
nonterminating form, 92
Poisson summation formula, 623
g-beta integrals, 514
^-binomial coefficients, 483
g-binomial theorem, 488
g-difference operator, 488
g-Dixon formula, 588
g-Dougall formula, 588
g-gamma function, 493
q -Gauss summation formula, 521
g-Hermite polynomials, 530, 531
generating function, 532
linearization formula, 532
Poisson kernel, 532
^-integral, 486
g-Jacobi polynomials, 592, 593
big, 593
little, 592
g-Kummer sum, 522
g-Laguerre polynomials, 593
q-Pfaff-Saalschutz identity, 524
g-ultraspherical polynomials, 527, 530
connection coefficient formula, 531
linearization formula, 531
quadratic reciprocity law, 53
quintuple product identity, 545
Racah polynomials, 344
radial functions, 464
Ramanujan's i ф\ formula, 505
reciprocal polynomials, 150
resultant, 414
Riemann zeta function, 15, 16
functional relation, 16
Rogers-Ramanujan identities, 565, 578
partition theoretic interpretation, 565
Schrodinger equation, 199
Schur's inequality, 423
Sears transformation, 524
Selberg sum, 435
Selberg's integral formula, 402
Aomoto's extension, 402
shifted factorial, 2
Siegel's inequality, 420
sine integral, 235
Sonine's integrals, 218
spherical harmonics, 451, 456
addition theorem, 456
Stieltjes's problem, 415
Stieltjes-Wigert polynomials, 594
Stirling numbers, 356
Stirling's formula, 18
Sturm's comparison theorem, 227
theta functions, 509
triple product identity, 497
Turan's inequality, 342
ultraspherical polynomials, 302, 303, 316, 319,
462
addition formula, 462
generating function, 302
Laplace integral, 316
linearization formula, 319
Rodrigues's formula, 303
Vietoris's inequalities, 377
W-Z method, 167
Watson's 807 transformation, 587
Watson's lemma, 614
Weber's equation, 198
Whipple's^ transformation, 140
Whipple's jFf, transformation, 143
Whittaker functions, 195
Whittaker's equation, 195
Wilson polynomials, 158
orthogonality, 158
Wilson's integral, 151
Wronskian, 136
zonal harmonics, 455

Symbol Index
(а)пЛ 1а(х),222
(a;q)k,48l /(/,»?), 39
B(x,y),4 Ja(x),200
В„,\2 /я, 325
Bn(x),20 K(k),tt2
Bq{a,p),494 Ка(х),22Ъ
C(x), 235 Kn(x;p,N), 347
Ci(x),235 ЦХ, 0,433
сп(нД),511 И, 197
Cnx(x),302 Li2(x), 102
Cn(x;p\q),521 Umx, 106
С„(х;а),347 Lan{x), 283
C*,37 L^(x;?),594
4,-(n),506 L<")(x;?),593
dn(H,*), 511 M(a,b), 134
D«^)-322 М„(х;Ь,с),346
D"(x)'198 Mtm(x), 195
D4/,529 WWF 430
E(k), 136 ';¦ ' ч
erf, 196 PSfll *' Cl xb75
erfc, 196 \аг bl °2 )
Eq(x),492 p(N,M,n), 561
F(a+), 96 p(m,n), 562
pFq(ai ap; b,...., bq; x), 62 PA(n), 554
g(<t>.e. vH.470 р„^х;и,р, q,,jy
Gi(V').466 /f(x),457
W^'i.ti. 208 Pi"'^ (Jt), 99
//i:'m. 208 4(x),326
Я,. 472 б(«, М,п),561
//„1Д1.446 em(n), 559
//„(.» i. 2"8 QmU, n), 560
Я„1л- </». 530 6m'')(n),559
//<¦„(«!. 324 Qn(x;a,8,N),3'
663

664 Symbol Index
p (p), 37
R(F,G), 414 ZQ>)*,38
»•*("). 506 Г f(x)dljx,4%6
Rn(k(x)),346 fx,y), 467
Rn(}.(xy,y,S,N),344 ["i 483
S(*),235 L*/2J>15
sn(«,t),510 {а,к),209
Si(x),235 a(G), 324
5t2(C),471 у з
,476 у'(в,л),197
, 466 Г(в,л),197
458
5„(Л ;с,Ь,с),331 Г,(л),493
Cte).473 д*/>488
^.^(^,472 f(*) 15
Тп(х),Ш <(л,*),15
r^/F. 430 ч(г)) 538
С/"(л)' 101 Si(z,q),i = 1.....4, 510
Vn(cose), 102 ПС*) 3
^C-1.3-2), 136 f0ii523
Wn(cos6>), 102 й-(л) 13
Wt.-W.196 ,^,505
Ya(x),200 fi>>556
Zp-45 ft(z, 9),i = 1,..., 4, 509