Ramanujan's Notebooks
Part III

Bruce C. Berndt
Ramanujan's Notebooks
Part III
Springer-Verlag
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Bruce C. Berndt
Department of Mathematics
University of Illinois at Urbana-Champaign
Urbana, IL 61801
USA
An earlier and shorter version of Chapter 16 was published in "Chapter 16 of Ramanujan's second
notebook: Theta-functions and ^-series," by C. Adiga, В. С Berndt, S. Bhargava and G. N.
Watson, Memoirs of the American Mathematical Society, Volume 53, Number 315, (January 1985).
The revised version in this book appears by permission of the American Mathematical Society.
AMS Subject Classifications: 10-00,10-03,01A60,01A75, lOAxx, 33-xx
Library of Congress Cataloging-in-Publication Data
(Revised for Part 3)
Ramanujan Aiyangar, Srinivasa, 1887-1920.
Ramanujan's notebooks.
Includes bibliographical references and indexes.
1. Mathematics. I. Berndt, Bruce C, 1939-
II. Title.
QA3.R33 1985 510 84-20201
ISBN 0-387-96110-0 (v.l)
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Dedicated to
S. Janaki Ammal
(Mrs. Ramanujan)

S. Janaki Ammal
Photograph by B. Berndt, 1987

A significant portion of G. N. Watson's research was influenced by Ramanu-
jan. No less than thirty of Watson's published papers were motivated by
assertions made by Ramanujan in his letters to G. H. Hardy and in his
notebooks. Beginning in about 1928, Watson invested at least ten years to the
editing of Ramanujan's notebooks. He never completed the task, but for-
fortunately his efforts have been preserved. Through the suggestion of R. A.
Rankin and the generosity of Mrs. Watson, all material pertaining to the
notebooks compiled by Watson was donated to the library of Trinity College,
Cambridge. These notes were invaluable to the author in the preparation of
this book. In particular, many proofs in Chapters 19-21 are due to Watson.
We are grateful to the Master and Fellows of Trinity College, Cambridge for
providing us a copy of Watson's notes. For an engaging biography of Watson,
see Rankin's paper [1].

G. N. Watson
Reprinted with courtesy of the London Mathematical Society.

Preface
During the years 1903-1914, Ramanujan recorded most of his mathematical
discoveries without proofs in notebooks. Although many of his results were
already in the literature, more were not. Almost a decade after Ramanujan's
death in 1920, G. N. Watson and В. М. Wilson began to edit his notebooks
but never completed the task. A photostat edition, with no editing, was
published by the Tata Institute of Fundamental Research in Bombay in 1957.
This book is the third of five volumes devoted to the editing of Ramanujan's
notebooks. Part I, published in 1985, contains an account of Chapters 1-9 in
the second notebook as well as a description of Ramanujan's quarterly re-
reports. Part II, published in 1989, comprises accounts of Chapters 10-15 in
Ramanujan's second notebook. In this volume, we examine Chapters 16-21
in the second notebook. For many of the results that are known, we provide
references in the literature where proofs may be found. Otherwise, we give
complete proofs. Most of the theorems in these six chapters have not previous-
previously been proved in print. Parts IV and V will contain accounts of the 100 pages
of unorganized material at the end of the second notebook, the thirty-three
pages of unorganized results comprising the third notebook, and those results
in the first notebook not recorded by Ramanujan in the second or third
notebooks. The second notebook is chiefly a much enlarged and somewhat
more organized edition of the first notebook.
Urbana, Illinois Bruce C. Berndt
May, 1990

Contents
Preface xi
Introduction 1
CHAPTER 16
q -Series and Theta-Functions 11
CHAPTER 17
Fundamental Properties of Elliptic Functions 87
CHAPTER 18
The Jacobian Elliptic Functions 143
CHAPTER 19
Modular Equations of Degrees 3, 5, and 7 and Associated
Theta-Function Identities 220
CHAPTER 20
Modular Equations of Higher and Composite Degrees 325
CHAPTER 21
Eisenstein Series 454
References 489
Index 505

Introduction
In der Theorie der Thetafunctionen ist es leicht, eine beliebig grosse Menge von
Relationen aufzustellen, aber die Schwierigkeit beginnt da, wo es sich darum
handelt, aus diesem Labyrinth von Formeln einen Ausweg zu finden.
G. Frobenius
The content of this volume is more unified than those of the first two volumes
of our attempts to provide proofs of the many beautiful theorems bequeathed
to us by Ramanujan in his notebooks. Theta-functions provide the binding
glue that blends Chapters 16-21 together. Although we provide proofs here
for all of Ramanujan's formulas, in many cases, we have been unable to find the
roads that led Ramanujan to his discoveries. It is hoped that others will at-
attempt to discover the pathways that Ramanujan took on his journey through
his luxuriant labyrinthine forest of enchanting and alluring formulas.
We first briefly review the content of Chapters 16-21. Although theta-
functions play the leading role, several other topics make appearances as well.
Some of Ramanujan's most famous theorems are found in Chapter 16. The
chapter begins with basic hypergeometric series and some q -continued frac-
fractions. In particular, a generalization of the Rogers-Ramanujan continued
fraction and a finite version of the Rogers-Ramanujan continued fraction are
found. Entry 7 offers an identity from which the Rogers-Ramanujan identities
(found in Section 38) can be deduced as limiting cases, a fact that evidently
Ramanujan failed to notice. The material on q-series ends with Ramanujan's
celebrated 1ф1 summation. After stating the Jacobi triple product identity,
which is a corollary of Ramanujan's 1ф1 summation, Ramanujan commences
his work on theta-functions. Several of his results are classical and well known,
but Ramanujan offers many interesting new results, especially in Sections
33-35. For an enlightening discussion of Ramanujan's contributions to basic

2 Introduction
hypergeometric series, as well as to hypergeometric series, see R. Askey's
survey paper [8].
Chapter 17 begins with Ramanujan's development of some of the basic
theory of elliptic functions highlighted by Entry 6, which provides the basic
inversion formula relating theta-functions with elliptic integrals and hyper-
hypergeometric functions. Section 7 offers many beautiful theorems on elliptic
integrals. The following sections are devoted to a catalogue of formulas for
the most well-known theta-functions and for Ramanujan's Eisenstein series,
L, M, and N, evaluated at different powers of the argument. These formulas
are of central importance in proving modular equations in Chapters 19-21.
Several topics are examined in Chapter 18, although most attention is given
to the Jacobian elliptic functions. Approximations to n and the perimeter of
an ellipse are found. More problems in geometry are discussed in this chapter
than in any other chapter. The chapter ends with Ramanujan's initial findings
about modular equations.
Chapters 19 and 20 are devoted to modular equations and associated
theta-function identities. Most of the results in these two chapters are new and
show Ramanujan at his very best. It is here that our proofs undoubtedly often
stray from the paths followed by Ramanujan.
Chapter 21 occupies only 4 pages and is the shortest chapter in the second
notebook. The content is not unlike that of the previous two chapters, but
here the emphasis is on formulas for the series L, M, and N.
Since Ramanujan's death in 1920, there has been much speculation on the
sources from which Ramanujan first learned about elliptic functions. In com-
commenting on Ramanujan's paper [2] in Ramanujan's Collected Papers [10],
L. J. Mordell writes "It would be extremely interesting to know if and how
much Ramanujan is indebted to other writers." Mordell then conjectures that
Ramanujan might have studied either Greenhill's [1] or Cayley's [1] books
on elliptic functions. Greenhill's book can be found in the library at the
Government College of Kumbakonam, but we have been unable to ascertain
for certain if this book was in the library when Ramanujan lived in Kumbako-
Kumbakonam. Hardy [3, p. 212] remarks that these two books were in the library
at the University of Madras, where Ramanujan held a scholarship for nine
months before departing for England. Hardy then quotes Littlewood's
thoughts: "a sufficient, and I think necessary, explanation would be that
Greenhill's very odd and individual Elliptic Functions was his text-book."
Mordell, Hardy, and Littlewood surmised that Greenhill's book served as
Ramanujan's source of knowledge partly because Greenhill's development
avoids the theory of functions of a complex variable, a subject thought to have
been never learned by Ramanujan. In particular, the double periodicity of
elliptic functions is not mentioned by Greenhill until page 254. In the un-
unorganized portions of the second notebook and in the third notebook, there
is some evidence that Ramanujan knew a few facts about complex function
theory. (See Berndt's book [11].) However, Ramanujan's development of the
theory of elliptic functions did not need or depend on complex function theory.

Introduction 3
Ramanujan also never mentions double periodicity. Because Cayley's book
contains several sections on modular equations, it is reasonable to conjecture
that this book might have been one of Ramanujan's sources of learning.
The origins of Ramanujan's knowledge of elliptic functions are probably
not very important, since Ramanujan's development of the subject is uni-
uniquely and characteristically his own without a trace of influence by any other
author. Ramanujan does not even use the standard notations for elliptic
integrals and any of the classical elliptic functions. The content of Ramanu-
Ramanujan's initial efforts overlaps with some of Jacobi's findings in his famous
Fundamenta Nova [1], [2]. However, it is unlikely that Ramanujan had access
to this work. Moreover, while the Jacobian elliptic functions were central in
Jacobi's development, they play a far more minor role in Ramanujan's theory.
(Our proofs in the pages that follow undoubtedly employ the Jacobian elliptic
functions more than Ramanujan did.) Both Jacobi and Ramanujan exten-
extensively utilized theta-functions, but the evolution of Ramanujan's theory is
quite different from that of Jacobi. The classical, general theta-function <93(z, q)
may be defined by
Uz,q)= f q°2e2i"\ (II)
n= — oo
where \q\ < 1 and z is any complex number. Ramanujan's general theta-
function f(a, b) is given by
f(a,b)=
и=-оо
where \ab\ < 1. The generalities of (II) and A2) are the same. To see this, set
a = q expBiz) and b = q expBiz). For many purposes, the definition (II) is
superior. However, for Ramanujan's interests and theory, A2) is definitely the
preferred definition and was strongly instrumental in helping Ramanujan
discover many new theorems in the subject.
Upon studying Ramanujan's development of the theory of modular equa-
equations in Chapters 18-21, we now are able to understand more clearly the
rationale for Ramanujan's introduction of "modular equations" in Sections
15 and 16 of Chapter 15 of his second notebook [9], which we have previously
described in Part II [9]. Before returning to this material, we need to define
the generalized hypergeometric function P+1FP by
-Y (Otl)»((X2)n---(«p+l)n Z"
l
_ - .a a
l> a2> •••> ap+b Pi. P2> ¦¦¦>
la
»=o
where p is a nonnegative integer, a1,a2,...,ap+1, рх,Р2,---,Рр are complex
numbers, \z\ < 1, and
for each nonnegative integer n.

4 Introduction
Ramanujan begins his study of "modular equatons" in Chapter 15 by
defining
00 1X\
p/v\ (Л v\—1/2 V У2/п n p (i \ || -i /Т'ЗЧ
r(X).— (l—X) — 2_, —ГХ ~ lA>l2> xh |X| < 1. (U)
n=0 nl
He then states the trivial identity
21 ' ¦¦ <№).
After setting a = 2t/(l + t) and /? = t2, Ramanujan offers the "modular equa-
equation of degree 2,"
PB - aJ = a2, A5)
which is readily verified. The factor A + i) in A4) is called the multiplier. He
then derives some modular equations of higher degree and offers some general
remarks. We emphasize that this definition of modular equation has no
connection with any of the standard definitions, but we shall draw some
parallels shortly.
There are many definitions of a modular equation in the literature. See
Ramanathan's paper [10] or our expository introduction to Ramanujan's
modular equations [7] for discussions of some of these alternative definitions.
We now give the definition of a modular equation that Ramanujan employed
and the one that we shall use in the sequel. First, the complete elliptic integral
of the first kind K(k) is defined by
- Г
Jo
Ktk) - Г d(p - U f (i)" k2n - П F A i-
J /l k2 2 i(n!) I
12 ¦2
y/ — k2 sin2 (p
where 0 < к < 1 and where the series representation in A6) is found by
expanding the integrand in a binomial series and integrating termwise. The
number к is called the modulus of K, and k' := y/l — k2 is called the comple-
complementary modulus. Let K, K', L, and L' denote complete elliptic integrals of
the first kind associated with the moduli к, к', ?, and t', respectively. Suppose
that the equality
K> L' /Г7Ч
(I7)
holds for some positive integer n. Then a modular equation of degree и is a
relation between the moduli к and t which is implied by A7). Ramanujan
writes his modular equations in terms of a and /?, where a = k2 and /? = t2.
We shall often say that /? has degree n. As we shall see in Section 6 of Chapter
17, modular equations can alternatively be expressed as identities involving
theta-functions. In fact, often one first proves a theta-function identity and
then transcribes it into an equivalent modular equation by using the formulas
in Entries 10-12 in Chapter 17. Ramanujan undoubtedly used this procedure

Introduction
in proving most of his modular equations, and we shall proceed in the same
fashion. The multiplier m for a modular equation of degree n is defined by
A8)
Ramanujan also established many "mixed" modular equations in which
four distinct moduli appear. See the introduction of Chapter 20 for the
definition of "mixed" modular equation.
For those not familiar with modular equations, these definitions may
appear to be arbitrary and unmotivated. The raison d'etre can be found in
the first six sections of Chapter 17. In particular, we note that the base q in
the classical theory of elliptic functions is defined by q = exp( — nK'/K). Often
one seeks relations among theta-functions where the arguments appearing are
q and q", for some interger n. Further motivation can be found in two survey
articles (Berndt [7], [8]).
Before offering some historical remarks about modular equations, we point
out the analogies between Ramanujan's definition of a "modular equation"
in Chapter 15 and the standard definition arising from A7) that we have given
above. The function F(x) in A3) is an analogue of K(k) in A6). Note that if one
of the parameters \ of 2f 1E,1', U k2) in A6) is replaced by 1, then this hyper-
geometric function reduces to tF0(%; k2), which appears in A6) with x = k2.
Observe that A5) is a relation between the "moduli" a and /?. Furthermore,
note that the multiplier 1 + ^/fi in A4) is analogous to the multiplier defined
in A8).
One could argue, as we did in [7], that the theory of modular equations
began in 1771 and 1775 with the appearance of J. Landen's two papers [1],
[2] in which Landen's transformation was introduced. Strictly speaking, the
theory commenced when A. M. Legendre [2] derived a modular equation of
degree 3 in 1825 and С G. J. Jacobi established modular equations of degrees
3 and 5 in his Fundamenta Nova [1], [2] in 1829. Subsequently, in the century
that followed, contributions were made by many mathematicians including
C. Guetzlaff, L. A. Sohncke, H. Schroter, L. Schlafli, F. Klein, A. Hurwitz,
E. Fiedler, A. Cayley, R. Fricke, R. Russell, and H. Weber. Classical texts
containing much material on modular equations include those of Enneper [1],
Weber [2], [3], Klein [2], [3], and Fricke [3]. Enneper's book [1] and
Hanna's paper [1] contain many references to the literature. As we shall see
in the remainder of this book, Ramanujan's contributions in the area of
modular equations are immense. He discovered many of the classical modular
equations found by the aforementioned authors, but he derived many more
new ones as well. With little or no exaggeration, we suggest that perhaps
Ramanujan found more modular equations than all of his predecessors dis-
discovered together. After approximately a half century of dormancy, modular
equations have become prominent once again. They arise in the theory of

6 Introduction
elliptic curves, in the hard hexagon models of lattice gases (Joyce [1]), and in
algorithms for the rapid calculation of n (J. M. Borwein [1]; J. M. and P. B.
Borwein [l]-[6]; J. M. Borwein, P. B. Borwein, and D. H. Bailey [1]).
H. Cohn [l]-[8] and Cohn and J. Deutsch [1] have returned to the classical
viewpoints but with a more modern approach and with computer algebra.
Further references and applications of modular equations are discussed in our
expository survey paper [7]. A briefer and more elementary introduction to
modular equations has been given by us in [8]. T. Kondo and T. Tasaka [1],
[2], G. Kohler [1], [2], and I. J. Zucker [3] have recently discovered some
new beautiful theta-function identities in the spirit of those arising in the
theory of modular equations.
Many algebraic, analytic, and elementary methods have been devised to
prove modular equations. Except for H. Schroter, we have not found the
methods of others helpful in proving Ramanujan's modular equations. Wat-
Watson (Hardy [3, p. 220]) has declared that "when dealing with Ramanujan's
modular equations generally, it has always seemed to me that knowledge of
other people's work is a positive disadvantage in that it tends to put one off
the shortest track."
In attempting to establish Ramanujan's modular equations, we have uti-
utilized three approaches. The first relies on the theory of theta-functions and
frequently employs Schroter's formulas, first established in his dissertation [1]
in 1854. Schroter's primary theorem is a formula representing a product of
theta-functions as a linear combination of products of other theta-functions.
Schroter's formulas can be found in the books of Hardy [3, p. 219], Tannery
and Molk [1, pp. 163-167], Enneper [1, p. 142], and J. M. and P. B. Borwein
[2, p. Ill], as well as in a recent paper by Kondo and Tasaka [1]. In our
applications, we need to slightly modify Schroter's formulas and obtain related
representations for/(a, b)f(c, d) ±/(—a, —b)f(—c, —d). All of the requisite
formulas are proved in detail in Section 36 of Chapter 16. Schroter [l]-[4]
utilized his formulas to find several modular equations, although, except for
his thesis [1], he never published complete proofs of his results. Ramanujan,
to our knowledge, has not explicitly stated Schroter's formulas in any of his
published papers, notebooks, or unpublished manuscripts. However, it seems
clear, from the theory of theta-functions and modular equations that he did
develop, that Ramanujan must have been aware of these formulas or at least
of the principles that yield the many special cases that Ramanujan doubtless
used. However, Schroter's formulas are applicable in only a small minority of
instances. We conjecture that Ramanujan possessed other general formulas
or procedures involving theta-functions that are unknown to us. In particular,
we think that he had derived a formula involving quotients of theta-functions
that he did not record in his notebooks and that we have been unable to find
elsewhere in the literature as well. Watson [5, p. 150] asserted that "a pro-
prolonged study of his modular equations has convinced me that he was in
possession of a general formula by means of which modular equations can be
constructed in almost terrifying numbers." Watson then intimates that Rama-

Introduction 7
nujan's "general formula" is, in fact, Schroter's most general formula. How-
However, as pointed out above, Schroter's formulas cannot be used in most
instances. Further efforts should be made in attempting to discover Ramanu-
jan's analytical methods.
The second method exploits previously derived modular equations and
may involve a heavy dosage of elementary algebra. The primary idea is to find
parametric representations for a and /? which are then employed along with
elementary algebra to verify a given modular equation. Ramanujan probably
used such methods, especially for small values of the degree n. The algebraic
difficulties normally increase very rapidly with n. Some of our algebraic proofs
are very tedious, and it is doubtful that Ramanujan would have employed
such drudgery. Ramanujan, with his great skills in spotting algebraic relation-
relationships, could undoubtedly discover modular equations using algebraic mani-
manipulation, but, particularly in Chapters 19 and 20, the reader will see that some
of the proofs presented here could not have been accomplished without
knowing the modular equation in advance.
Our third method employs the theory of modular forms. In some ways, this
represents the best approach. First, the theory of modular forms provides the
theoretical basis which explains why certain identities among theta-functions
exist. Second, this approach usually does not become too much more compli-
complicated with increasing n, and so proofs remain comparatively short, after the
requisite theory has been developed. The primary disadvantage to this method
is that the modular equation must be known in advance, and so, as in the
second approach, the proofs are more properly called verifications. The princi-
principal idea is to show that the multiplier systems of certain modular forms agree
and that the coefficients in the expansion of a certain modular form are equal
to zero up to a certain prescribed point. We then can conclude that the
modular form must identically be equal to zero. This approach has been used
by A. J. Biagioli [1], S. Raghavan [1], [2], Raghavan and S. S. Rangachari
[1], and R. J. Evans [1] in establishing several of Ramanujan's theta-function
identities. It might be argued that Ramanujan used a variant of this method
by comparing coefficients in the expansions of theta-functions. This is ex-
extremely doubtful, however, because Ramanujan would not have discovered
the identities by this procedure.
An earlier version of Chapter 16, coauthored with С Adiga, S. Bhargava,
and G. N. Watson, was published in "Chapter 16 of Ramanujan's second
notebook: Theta-functions and ^-series," Memoirs of the American Mathema-
Mathematical Society, vol. 53, no. 315, 1985. The revised version appears here by
permission of the American Mathematical Society. A substantial majority of
the theorems and proofs appearing in Chapters 17-21 have not heretofore
appeared in print. В. С Berndt, A. J. Biagioli, and J. M. Purtilo [l]-[3] have
proved some of Ramanujan's modular equations in journals commemorating
the centenary of Ramanujan's birth. A brief description of Ramanujan's work
on Eisenstein series in Chapter 21 was given by us in [10]. Some of Ramanu-
Ramanujan's work on modular equations has also been examined by K. G. Rama-

о Introduction
nathan [9], [10], V. R. Thiruvenkatachar and K. Venkatachaliengar [1], and
K. Venkatachaliengar [1].
To help readers find modular equations of certain degrees, we offer a table
indicating the chapter and sections where the desired modular equations may
be found.
Degree Chapter Sections
3
5
7
11
13
15
17
19
23
31
47
71
3,9
5,25
3, 5,15
3, 7,21
3,9, 27
3,11,33
3, 13, 39
3, 21, 63
3, 29, 87
5, 7, 35
5,11,55
5,19,95
5,27,135
7,9,63
7,17,119
7, 25,175
9,15,135
9, 23,207
11,13,143
11,21,231
13,19, 247
15,17, 255
19
19
19
20
20
20
20
20
20
20
20
20
20
20
19
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
5,7
11,13
18,19
21
7
8
21
12
16
15
22
23
23
3
15
11
13
5
14
19,21
20
24
18,19
19,21
20
24
19,21
20
24
20
24
20
24
24
24
Each of Chapters 16-20 in the second notebook contains 12 pages, while
Chapter 21 has only 4 pages. The number of theorems, corollaries, and
examples found in each chapter is listed in the following table.

Introduction
Chapter
16
17
18
19
20
21
Total
Number of Results
134
162
135
185
173
45
834
Many of the theorems that Ramanujan communicated in his letters of
January 16, 1913 and February 27, 1913 to G. H. Hardy may be found in
Chapters 16-21. We list these results in the following table.
Location in Collected Papers Location in Notebooks
p. xxviii, A) Chapter 16, Entry 15 and corollary, Entry 39 (i)
p. xxviii, F) Chapter 20, Entry 20 (i)
p. xxix, A5) Chapter 18, Corollary in Section 12
p. xxix, B0) (i), (v) Chapter 20, Entries 11 (i), (ii), (xiv)
p. xxix, B1) Chapter 20, Entry 19 (iii)
p. 350,C) Chapter 18, Entry 12 (ii)
p. 353, B0) (ii), (iii), (iv), (vi) Chapter 20, Entries 11 (iii), (iv), (v), (xv)
p. 353, B1) Chapter 20, Entry 19 (iii)
p. 353, B2) Chapter 20, Entry 24 (i)
A few of Ramanujan's published papers and questions posed to readers of
the Journal of the Indian Mathematical Society have their origins in Chapters
16-21 of the second notebook. In some cases, only a small portion of the paper
actually arises from material in the notebooks. The following table lists those
papers and the corresponding locations in the notebooks.
Paper Location in Notebooks
Squaring the circle Chaper 18, Entry 20 (i)
Modular equations and Chapter 18, Entry 3, Corollary in
approximations to n Section 3; Chapter 21
Question 584 Chapter 16, Entries 38 (i), (ii)
Some definite integrals Chapter 16, Entry 14
Question 662 Chapter 19, Entry 7 (iv) (first part)
On certain arithmetical Chapter 16, Section 35; Chapter
functions 17, Entry 13
Question 755 Chapter 18, Corollary (ii) of
Section 19
Proof of certain identities in Chapter 16, Entries 38 (i), (ii)
combinatory analysis

10 Introduction
In the sequel, equation numbers refer to equations in the same chapter,
unless another chapter is indicated. Unless otherwise stated, page numbers
refer to pages in the pagination of the Tata Institute's publication of Ramanu-
jan's second notebook [9]. Page numbers unattended by any reference num-
number always refer to Ramanujan's second notebook. Parts I and II refer to the
author's accounts [5] and [9], respectively, of Ramanujan's notebooks.
We mention some standard notations that will be used in the sequel. The
rational integers, the rational numbers, the real numbers, and the complex
numbers are denoted by D, Q, R, and % respectively. The residue of a mero-
morphic function / at a pole a is denoted by Ra, if the identity of the function
/ is understood.
I am very grateful to many mathematicians for the proofs and suggestions
that they have supplied. I am most indebted to G. N. Watson for the notes
that he compiled on Chapters 16-21. In particular, many of the proofs in
Chapters 19-21 are due to Watson. F. J. Dyson [1, p. 7] has affirmed that
"Watson was chief gardener in the 1930's and worked hard to develop and
elucidate Ramanujan's ideas." Evidently, Watson was very careful about
whom he would permit to stroll through this garden. However, through the
extensive notes that he left behind, he has allowed me to view many of the
flowers in the garden, and I am very appreciative.
I owe special thanks to the following mathematicians. С Adiga and S.
Bhargava made many contributions in their coauthoring an earlier version of
Chapter 16 with me. The quality of Chapter 16 has greatly been enhanced by
the many suggestions offered by R. A. Askey. A. J. Biagioli and J. M. Purtilo
provided invaluable and necessary help in the theory of modular forms and
MACSYMA, respectively. R. J. Evans [1] furnished beautiful proofs of some
of Ramanujan's most intractable theta-function identities, and we have re-
reproduced in the sequel much of his paper. L. Jacobsen has contributed several
helpful remarks and suggestions on continued fractions.
For their comments and suggestions, I am also obliged to G. Almkvist,
G. E. Andrews, J. M. and P. B. Borwein, J. Brillhart, R. L. Lamphere,
R. Miiller, C. Rama Murthy, K. G. Ramanathan, K. Stolarsky, M. Villarino,
H. Waadeland, J. Wetzel, and I. J. Zucker.
The author bears the responsibility for all errors and wishes to be notified
of such, whether they be minor or serious.
Most of the manuscript for this book was typed by Dee Wrather, and I
thank her for her very accurate and rapid typing.
The figures in Chapters 19 and 20 were drawn by Jonathan Manton using
the graphics of Mathematica.
A perusal of the references at the conclusion of this book indicates that
several are obscure. Nancy Anderson, the mathematics librarian at the Uni-
University of Illinois, helped to unearth many of these, and I owe her special
thanks.
Lastly, I express my deep gratitude to James Vaughn and the Vaughn
Foundation, and to the National Science Foundation for their financial
support during several summers.

CHAPTER 16
«/-Series and Theta-Functions
In Chapter 16, Ramanujan develops two closely related topics, ^-series and
theta-functions. The first 17 sections are devoted primarily to <j-series, while
the latter 22 sections constitute a very thorough development of the theory of
theta-functions.
Ramanujan begins by stating some mostly familiar theorems in the theory
of ^-series. In particular, Ramanujan rediscovered some of Heine's famous
theorems including his <j-analogue of Gauss' theorem. However, several re-
results appear to be new. Perhaps most noteworthy in this respect are the
continued fractions in Sections 10-13. (Entry 10 is not a q-continued frac-
fraction and is more properly placed in Chapter 12 among other theorems of this
type.) Entry 13 was later generalized by Ramanujan in his "lost notebook"
[11]. Entry 16 is a "finite" form of what is now generally known as the
"Rogers-Ramanujan continued fraction" and was first established in print
by Hirschhorn [1] in 1972 while being unaware that the result is found in
Ramanujan's notebooks.
As is to be expected, Ramanujan's findings in the theory of theta-functions
contain many of their classical properties. In particular, he rediscovered
several theorems found in Jacobi's epic Fundamenta Nova [1], [2]. In Entry
27, Ramanujan records transformation formulas for the modular transforma-
transformation: T(t) = — 1/t. He did not discover more general transformation formulas.
In Entry 19, Ramanujan gives the famous Jacobi triple product identity of
which he made numerous applications. Because several of our proofs employ
Watson's quintuple product identity, it would seem that Ramanujan had
discovered it. Indeed, the quintuple product identity can be found in Ramanu-
Ramanujan's "lost notebook" [11]. Results in the last part of Chapter 16 indicate that
Ramanujan had found Schroter's formulas [1]. Although Ramanujan does
not give these formulas in their most general form, he does offer several special
cases and deductions from them.

12 16. q-Series and Theta-Functions
But more importantly, Ramanujan discovered several new and deep theo-
theorems in the theory of theta-functions. For example, the beautiful theorems in
Sections 33-35 appear to be new, as well as Entry 38(iv) and the corollaries
in Section 37.
In closing our brief survey of the content of Chapter 16, we would like to
mention that this chapter contains four results that are due originally to
Ramanujan and for which he is justly famous. Entry 14 offers Ramanujan's
^-analogue of the beta-function. The evaluation of this integral was first
recorded by Ramanujan in [4], [10, p. 57]. There are now at least four distinct
verifications. In Entry 17 we find "Ramanujan's 1ф1 summation." Several
proofs, including a new one offered here, now exist. Ramanujan found many
applications for his 1i/^1 summation, including a proof of Jacobi's triple pro-
product identity. The remarkable Rogers-Ramanujan identities are found in
Entries 38(i), (ii), and the "Rogers-Ramanujan continued fraction" in Entry
38(iii). It might be remarked that this continued fraction is the only continued
fraction proved in Ramanujan's published papers. However, he did submit
several formulas containing continued fractions to the problems section of the
Journal of the Indian Mathematical Society. Also, Ramanujan's letters to
Hardy contain many beautiful theorems on continued fractions.
We conclude our introduction with several remarks on notation. For those
reading this book in conjunction with the notebooks, it seems best to retain
Ramanujan's notation f(a, b) for the theta-functions (see A8.1)). We remark
that f(a, b) = S3(z, z), where ab = e2niz, a/b = e*iz, and 93{z, z) denotes the
classical theta-function in the notation of Whittaker and Watson [1]. Most
of the results in the sequel are, in fact, more easily stated in the notation f(a, b)
rather than in the notation S3(z, т). Ramanujan uses x to denote his primary
variable. Since q is almost universally used today instead of x, we have adopted
the more standard designation. It is assumed throughout the sequel that
\q\ < 1. As usual, for any complex number a, we write
(a)k := (a; q)k := A - a){\ - aq){\ - aq2)---{\ - aq"'1)
and
(a)m := (a; q)m := f[ A - aqk).
k=0
Ramanujan writes ]~[ (— a, x) for (а)и, where x = q. The basic hypergeometric
series s+i(ps is defined by
аиа2,...,а$+1 1 _ » (aAjaA^VHi)* х*_
Л,ь2,...Л 'xJ h (bA(b2)k-(bs)k («)»' m)
where |x| < 1 and al,a2, ..., as+1, bub2, ..., bs are arbitrary, except that,
of course, (bj)k ?=0, 1 <j < s, 0 < к < оо. If s is "small," we shall write
s+i<Ps(au ¦¦¦>as+ubi, ¦-., bs;x) in place of the notation at the left side of
@.1). Finally, to denote the dependence on the base q, we may write
s+1(ps{a1,...,as+l;bu...,bs;q;x).

16. q -Series and Theta-Functions 13
Entry 1. Let q be real with \q\ < 1, and suppose that a and x are any complex
numbers. Let the principal branches of A — a)x and A — q)x be chosen. Then
(in) Woo = П
*o
and
П
Proof. First assume that \a\ < 1. Apply B.1) below with a and t replaced by
aqx and g"~*, respectively. Hence,
g'-"<-
by the binomial theorem. The general result follows by analytic continuation.
The following proof of (ii) is due to R. W. Gosper, Write
к
Identity (iii) follows easily by regrouping the factors on the left side. To prove
(iv), let n = 2 in (iii) and replace q /
The q -gamma function Г,(х) is defined by
Thus, Entry l(ii) may be rewritten in the form
lim Г,(х + 1) = Г(х + 1).
Gosper's proof of Entry l(ii) may also be found in Andrews' monograph [18,
p. 109]. Our proofs of Entries l(i), (ii) are not completely rigorous, because
limits were taken without justification under the summation and product
signs, respectively. Т. Н. Koornwinder [1] has indeed justified these formal
processes and provided rigorous proofs.
Ramanujan's proof of Entry 2 below can be found in his paper [4] [10,
pp. 57-58].

14 16. q-Series and Theta-Functions
(-b)«, _ S (-b/a)kak
(a). ?o («)* '
The earliest known reference for Entry 2, the <j-binomial theorem, is the
work of Rothe [1]. Entry 2 was also discovered by Cauchy [1], [2, pp. 42-50]
and has been attributed to him, Euler, Gauss, and Heine. If we put —b = at,
Entry 2 may be written in the form
B.1)
(*). *=o (q)k
Entry 3. // a is arbitrary and \q\ < 1, then
Entry 3 is normally attributed to Cauchy [1], [2, pp. 42-50]. However,
C.1) can be found in Jacobi's Fundamenta Nova [1], [2, p. 232] published 14
years earlier. We defer a proof of Entry 3 until Section 9 where a generalization
will be proved.
Entry 4. If \abc\ < 1, then
(abUac),» _ g (l/b)k(l/c)k(abc)k
{a)m(abc)m k% {q)k(a)k
Entry 4 is a famous result of Heine [1] and is the ^-analogue of Gauss'
summation of the ordinary hypergeometric series. For a proof of Entry 4, see
Andrews' text [9, p. 20].
Observe that if we replace b by 1/t, с by 1/c, and then a by ate and lastly
put с = 0, we obtain B.1). Letting b and с tend to 0 and replacing a by aq in
Entry 4, we deduce Entry 3.
As Askey [8, p. 69] has observed, Ramanujan formulates his discoveries
on basic hypergeometric series to emphasize the symmetry of the value of the
sum, of which the statements of Entries 4 and 5, for example, attest.
Entry 5. If\q\, \abcd\ < 1, then
f (a/qW/bW/c)k(l/d)k(l - aq
(ab)k(ac)k(ad)k(q)k(l - a/q)
Entry 5 was first found by L. J. Rogers [2, p. 29] (where v — J~q should be
replaced by 1 — vj~q in one factor). It is a limiting case of a more general
identity found by F. H. Jackson [1]. Another proof of Jackson's theorem has
been given by Andrews and Askey [1].

16. q -Series and Theta-Functions 15
Proof. The aforementioned identity of Jackson is the q-analogue of Dougall's
theorem and is given by (Bailey [4, p. 67])
'a, q^/a, -q^/a, b, c, d, e, q~N
l~a, —^fa, aq/b, aq/c, aq/d, aq/e, aqN+1'
(aq)N(aq/cd)N(aq/bd)N(aq/bc)N
1
1> qj
(aq/b)N(aq/c)N(aq/d)N(aq/bcd)N'
where N is a positive integer and a2qN+1 = bcde. Observe that
E.1)
E2)
We now let N tend to oo and e tend to 0 in E.1). Since
Щд-\ _ (e)k(a2q/bcde)k _ ( a V
...o (aq/e)k(aqN+1)k e^0(aq/e)k(bcde/a)k \bcd)'
we find that
? (a)k(b)k(c)k(d)k(\ - aq2k)
k% (aq/b)k{aq/c)k(aq/d)k(q)k(l - a)\bcdj
= (aq/bUaq/cUaq/dUaq/bcd^' E-3)
Replacing a by a/q, b by l/b, с by 1/c, and d by l/d in E.3), we deduce Entry
5 at once.
Entry 6.//|a|,|c|,|e|<l, then
2(Pi(b/a, c; d; a) ~ ,^^ 2<Pi(d/c, a; b; c).
This beautiful theorem is due to Heine [1], and a simple proof based on
Entry 2 may be found in Andrews' book [9, p. 19]. In the other direction,
setting с = d in Entry 6, we obtain Entry 2.
Applying Entry 6 three times, we find that
h (c)k(q)k (c)Jt)m ,4 {at)k(q)k
_ (bH0(atH0 (c/fynibt)^ S (b)k(abt/c)k (cV
~ (c)oo(Ooc ~(at)m(b)m ho (bt)k{q)k \b)
(c/a)k(c/b)k(abe\k
(c/a)k(c/b)k(abt
W. &> (c)k(q)k \c/'
which we use below.

16 16. q -Series and Theta- Functions
Entry 7. If\q\ < I, then
f (a)k(d/b)k(d/c)k(d/q)k(l ~ dq2k-1)(bc
*=o (b)k(c)k(d/a)k(q)k(l - d/q)
(a)Jd)m " (b/a)k(c/a)k
Шс)„ h (d/a)k(q)k
k
Proof. By a theorem of Watson [2] (Bailey [4, p. 69]),
Га, q^a, -qfa, c, d, e,f, q~N . a2qN+2
L
v cdef
[aq/cd, ej, q~N
(aq/eUaq/fUaqN+l Uaq^/ef^ 4ГЗ \ejq^/a, aq/c, aq/d' ч J' v '
where JV is a positive integer. Short calculations show that
and
lim
;Zl (efq-»/a)k ~ \ef) '
I) and using the calculat;
(a)k(c)k(d)k{e)k{f)k(l - aq2k)(-a2/cdef)kq«k+3)'2
Letting N tend to oo in G.1) and using the calculations above as well as E.2),
we find that
(aq/eUaq/Л*
We next let d tend to oo in G.2). Since
(aq/c)k(aq/d)k(aq/e)k(aq/f)k(q)k(l - a)
d,e,faql
,aq/d'ef\ (?'2)
we find that
(fl)*(c)*(e)*(/)t(l - aq2k)(a2/cef)kq«k+»
V
% (aq/c)k(aq/e)k(aq/f)k(q)k(l - a)
(aqUaq/ef) f. (e)k(f\ (щ\ п Ъ
(aq/c)k(aq/e)k(aq/f)k(q)k(l - a)
f. (e)k(f\ (
= f (
(aq/eUaq/f)m kk (aq/c)k(q)k\efj '
Replacing a, c, e, and/by d/q, a, d/b, and d/c, respectively, in G.3), we find that
f (a)k(d/b)k(d/c)k(d/q)k(l - dq^Kbc/afqW-»
(b)k(c)k(d/a)k(q)k(l - d/q)
(d)m(bc/d)K « (d/b)k(d/c)k(bc\k
h (d/a)k(q)k \d

16. q -Series and Theta-Functions 17
_(dUbcld)w (a)m » (ЬЩс/а)к k
(b)Jc)x (bcld^h (d/a)k(q)k '
where we have applied F.1). This completes the proof of Entry 7.
An important application of Entry 7 will be made in Section 38.
We now prove a lemma from which Entries 8 and 9 will follow as limiting
cases.
Lemma. For \de/abc\, \e/a\, \q\ < 1,
3cp2(a, b, c; d, e; de/abc) = y^J^ ъ<Рг{а, d/b, d/c; d, de/bc; e/a). (8.1)
Proof. Using Entry 2 and F.1), we find that, for \a\, \e/a\, \de/abc\ < 1,
f (a)k(b)k(c)k(de\k JdLf (Ъ\(с)к(едк)о> ( de V
h (d)k(e)k(q)k \abc) (e), k% (d)k(q)k(aqk)n \abc)
(e/a)
m k
% (q)m {Ч'
f Шт m f (b)k(c)k(deqm\k
- (q)m kk(d)k(q)k\abc)
(e) 4b
(e), „4b («L (deq^/abc^ k% (d)k(q)k \ a
(flUe/fl), у (d/b)k(d/c)kfe\k « (Ле/аЬс)
eUde/abcb h (d)k(q)k \aj m% (q)m [ 9 ]
by Entry 2 again. The restriction \a\ < 1 may now be removed by analytic
continuation, and the lemma easily follows.
Entry 8. 7/|a|, |^| < 1, then
(c)k(b/a)k k =
{b)xk% (d)k(q)k h (b)k(d)k(q)k
Proof. Let b tend to oo in (8.1). Then replace a, c, and e by b/a, d/c, and b,
respectively, to achieve the desired result.
Observe that Entry 8 is a q-extension of Pfaff's transformation (Bailey
[4, p. 10])
2Fx{a, b; c; x) = A - xY^F^a, c-b;c; -x/(l - x)).

18 16. q -Series and Theta- Functions
Entry 9. If\q\ < I, then
First Proof. Letting a and b tend to oo in (8.1), we deduce that
у {c)kqk2~k /de\k _ 1 ™ (-lf(d/c)kekqkik-m
k%(d)k(e)k(q)k\c~) 'VL&o (d)k(q)k '
Next, let d = bc/a and e = aq. Letting с tend to 0, we complete the proof of
Entry 9.
Second Proof. Using Entry 6 twice, we find that
2<Pi(a, o; c; x) = гфЛФ, *', ax; b)
(Х)тУЧт
(bx^ic/b)^
= -тттт—2(рЛаЬх/с, b; bx; c/b).
(x)m(c)m
Now replace x by x/ab and let a and Ь tend to oo. This yields
»xV2-*_ 1 » (-1)*(х/с)кс^*(к-1)/2
Replacing x by bq and с by a<j, we obtain Entry 9.
V. Ramamani [1] has given a proof of Entry 9 by obtaining two functional
relations for the right side. Andrews [10] has shown that Entry 9 is a limiting
case of an identity due to Rogers. V. Ramamani and K. Venkatachaliengar
[1] have established Entry 9 by showing that it is a limiting case of Heine's
transformation, Entry 6. A generalization of Entry 9 has been discovered by
Bhargava and Adiga [3]. H. M. Srivastava [1] subsequently established an
equivalent form of their result. The particular case a = — 1 of Entry 9 was
posed as a problem by Carlitz [2].
Observe that if we let a = b, then Entry 9 reduces to Entry 3.
Corollary (i). If\q\ < I, then
oo Мк+i.) oo
Proof. Put a = 1 and b = q in Entry 9.
Corollary (ii). If\q\<l, then
nkBk+l) oo
ЩJк к=0

16. q-Series and Theta-Functions 19
Proof. Replace q by q2 in Entry 9 and then set a = l/q and b = q.
Entry 10. Let x, t, m, and n denote complex numbers. Define
T
~ T
Then, if either ?, m, or n is an integer or if Re x > 0,
1 - P _ Itmx 4(x2 - 12)(/2 - I2)(m2 - I2)
1 + P ~ x2 + t2 + m2 - n2 - 1 + 3(x2 + t2 + m2 - n2 - 5)
4(x2 - 22)(/2 - 22)(m2 - 22)
+ 5(x2 + /2 + m2 - n2 - 13)
4(x2 - (k - 1J)(/2 - (fc - lJ)(m2 - (fc - IJ)
+ •••+ Bfc - l)(x2 + /2 + m2 - n2 - 2k2 + 2fe - 1) +•••'
Proof. We apply Entry 40 from Chapter 12 in Ramanujan's second notebook
[9, p. 163] (Part II [9, pp. 151-152]), which was initially proved by Watson
[6]. Let
R = ПГ(|(а ± р ± у ± д ± e + 1)),
where the product contains eight gamma functions and where the argument
of each gamma function contains an even number of minus signs. Let
Q = ПГ(|(а + p ± у ± S ± e + 1)),
where the product contains eight gamma functions and where the argument
of each gamma function contains an odd number of minus signs. Suppose that
at least one of the parameters /?, y, S, e is equal to a nonzero integer. Then
1 + Q/R
= 1{2(а4 + P* + y* + S* + e* + 1) - (a2 + p2 + y2 + S2 + e2 - IJ - 22}
64(a2 - 12)(P2 - I2)(y2 - 12)(E2 - 12)(?2 - I2)
+ 3{2(a4 + p* + y* + S* + e* + 1) - (a2 + p2 + y2 + S2 + e2 - 5J - 62}
64(a2 - 22)(P2 - 22)(y2 - 22)(S2 - 22)(e2 - 22)
+ 5{2(a4 + P* + y* + 5* + e+ + 1) - (a2 + p2+ y2 + 52 + e2 - 13J - 142} + •••'
A0.1)
In A0.1), let a = x, )S = n — e, у = /, and E = m, where e is a positive integer.
In the quotient Q/R of 16 gamma functions, we observe that eight are in-
independent of e and eight depend on e. The quotient that is independent of e
is precisely equal to P, while the quotient that depends on e is equal to

20 16. q-Series and Theta-Functions
By Stirling's formula, the quotient above tends to 1 as e tends to oo. Hence,
We next examine the right side of A0.1) as e tends to oo. An elementary
calculation shows that
2(x4 + {n- eL + /4 + m4 + e4 + 1) - (x2 + (n - eJ + /2 + m2 + e2
- Bj2 + 2/ + I)J
= 4(n2 - x2 - t2 - m2 + 2/2 + 2/ + l)e2 + O(e),
as e tends to oo, where 0 <j < oo. Hence, the continued fraction on the right
side of A0.1) is equal to
0F)
1{4(и2 - х2 - /2 - т2 + 1)г2 + 0(е)}
64(х2 - 12)(/2 - I2)(w2 - 12)е4 + 0(е3)
+ 3{4(n2-x2-/2-m2 + 5)e2 + O(e)}
64(х2 - 22)(/2 - 22)(w2 - 22)г4 + 0(г3)
+ 5{4(и2 - х2 - f2 - т2 + 13)е2 + 0(е)} + •••'
as e tends to oo. Successively dividing the numerators and denominators above
by — 4e2 and letting e tend to oo, we find that the foregoing continued fraction
tends termwise to
2xtm 4(x2 - 12)(/2 - I2)(m2 - I2)
A0.3)
x2 + t2 + m2 - n2 - 1 + 3(x2 + e2 + m2 - n2 - 5)
4(x2 - 22)(/2 - 22)(m2 - 22)
+ 5(x2 + /2 + m2 - n2 - 13) +•••"
Combining this with A0.2), we complete the proof of Entry 10, except for an
examination of the convergence of A0.3).
If either /, m, or n is an integer, then the continued fraction terminates, and
the limiting process is easily justified. If none of these parameters is an integer,
then the convergence for Re x > 0 follows from an application of the uniform
parabola theorem. The details are similar to those in proving Entry 35 of
Chapter 12 [9, p. 135] (Part II [9, pp. 156-158]). We refer the reader to
Jacobsen's paper [1, pp. 427-429] for all of these necessary details.
The following beautiful theorem has some resemblance to Entry 33 in
Chapter 12 [9, p. 149] (Part II [9, p. 155]). Our first proof below has also been
given by Ramanathan [6].

16. q -Series and Theta-Functions 21
Entry 11. Suppose that either q, a, and b are complex numbers with \q\ < 1, or
q, a, and b are complex numbers with a = bqm for some integer m. Then
_ a - b (a - bq)(aq — b) q(a — bq2)(aq2 - b)
~l-q+ l-q3 + l~qS +¦¦•'
We give two proofs for the case а ф bqm for all integers m.
First Proof. We employ Heine's [1] continued fraction for a quotient of two
contiguous basic hypergeometric series, namely, for \q\ < 1 and \z\ < 1,
2(pi(cc,Pq;yq;q;z)_l ax a2 a3
П^Г~Т + Т + Т + Т + - ¦¦' AU)
where
zafl*(l — 6qk)(l — yqkla)
a2k = 2t-iw< ш—> k> 1,
A — yq )A — yq )
and
A - yq")(l - yq*
Now replace a, fl, y, q, and z by bq/a, b/a, q, q2, and a2, respectively. Then
_ «,., (a ~ bg2k)(b - flg»)
2k q (l _9«-i)(i -^4*+1)' - '
and
^(a-b<Z2*+1)(b-a<z24+1)
• /C ^ U.
In summary,
It follows from A1.1) that, for \a\ < 1,
a-b 2(Pi{bqla, bq2/a; q3; q2; a2)
1 - q 2(pt{bq/a, b/a; q; q2; a2)
_a-b (a- bq){aq - b) q(a - bq2)(aq2 - b)
~ 1 -q+ 1 -q3 + 1 -q5 +•
• (И.2)

22 16. q-Series and Theta-Functions
Letting
A = 2(pi(bq/a, b/a; q; q2; a2) and В = 2(Pi{bq/a, bq2/a; q3; q2; a2),
1 -q
we observe that, for \a\ < 1,
a-b (a- b)(a - bq) (a- b)(a - bq)(a - bq2)
A±a ^ + ^ +
by Entry 2. It follows that
(bU(a)oo ~ (-b)J(-a)^ _A + B-(A-B)_B
(*)./(*). + (- *)„/( - a). A + В + (A - B) A'
Combining A1.2) and A1.3), we complete the proof for \a\ < 1.
A1.3)
If |<j| < 1, the given continued fraction is easily seen to be equivalent to a
continued fraction of the form K(ck/1), where ck tends to 0 locally uniformly
with respect to either a, b, or q, and where we have used a familiar notation
K(ak/bk) for a continued fraction with kth partial numerator ak and kth partial
denominator bk, к > 1. By analytic continuation, equality holds for all a, b,
and q such that а ф bqm for all integers m (Jacobsen [1, pp. 418, 435]).
Second Proof. For \q\ < 1 and 0 < \a\ < 1, define the sequence {Pm} by
Po-i^o» A1.4)
»=o («)„
and
Then {Pm} satisfies the recurrence relation
Pm = (I — q2m+1)Pm+1 + qm(a— bqm+1)(aqm+1 — b)Pm+2, m > 0, A1.6)
since by A1.4) and A1.5), Po - A - q)Pi = (a - bq)(aq - b)P2 and, for m > 1,
m+1 (qhm-i (gJm-in=i D2тJ»
1 ~ q2m+1 {q2. q2) 1 - q2m+l j. (bqm+1/aJn
п-1Гя2п+2. л2^
„2m+2\
n^i (q2m+2Jn-
-Щ i 4 )m-l
n-2

16. q -Series and Theta-Functions 23
2
_ ~2m+2n+l
2
л2п+2
qm(qm+l _ ^bj _ ^2н+2ч( j _
ЩJт+3 1=0
x (n2n+2. 2\ 2n
= <jm(a — b<jm+1)(a<jm+1 — b)Pm+2.
Hence, by A1.6),
Po , . (a - bq)(aq - b)
(a - bg)(aq - b) g(fl - bg2)(flg2 - b)
-* *T l-«3 + P2/P3
and so on. The continued fraction thus generated is limit periodic. Indeed, the
feth partial numerator approaches 0 and the feth partial denominator ap-
approaches 1 as fe tends to oo. Hence, the continued fraction converges to Ро/Л
if lim^,^ PJPm+1 ф -1. From the definition A1.5) of Pm, we observe that
limm4oo PJPm+i = 1. Thus, for \q\ < 1 and 0 < \a\ < 1, we conclude that
Po = 1_ +M#H) g(« ~ bg2)(flg2 ~ b)
Therefore the continued fraction in this entry converges to
? Ф/аJп+1 ,.+1
% (q)
2n+i
Pol Pi f, (b/dhn агп (b)J(a)a + (- b)J( - a)a'
п=о
where the last equality follows from Entry 2. Multiplying the numerator and
denominator of this last expression by (aH0(—a)m, we complete the second
proof of Entry 11 for |^r| < l,0< \a\ < 1, and a±bqm. The result follows for
all complex a by analytic continuation.
It remains to prove Entry 11 for a = bqm, when the continued fraction
terminates. In such a case, both sides are rational functions of q and b. We
refer the reader to L. Jacobsen's paper [1, p. 435] for complete details.

24 16. q-Series and Theta-Functions
The second proof we have given is a simplification, due to Jacobsen, of that
given in the monograph by Adiga, Berndt, Bhargava, and Watson [1].
Entry 12. Suppose that a, b, and q are complex numbers with \ab\ < 1 and \q\ < 1
or that a = b2m+1 for some integer m. Then
(a2q3; q4)x(b2q3; q*)m 1 (a — bq)(b — aq)
A2.1)
(a2q; q*Ub2q; «*)„, 1 - ab + A - ab)(q2 + 1)
(a - Ъдъ){Ъ - aq3)
+ A -ab)(q4 + 1) +•••'
The following proof is due to L. Jacobsen [1]. It supplants the more
complicated original proof found in the monograph by Adiga, Berndt, Bhar-
Bhargava, and Watson [1]. Jacobsen's proof, although it has some time-consuming
calculations, has two advantages. First, by using Vitali's theorem, she [1,
p. 424] shows that it is sufficient to establish A2.1) for a certain discrete set of
values of a as well as for a limit point of this set. In fact, the continued fractions
on this set are terminating. Second, the proof uses three other entries from
this chapter, Entries 2, 8, and 15.
Lemma 12.1. For any complex numbers a and q and nonnegative integer n,
(„\ \T / 1 Vt ^* ^ ,i(fc2~fc)/2 rtk /II *>\
[a)n — 2a \—*/ —7~\ 4 Q • \L^"^)
Proof. Since
\q h — \~ 4 \q )kq , к 2. и,
the right side of A2.2) may be written in the form
« (q-n)k(aqn)k
k% (q)k '
Applying B.1) with t = q~" and a replaced by aq", we complete the proof.
Proof of Entry 12. Following Jacobsen [1, p. 425], we first establish
A2.1) for a = bq2m+1, where m is any fixed nonnegative integer and \ab\ =
\b2q2m+1\ < 1. Since A2.1) is trivial when bq = 0, we assume that bq ф 0 in
the sequel. Observe that the left side of A2.1) equals
by Lemma 12.1.
m
*=(
m+1
V |
(-1)"
)
(qMm
(c
f(m +
/~4
: + l *
f;q'
2-*);
¦. ~4\
\
q\
q2k\b
2k2_2k
2q)k
2 fc
\" 4.7

16. q-Series and Theta-Functions 25
When a = bq2m+1, the right side of A2.1) is the terminating continued
fraction
1 b2q{\ - q2m)(l - q2m+2) b2q3(l - q2m~2){l - q2m+4)
1 _ b2q2m+1 - A - b2q2m+i){l + q2) - A - b2q2m+1){l + q4)
m+1)(i + q2m)
dxb2q d2b2q dmb2q
1 - b2q2m+1 + 1 - b2q2m+i + 1 - b2q2m+1 + ••• + 1 - b2q2m+1'
A2.4)
where
A - g*")(l - q2m+2)
d^ ГТ7
and
A + q2"){l + q2"-
for 2 < n < m.
Let
and
oo („Чт+2-п-к). _4\ ( _„2. „2\ /_ „2m-2k+2. „2
. -2\
'g >k
L, /_4. _4\ / „2n. _2\ I _2
k=o Щ ,q h(-<l ,1 h\-<l
п~ L, \ l> /_4. _4\ / „2n. _2\ I _2m-2n-2fc+4. -2
ko Щ q h(-<l 1 h\<l 4
where 1 < n < m + 1. By a somewhat tedious calculation, it can be shown
that, for n = 0, 1, 2,..., m — i,Fn satisfies the recurrence relation
Fn = A - q2m+1b2)Fn+1 + dn+1b2qFn+2. A2.5)
It should be pointed out that the verification for n = 0 must be accomplished
separately from the cases 1 < n < m — 1, because for n = 0 the definitions of
Fn and dn+1 are different from those when n > 0. From A2.5),
Fo _ 2m+1 dxb2q
l2 2m+i d±b2q d2b2q
9 + 1 - ЬУт+1 + JJF3
2„2т+1 , ^24 dmb2q
= i - ь vm+1 +
= 1 - b2q2m+1 +

26 16. q-Series and Theta-Functions
since Fm = 1 — b2q2m+1 and Fm+1 = 1. Hence, the continued fraction in A2.4)
equals F1/Fo. But this is exactly the expression that we found in A2.3) for the
left side of A2.1). Hence, we have proved A2.1) for the set {a = bq2m+1:0 <
m < oo}.
This set has a limit point a = 0. Next, we must establish A2.1) for a = 0.
In this case, A2.1) reduces to the equality
(b2g3;q*)ao_l b2g b2g3 b2q5
However, by applying Entry 15 below with a = —l,q replaced by q2, and b
replaced by —b2/q, we find, after some simplification, that
.V 1; («*;Л _i J!i_ #t_ #l_ A27)
By letting n tend to oo in Lemma 12.1 with q replaced by qA and a = b2q3 and
b2q, respectively, we find that the left side of A2.7) equals
(Alternatively, we may draw the same conclusion by applying Entry 8 with q
replaced by q4, b = 0, с = 1, d = 0, and a replaced by b2q3 and b2q, respec-
respectively.) Hence, we have shown that A2.7) reduces to A2.6), as desired.
Next, if we write the continued fraction of A2.1) in the form
where ck = akjbk^\, к > 1, and b0 = 1, we must show, for к > m + 1 and m
sufficiently large, that the elements ck+1 lie in the parabolic region
{z: \z\ - Re z < i} A2.8)
(Jacobsen [1, pp. 417, 424]). A brief calculation yields
C"+1 = A - b2q2m+1J A + q2k){\ + q2k~2) "
Clearly, for k> m + 1 and m sufficiently large, ck+1 lies in the region A2.8).
Lastly, we must examine the convergence of the continued fraction in A2.1).
Now, as к tends to oo,
(a - bq2k~l){b - aq2k~l)
°k+1 ~ A - abJ(q2k + l)(q2k-2 + 1)
tends to ab/{l — abJ locally uniformly with respect to a, b, and q. For con-
convergence, we need that ab/(l — abJ ${-oo, —?], and this is so if and only if
\ab\ < 1. This completes the proof.

16. g-Series and Theta-Functions 27
We have not described here the entire theoretical background for the proof
above; we refer to Jacobsen's paper [1] for the remainder of the theory.
If \ab\ > 1, the continued fraction in A2.1) is equivalent to
-1/И) (Vfl - q/b){l/b ~ g/a) (I/a - g3/b)(l/b - q3/a)
l-l/(ab)+ A - l/(ab))(q2 + 1) + A - l/(ab))(e* + 1) +•••'
which, by Entry 12, converges to
which, in general, does not equal the left side of A2.1). Since the continued
fraction diverges for \ab\ = 1, the hypotheses on a and b in Entry 12 cannot
be relaxed.
If \ab\ < 1 and \q\ > 1, the continued fraction in A2.1) converges to
(a2/q3; W)Jb2lq3; Ve*)«
(a2/q; l/q*
which furthermore evinces the beautiful symmetry in this wonderful entry.
Entry 13. lf\q\ < I, then
Proof. Let
/(М) = МЛт^г. A3-2)
Then it is easy to verify that
f(b,a)=f(b,aq)-aqf(bq,aq) A3.3)
and
f(b, a) = f(bq, a) + bqf(bq2, aq). A3.4)
From A3.3) and A3.4),
f(bq,a)=f(bq2,a) + bq2f(bq\aq)
= f(bq2, aq) + (bq2 - aq)f(bq3, aq).
Using A3.4), the equality above, and iteration, we find that
f(b, a) _ bq _ bq bq2 - aq
f(bq,a) + f(bq,a) ^ +f(bq2,aq)
= 1 +
f(bq2,aq) * "" f{bq\aq)
bq bq2 — aq bq3 bq* - aq2
f(bq5,aq2)

28 16. <j-Series and Theta-Functions
= ^ bq bq2 - aq bq*_ bq* - aq2
1 + 1 + 1 + 1 +•••'
This continued fraction is of the form
с„ + bd,
1 + K
1
where с„ and dn tend to 0 as и tends to oo. The continued fraction therefore
converges. Furthermore, since also f(bq2n, aqn)/f{bq2n+1, aqn) = 1 + O(q2n+1)
as n tends to oo, the continued fraction converges to f(b, a)/f(bq, a).
Now set b = a in A3.5) and employ Entry 9. Observe that, by Entry 9,
f(a, a) = 1. Taking the reciprocal of both sides of A3.5) we deduce A3.1).
Entry 13 is originally due to Eisenstein [1], [2, pp. 35-39]. However, the
special case a = 1 can be found in an entry of Gauss' diary [1, p. 68], dated
February 16, 1797. See also J. J. Gray's paper [1, p. 114], which provides a
translation of Gauss' diary. A generalization of Entry 13 appears in Ramanu-
jan's "lost notebook" [11]. For proofs of this more general theorem, see papers
of Andrews [10, Eq. A.4)], Hirschhorn [3], and Ramanathan [4]. Bhargava
and Adiga [1] have established, with a unified approach, some continued
fraction expansions in Ramanujan's "lost notebook," including those men-
mentioned above as well as a related continued fraction of Hirschhorn [2]. R. Y.
Denis [5] and N. A. Bhagirathi [1] have proved some very general continued
fractions for certain basic bilateral hypergeometric series which include Entry
13 as special cases.
After stating Entry 13, Ramanujan gives formulas for the denominator Dn
of the nth convergent of A3.1), namely,
D2n = t P~ fl A - qn+1-j), n > 1, A3.6)
k=o (q)k j=i
and
D2a+1 = I -f— П A - <?"+W)> « ^ 0. A3.7)
*=o (q)k j=i
To prove A3.6) and A3.7), we employ a familiar recursion formula for denomi-
denominators (Wall [1, p. 15]) along with induction on n.
First, from A3.1), it is obvious that Dj = 1 and D2 - 1 + aq, which are in
agreement with A3.7) and A3.6), respectively. Proceeding by induction and
using the aforementioned recursion formula, we find that
= D2n
л-1
= 1 +
-1 +
akqnk
\Ч)к
П s
I-
к
tkqnk
(a\
: к
п п
,1(
n fl*-lfl(n-l)(k-l) k-1
k=i (q)k-i j=i

16. q -Series and Theta-Functions 29
= i + t Pv- {(i - «""*) + (<rk - 9я)} П (i - «""')
= I ^- П a - <T J>
fc=o (qh J=o
Replacing j byj — 1, we complete the proof of A3.6).
The proof of A3.7), which begins with the recursion formula
D2n+i = D2n + a{q2n - qn)D2n-u
is very similar to the proof of A3.6), and so we omit the details.
Entry 14. If n<\ and 0 < a < q1'", then
— dt = — ——j—.
Jo t"( — t)^ sm{nn) (q)a:i(aqn ^
This beautiful integral formula was stated by Ramanujan in his paper [4],
[10, p. 57]. Acknowledging that he did not possess a rigorous proof, Ramanu-
Ramanujan confessed: "My own proofs of the above results make use of a general
formula, the truth of which depends on conditions which I have not yet
investigated completely. A direct proof depending on Cauchy's theorem will
be found in Mr. Hardy's note which follows this paper." (That paper is [1],
[2, pp. 594-597].) The special case a = 0 of A4.1) is found in Ramanujan's
quarterly reports. To see how Ramanujan "proved" A4.1), consult Hardy's
book [3, p. 194] or Berndt's account of the quarterly reports [5].
Askey [2] has found a simple proof of Entry 14 and has demonstrated why
A4.1) is a g-analogue of the beta function. Recalling the definition of the
g-gamma function in A.1), we may rewrite A4.1) in the form
Г
Jo
rf(i - «)r> + js)'
where a, /? > 0 (Askey [2], [8]). It is thus clear that A4.2) is an extension of
the beta function
Г(а)Г(/?)
I,
? v+fidt =
0 v ' )
R. L. Lamphere [1] has found a very elementary proof of A4.1).
For further g-beta integrals, see papers by Askey [1], [3], [5], [6], Andrews
and Askey [3], Askey and Roy [1], and Rahman [1] as well as Andrews'
monograph [18]. An asymptotic expansion for Tq{x), uniform in q as q tends
to 1 —, that is, an analogue of Stirling's formula, has been proved by N. Koblitz
[1] and by D. S. Moak [2], who [1] has also studied orthogonal polynomials
with respect to weight functions akin to the integrand in A4.1). More general
work in this direction has been accomplished by Askey and Wilson [1], where
a plethora of references may be found.

30 16. q-Series and Theta-Functions
Entry 15. If \q\ < 1, then
=0 {aq)k{q)k = { bq bq2 bq3
« bkqk{k+l) " ' 1 ~ aq + 1 - aq2 + 1 - aq* + •¦¦'
Proof. Let/(b, a) be defined by A3.2). Replacing a by aq in A3.4) and adding
the result to A3.3), we find that
f(b, a) = A - aq)f(bq, aq) + bqf(bq2, aq2).
Replacing a by aq"'1 and b by bq", we may rewrite the previous equality in
the form
f(bq"+2,aqn+1)
Using A3.4), A5.1), and iteration, we deduce that
f(b,a) =J+^=1+ ^ bq2
f(bq, a) f(bq, a) I ~ aq + f(bq2, aq)
f(bq2, aq) f(bq\ aq2)
1 - aq + 1 - aq2 + 1 - aq3 + ¦ • ¦'
That this continued fraction converges and that it converges, indeed, to
f(b, a)/f(bq, a) follow as in the proof of Entry 13. If b Ф 0 and aq" = 1 for some
positive integer n, then equality holds with the convention that we take the
limit of both sides as a tends to l/q". If b ~ 0 and aq" = 1, we interpret both
sides as equaling 1. This completes the proof.
Corollary. If\q\ < 1, then
V " q
k=o (q)k _ 1 ОД ОД2 ОД3
f ^^' 1+T + T + T + ---'
k=0 (<?)k
Proof. Set a = 0 in Entry 15 and then replace b by a. The corollary now
readily follows.
The continued fractions of both Entry 15 and its corollary were mentioned
by Ramanujan [10, p. xxviii] in his second letter to Hardy. The corollary was
established earlier by Rogers [1, p. 328, Eq. D)] and then later by Watson [3].
The special case a — 1 is Entry 38(iii) and is discussed in detail in Section 38.

16. ^-Series and Theta-Functions 31
Ramamani [1] has given a similar proof of Entry 15 by obtaining functional
relations for the function
A) (q)k
and using Entry 9. Entry 9 is not used in our proof. Entry 16 below provides
a finite version of Entry 15. Thus, an alternative proof of Entry 15 is obtained
by letting n tend to oo in Entry 16. Hirschhorn [1], [4] has given a proof of
Entry 16; perhaps our proof is somewhat simpler.
Entry 16. For each positive integer n, let
and
Then
v 1+1 +¦¦•+ 1
Proof. For each nonnegative integer r, define
k=o {q)k(q)n-r-2k+i
Observe that Fo = ц, Ft = v, ?„ = 1, and Fn_t = 1 + aq". A straightforward
calculation shows that
Fr - Fr+1 = aq'+1Fr+2, r > 0. A6.1)
Using A6.1), iteration, and the special cases pointed out above, we find that
FJF2 1 + F2/F
1+ i + i +...+ i + i'
which is the required result.
Entry 17 offers another famous discovery of Ramanujan known as "Rama-
nujan's summation of the ^i^." It was first brought before the mathematical
world by Hardy [3, pp. 222, 223] who described it as "a remarkable formula

32 16. ^-Series and Theta-Functions
with many parameters." Hardy did not supply a proof but indicated that a
proof could be constructed from the g-binomial theorem. The first published
proofs appear to be by W. Hahn [1] and M. Jackson [1] in 1949 and 1950,
respectively. Other proofs have been given by Andrews [2], [3], Andrews and
Askey [2], Askey [2], Ismail [1], Fine [1, pp. 19-20], and Mimachi [1]. The
short proof of Entry 17 that we offer below has been motivated by Askey's
paper [2] and has been discovered independently by K. Venkatachaliengar
[1]. See also his monograph with V. R. Thiruvenkatachar [1]. Askey [4] has
discussed our proof along with a "proof" of a "false theorm" to illustrate
certain pitfalls in formally manipulating Laurent series.
We emphasize that Entry 17 is an extremely useful result, and several
applications of it will be made in the sequel. Fine [1] and Bhargava and Adiga
[5] have employed Entry 17 in their work on sums of squares. For a connec-
connection between Entries 14 and 17, see Askey's paper [2]. Further applications
of Entry 17 have been made by Andrews [10], [18, Chap. 5], Askey [3], [5],
and Moak [1]. A generalization of Entry 17 has been found by Andrews [12,
Theorem 6].
As we shall see in Section 19, the Jacobi triple product identity is a special
case of Ramanujan's 1ф1 summation. In 1972,1. Macdonald [1] found multi-
multidimensional analogues of the Jacobi triple product identity, which can also
be considered as analogues of Entry 17, and which are now called the Mac-
Macdonald identities. One of the Macdonald identities is, in fact, the quintuple
product identity, discussed in detail in Section 38. More elementary proofs of
some of Macdonald's identities have been found by S. Milne [1]. These
considerations partly motivated Milne [2]-[4] to develop multiple sum gen-
generalizations of Ramanujan's 1ф1 sum. R. Gustafson [1] has found further
analogues of the ^ summation. Lastly, we mention that D. Stanton [1] has
developed an elementary approach to the Macdonald identities.
Entry 17. Suppose that \Pq\ < \z\ < l/|ag|. Then
j + f №g2)k(-ccgf=k + - (ф,д\(-Рд)к:_к
ел (Pq2;q\ »=i («<?2;<z2)*
{-qz; q2)J-q/z; q2)x \ j"(q2; д2)х{ссРд2; q2)w\ A? „
{-xqz; q2)^-Pq/z; q2)^) \{щ2; q^JPq2; q2)x)
Proof. Let f{z) denote the former expression in curly brackets on the right
side of A7.1). Since/(z) is analytic in the annulus, \Pq\ < \z\ < l/|aq|, we may
set
/(z)= ? ckz\ \Pq\ < \z\ < 1/N|.
From the definition of/, it is easy to see that
(P + qz)f(q2z) = A + ctqz)f{z),

16. g-Series and Theta-Functions 33
provided that also \fiq\ < \q2z\. Thus, in the sequel we assume that
\P/q\ < \z\ < У\щ\.
Equating coefficients of zk, — oo < к < oo, on both sides, we find that
Pq2kck + <z2*-4-i =ck + *qck-i. A7.2)
Hence,
aq{\ - q2k~2/oi)ck_,
and
c_t= -
1 — a.q
where, to get the latter equality, we replaced к by 1 — к in A7.2). Iterating the
last two equalities, we deduce that, respectively,
_ (-щ)к(Щ;г2^
UKoo, A7.3)
ypq > q )k
and
(-f}qf{l/P;q2)kc0
C~k (a<j2; q\ ' ~ °°'
Examining A7.1), we see that, to complete the proof, it suffices to show that
0 (e2;e2U«fa2;eV
Now let q>{z) and ^(z) denote, respectively, the two infinite series on the left
side of A7.1). Now f(z) has a simple pole at z = —l/ctq, and since ф(г) is
analytic for \z\ > \Pq\, we find that
lim A + <xqz)f(z) = lim A + a.qz)q>(z) = lim ^if", A7.5)
by Abel's theorem. Using the definition of/(z) and A7.3), we may rewrite A7.5)
in the form
в2). (Й. 92). "
Equality A7.4) obviously follows, and so the proof of Entry 17 is complete for
\P/q\ < \z\ < l/\ctq\. By analytic continuation, A7.1) is valid for \Pq\ < \z\ <
\l\aq\.
Entry 17 can be reformulated in a more compact setting. We first extend
the definition of (c; q)k by defining
(c) -(c-a) - iC;q)°°
\C)k ~ lc, q)k - 7—5—T-,

34 16. q-Series and Theta-Functions
for every integer k. In Entry 17, now replace a, /?, and z by I/a, b/q2, and — az/g,
respectively. Lastly, replace q2 by q. Then A7.1) can be written in the form
where |b/a| < |z| < 1.
For another proof of A7.4), see the monograph of Thiruvenkatachar and
Venkatachaliengar [1].
Corollary. If \nq\ < \z\ < l/\nq\, then
l+k (nq2;q2)k
_ (-gz; g2U-g/z; д2Щ; g2Un2g2; q2)w
{-nqz;q2)J-nq/z;q2)Jnq2;q2)l
Proof. Set a = ft = n in Entry 17.
The remainder of Chapter 16 is devoted to the theta-function
f(a, b) = 1 + ? (abf«-m(ak + bk) = J дЮ+^ьВД-ид A8.1)
*=1 t=-oo
where |ab| < 1. If we set a = qe2iz, b = qe~2iz, and q = emr, where z is complex
and Im(r) > 0, then f(a, b) = 93(z, т), where ¦93(z, т) denotes one of the clas-
classical theta-functions in its standard notation (Whittaker and Watson [1,
p. 464]). Thus, all of Ramanujan's theorems on f(a, b) may be reformulated
in terms of 53(z, т). It seems preferable, however, to retain Ramanujan's
notation. Not only will the reader find it easier to follow our presentation in
conjunction with Ramanujan's, but Ramanujan's theorems are more simply
and elegantly stated in his notation.
Entry 18. We have
(i) f(a,b)=f(b,a),
(ii) /A, a) = If {a, a3),
(iii) /(-l,a) = 0,
and, if n is an integer,
(iv) f{a, b) = anin+1)l2b*n-mf{a(ab)n, b{abyn).
Ramanujan remarks that (iv) is approximately true when и is not an integer.
We have not been able to give a mathematically precise formulation of this
statement. Repeated use of (iv) will be made in the sequel.

16. ^-Series and Theta-Functions 35
Proof. First, (i) is obvious.
Second,
/A, a) = 2 + ? ?
fc=l k=2
Z
00 00 ^ \
k=i k=\ ) J
к even к odd
00 00
2, a + 2, a
= 2f 1 + У ak(k-1)l2(a3fik+1)l2 + У a
= 2f(a, a3).
Third,
t=2
upon the replacement of к by к + 1 in the first sum on the right side.
Fourth, replacing к by к + п on the far right side of A8.1), we find that
f(a, b) = У д№+»)(*+»
k=-m
*=-оо
which completes the proof of (iv).
Entry 19. We have
f(a, b) = (-a; ab)J-b; ab)Jab; ab)x.
Proof. In Entry 17, let qz = a, q/z = b, and а = /} = 0.
In the notebooks [9, Vol. 2, p. 197], Ramanujan informs us how he proved
Entry 19 by remarking: "This result can be got like XVI. 17 Cor. or as follows.
We see from iv. that if a(abf or b(abf be equal to — 1 then f(a, b) = 0 and
also if (ab)n = 1, f(a, b){l - {a/bf12} = 0 and hence f(a, b) = 0. Therefore
(— a; ab)m, (— b; ab)m, and (ab; ab)m are the factors off {a, b)" (We have slightly

36 16. ^-Series and Theta-Functions
altered Ramanujan's notation.) The product and series in Entry 19 converge
only when \ab\ < 1, but there is even a more serious objection to Ramanujan's
argument. It is not clear that the only factors of/(a, b) are (—a; ab)^, (—ft; ab)^,
and (aft; ab)^.
Entry 19 is Jacobi's famous triple product identity, established in his
Fundamenta Nova [1], [2] but, in fact, first proved by Gauss [3, p. 464]. See
the texts of Andrews [9, pp. 21, 22] and Hardy and Wright [1, pp. 282, 283]
for other proofs.
Entry 20. // a/? = л, Re(a2) > 0, and n is any complex number, then
Entry 20 is a formulation of the classical transformation formula for the
theta-function 53(z, т) (Whittaker and Watson [1, p. 475]). This entry is also
recorded in Chapter 14 [9, Vol. 2, p. 169, Entry 7]. A proof via the Poisson
summation formula is sketched in our book [9, p. 253].
Entry 21. If\q\,\a\,\b\<\,then
L°«-^-=lwh) BU)
and
a. (-\\k-l(ak + bk)
Log/(a, b) = LogH; аЬ)х + % k{l _ a^k) ¦ B1-2)
Proof. For \q\, \a\ < 1,
00 00 00 / Пк-1(ППП\к
Log(-a; e) = X Log(l + aq") = ^ Z ( } , ^ *
и=0 и=0 к=1 К
it=i к „=o k=i k(l — q )
Equality B1.2) follows immediately from Entry 19 and B1.1).
Entry 22. If\q\ < I, then
r\ i \ t-i \ i i т V к2 \Ч'Ч )ooM7 > 4 )o
(О Ф(в):=Л*в)=1 + 2?в
(ii)
(Ш) fi-q) ==/(-«, -<?2) = 1 (-1)У{3к~1I2 + ? (-1)У
k=0 t=l
= (q; 9H0,
<3'+1»2

16. g-Series and Theta-Functions 37
and
(tv) x(q)--={-q\q2)m.
Observe that <p(q) = 53@, т), where q = e™. If q = e2n", then f(-q) =
e~nil/12r](z), where >/(т) denotes the classical Dedekind eta-function. Equality
(iii) is a statement of Euler's famous pentagonal number theorem [1], [5]. See
Andrews' book [9, pp. 9-12, 14] for an elementary proof and further refer-
references. Note that (iv) is only a definition of %(q).
Proof of (i). The first equality follows immediately from the definition A8.1)
of Да, b).
From Entry 19,
f(q,q) = (-<z;«W;<Z2)oo. B2.1)
Now,
B22)
which is a famous identity of Euler. Substituting B2.2) into B2.1), we complete
the proof of (i).
Observe that B2.2) may be rewritten in the form
The equality B2.3) is the analytic equivalent of Euler's famous theorem: the
number of partitions of a positive integer и into distinct parts is equal to the
number of partitions of n into odd parts.
Using B2.3) in Entry 22(i), we derive the useful representations
B2.4)
Proof
which
OF (ll).
proves
For|g|
f(q,1
the first
<1,
?)=! +
00
«odd
k=0
equality.
00
У Q2k
k=l
qk(k+l)/2
00
2 , у
k=0
it even

38 16. q-Series and Theta-Functions
By Entry 19,
Ля, q3) = (-«; q%(-q3; «*).(«*; iX
= (-q-,q2U-q2;q2Uq2;q2)a>
_ (g2; g2)oc
(«; в2)»'
by B2.2).
Proof of (iii). The first equality follows immediately from the definition A8.1)
of/(a, b).
By Entry 19,
/(-«, -<z2) = («; «3U«2; «3).te3; e3). = («; «)«•
Entry 23. For |q\ < 1,
(iv)
and
Proof. Equalities (i)—(iv) follow easily from (i)—(iv), respectively, of Entry 22.
Since the proofs are very similar, we prove only (i). Thus, by Entry 22(i), for
\q\ < 1,
Log Ф(9) = X Log -^ - X Log -я
л = 1 1—9 и=1 1—9
00 (ntt\2k-l
Z
)t=l
?i {2k - 1)A + 92")'
Finally, (v) is an immediate consequence of Entries 22(i), (ii).

16. q-Series and Theta-Functions 39
Example.
Proof. Put q = 75 in Entry 22(ii), and the desired result readily follows.
Entry 24. We have
а) М-
/(-<?) t(-q) x(-q) V <?(-<?)'
(ii) p(-q) = cpH-qWiq) = ? (-tfP* + l)?k(*+1,
lt=O
(iv) /3(
Proof of (i). Using Entries 22(iii), (ii), (iv), and (i), respectively, we find that
each of the given ratios is equal to ( — q; q2)J(q; q2)x-
Proof of (ii). By Entries 22(i) and (ii),
by Entry 22(iii), where in the penultimate line we used B2.3).
The second equality in (ii) is another famous theorem of Jacobi [1], [2]
and is a limiting case of his triple product identity. We refer to the well-known
book of Hardy and Wright [1, p. 285] for a proof. An elegant generalization
of Jacobi's identity has been discovered by Bhargava, Adiga, and Somashe-
kara [4].
Proof of (iii). Each of the five displayed expressions is equal to {—q;q2)aD.
For the first, second, and fifth, this claim follows immediately from Entry 22.
For the third, B2.2) must also be used. Lastly, by Entry 22,
<?(<?)_ (-q;q2)w(q2;q2)x
f(q) (q',q2)x{—q2'^q2)x,(~q''>—q)x,
by B2.2).

40 16. ^-Series and Theta-Functions
Proof of (iv). By Entry 22,
where we have used B2.2) again.
Lastly, by Entry 22 (iv),
x(q)x(-q) = (-«; ч2
Entry 25. We
(i)
(ii)
(Hi) <p(q)<p(-q) = q>4-q2),
(iv)
(v)
(vi)
and
(vii)
Proof of (i). By Entry 22(i),
= 2 + 4 ? q4*2 = 2cp(q*).
Proof of (ii). By Entries 22(i) and (ii),
k=l
Proof of (iii). The first equality is a ready consequence of Entry 22(i). By
Entries 22(i) and (ii) and B2.2),

16. g-Series and Theta-Functions 41
On the other hand, by Entry 22(ii) and B2.2),
Hence, the second equality in (iii) follows.
Proof of (iv). By Entries 22 (i) and (ii),
(«;«2Ш-«2;«2и-«;«2).
Proof of (v). Multiply equalities (i) and (ii) and then employ equality (iv).
Proof of (vi). By Entry 22 (i),
= 2
m,n = — oo
m+n even
= 2 t q2+k2> = 2cp V),
j,k=-m
where we set m + n = 2} and m — n = 2k.
Proof of (vii). Multiply equalities (v) and (vi) together and use (iv).
Corollary. If
1 + t cp2(q) '
then
B5.1)
Proof. Let X denote the right side of B5.1). By Entries 25(iii) and (vi),
<p\-g2)_\ 2у{дЫ-д) I2 = 4Я
=
(A + IJ
after an elementary algebraic computation.
We have given above a slightly more explicit version of the corollary than
did Ramanujan.

42 16. <j-Series and Theta-Functions
In the next section, we write, for brevity,
G(q) = qe»-nmm+n)j-(qm^ qn^ щп>0.
In Entry 26 and its two corollaries, we quote from the notebooks [9, Vol. 2,
p. 198].
Entry 26. G(q) is a perfect, complete, pure, double series of \a degree.
Corollary (i). (p(q), %fq\li(q), and 2ffqf{q) are complete series of \a degree.
Corollary (ii). х(Ф12\[ч *s a complete series of 0 degree.
Proof of Entry 26. The definitions of "perfect," "complete," "pure," "double,"
and "degree" are found in Section 10 of Chapter 15 (pp. 186,187) (Part II [9,
pp. 320-321]).
Apply the Euler-Maclaurin summation formula (Whittaker and Watson
[1, p. 128]) to the function
q(x) '= a(m-n>2/8(m+")+m;
_ a {2(m+n)x+(m-n)}2/8(m+n)
on the interval — oo < x < oo. The series
where Bj denotes the jth Bernoulli number, terminates and, in fact, is iden-
identically equal to zero. Thus, according to Ramanujan, G(q) = ^^=-„0 g(k) is
perfect and complete, respectively.
Next, G(q) is pure because the coefficients (which are = 1) are homogen-
homogeneous.
As with Example 6, Section 10 of Chapter 15, which is the special case
m = n = 1 here, Ramanujan evidently intends "double" series to mean "bilat-
"bilateral" series in this example.
Lastly, M. E. H. Ismail has suggested to us that the degree of a series should
be equal to the order of an appropriate singularity on the boundary of con-
convergence of the series. This does not seem to be the definition given rather
hazily by Ramanujan. But Ismail's interpretation is more feasible for some
of Ramanujan's examples. By the Poisson summation formula (Knopp [1,
p. 40]),
G(q) = У e-*Hk+{m-n)l2(m+n)J
k=-oo
1 oo
_ V o-nk2lt+nik(m-nM(m+n)
— г /_, С ,
where e~nt = qim+n)l2. Thus, G(q) has a singularity at t = 0 of degree \.

16. g-Series and Theta-Functions 43
Proof of Corollary (i). The results follow immediately from Entries 22 and
26.
Proof of corollary (ii). This claim follows at once from Corollary (i) and
the fact x(q) = cp(q)/f(q) from Entry 24(iii).
Entry 27. It is assumed that a and fi are such that the modulus of each exponential
argument below is less than 1. If a/} = n, then
(i)
and
(ii)
If <xfi = л2, then
(Hi) e-
(iv)
and
(v)
Proof of (i). Set n = 0 in Entry 20.
Proof of (ii). Set n = a in Entry 20 and observe that /A, e~2) = 2ф(е~2)
by Entries 18(ii) and 22(ii).
Proof of (iii). By Entries 27(i), 27(ii), and 25(iii),
where аД = л. Multiplying both sides by ф(е~2р2) and using Entry 24(ii), we
find that
Interchanging a and ft, we deduce that
грф-РЩе-^Ще-2112) = ае/73(-е-2).
Divide the former equality by the latter and use Entry 27 (i) to conclude that
Taking the cube root of both sides and replacing a2 and f}2 by a and /},
respectively, we complete the proof.
Proof of (iv). Replace a2 and /f2 by a and /}, respectively, in Entry 27(i) and
multiply the resulting equality by Entry 27 (iii) to get
")/(-е-2"). B7.1)

44 16. gr-Series and Theta-Functions
Using the equality f2(q) = cp(q)f{—q2) from Entry 24(iii) and then taking the
square root of both sides of B7.1), we complete the proof.
Proof of (v). In Entry 27(i) replace a2 and ft2 by a and /?, respectively, and
then divide that equality by the equality of Entry 27 (iv) to obtain
f(e-") fie-»)'
where a/f = n2. Using the equality %{я) = Ф(<?)//(<?) from Entry 24(iii), we
complete the proof.
Of course, Entry 27(i) is the transformation formula for $3@, т) = <93(т), and
Entry 27(Hi) gives the transformation formula for n{x). Several proofs exist for
each of these transformation formulas. Moreover, Entries 27(i) and (iii) are
only special cases of more general transformation formulas under modular
transformations. See Knopp's book [1, Chap. 3] for a full discussion of these
transformation formulas. A unified approach to the transformation formulas
of n{x), 53(t), the other classical theta-functions, and many generalizations has
been presented by Berndt [1], [2], [4].
Entry 28. If p = ab and n is any natural number, then
' " ' f(-p)
Proof. Using Entries 19, 22(iii), and l(iii), we find that
= П (-«p*; p"U-bp"-k; PnUpn; pX
k=l
= П /(«p". bPn~k\
k=l
Corollary. We have
f(-q2, -q3)f(-q, ~q*) = /(-¦*)/(-V)
and
/(-«, -q6)fi-q\-q5)K-q\ -«*) = f(-q)f2(-q1\
After the last equality, Ramanujan [9, p. 199] remarks "and so on." By
these words, he implies that
fl f(~q\ -q2n+1~k) = /(-g)/"~1(-42"+1), B8.1)
k=l

16. q-Series and Theta-Functions 45
where n is any positive integer. The corollary records the cases и = 2, 3 of
B8.1).
We shall now establish B8.1). Employing Entries 19, l(iii), and 22(iii), we
find that
fl /(-«*, -ч2п+1-к) = П {(«*; q2n+1Uq2n+1-k; <z2n+1U<z2n+1; q2n+1U
k=l k=l
Entry 29. // ab = cd, then
(i) /(a, b)f(c, d)+f(-a,-b)f{-c-d) = 2f(ac, bd)f(ad, be)
and
(ii) f(a, b)f(c, d)-f(-a,- b)f{ -c,-d)
Many of the identities of Entry 25 above and Entry 30 below are instances
of Entry 29. Formula (ii) was discussed by Hardy [3, p. 223] who also briefly
sketched a proof. Since the proofs of (i) and (ii) are similar, we give only the
proof of (i). Less elementary proofs of Entries 29(i), (ii) may be found in the
treatise of Tannery and Molk [1].
Proof of (i). Letting p = ab = cd, we see that
f{a, b)f(c, d)= ? р(-+»2)/2-(т+")/2аися.
m,n=— oo
Thus, setting m — n = 2/ and m + n = 2k, we find that
f(a, b)f(c, d) + /(-«, -b)f(-c, -d)
— у р(т2+п2)/2-(т+п)/2атсп
m,n~—oo
m+n even
= 2 X ря+*1-*в'+*с*-'
M=-oo
= 2 J pk^\acf^+l
j,k = -m
= 2f(ac, bd)f(ad, be).
Several of the identities of Entry 25 are special cases of the formulas in
Entry 30.

46 16. ^-Series and Theta-Functions
Entry 30. We have
(i) f(a, ab2)f(b, a2b) = f(a, Ь)ф(аЬ),
(ii) f(a, b)+f(-a,-b) = 2f(a3b, ab3),
(iii) f(a, b) - f(-a, ) f,
(iv) f(a, b)f(-a, -b) = /(-a2, -Ь2)(р(-аЬ),
(v) f\a, b) + f2(-a, -b) = 2f(a2, b2)cp(ab),
and
(vi) /2(a, b)-f2(-a, -b) =
In the proofs below, we set p = ab.
Proof of (i). Using in turn Entries 28, 24(iv), and 24(iii), we deduce that
f(a, ЬЫ-р
= /(а,Ь)ф(р).
Я-Р)Я-Р2)
Proof of (ii). Using the definition A8.1) of/(a, b), we find that
f(a,b)+f(-a,-b) =
k=-aD
к even
= 2 У рЧ2к-\)а1к
k = -ao
= 2 ? (p*)k<k-»>2(a3b)k
k=-oo
= 2f(a3b, ab3).
Proof of (iii). Proceeding as in the proof above, we have
к — — оо
к odd
= 2 У" п*B*+1)/12*+1
к=-оо
= 2a

16. q -Series and Theta-Functions 47
Proof of (iv). By Entries 19 and 22(i) and B2.3),
f(a, b)f(-a, -b) = (a2; P2)Jb2; p2)Jp; p&
\P г P )a>
Alternatively, if we set с = —a and d = — b in Entry 29 (i), we easily obtain
the desired result.
Proof of (v). Putting с = a and d = b in Entry 29 (i), we easily achieve the
sought result.
Proof of (vi). Set с = a and d = b in Entry 29(ii) and use Entries 18(ii) and
Corollary. If ab = cd, then
f{a,b)f(c,d)f(an,b/ri)f(cn,d/n)
-/(-a, -b)f(-c, -d)f(-an, -b/n)f(-cn, -d/n)
= 2af(c/a, ad)f(d/an, acn)f(n, аЬ/п)ф{аЬ).
Proof. For brevity, set
a = f(a, b)f(c, d), a' = /(- a, - b)f( - c, - d),
p = f{an, b/n)f{cn, d/n), /?' = /(- an, - b/n)f( - en, - d/ri),
and
L = a)? - <x'P'.
By Entries 29(i) and (ii), we readily find that
a +¦«' = 2f(ac, bd)f(ad, be),
and
Substituting these in the obvious identity
2(a/? - а'Л = (a + a')(? - Л + (a - a')(? +

48 16. q-Series and Theta-Functions
and using Entries 29 (i) and (ii), we find that
L = af(ad, bc)f[~, -abed I \2nf(ac, bd)f[ —=, ~^-abcd
\d b /I
(b с . \ J , bd
If we apply Entry 30(i) and use the hypothesis ab = cd, we find that the
equality above may be written
L = 2af(c/a, ad№(ab)f(n, ^)/(^.
which is what we wanted to prove.
Entry 31. Let Un = а*№Щ«(«-т апд Vn = а<*п-1Щ«*+т for each integer
и. Then
"f () C1.1)
Ramanujan writes Entry 31 in the form
№» V,) = f(Un, VB) + l/x
-' Cu)
where the sum on the right side evidently contains n terms. However, by Entry
18(iv), for r > 1,
rJ \ V ' V I ~ n~rJ \ V3 ' U
\ 'r yr / \ yr un-r.
This shows that the sums on the right sides of C1.1) and C1.2) are equal.

16. q -Series and Theta-Functions 49
Proof. We have
oo n-1 /f/ W+D/2 /y \k(k-l)/2
I Е
k=-oo c=0
_ V V д(и*+г)(и*+г+1)/2?(п1к+г)<л1к+г-1)/2
k 0
k = -co r = 0
Bhargava and Adiga [4] have given a slightly different proof of Entry 31.
Corollary. We have
(i) <p(q) = (p(q9) + 2qf(q\q15)
= <p(q25) + 2qf(q15, q35) + V/(«5, «45)
(ii) ^(q) = /(q3, q6) +
Proof. The two equalities of part (i) follow from Entry 31 and C1.3) by setting
a = b — q and n = 3, 5, respectively.
The four equalities in part (ii) follow from Entry 31 by setting (a, b, n) =
A, q, 3), (q, q3, 2), A, q, 5), and (q, q3, 3), respectively.
Parts (i) and (ii) above are, in fact, special cases of the following general
formulas:
<p(q) = (p(q) + "z qr2f{qn{n~2r\ qnln+2r))
and
1 "~1
2 ,=0
n-l
_ у _rBl—1)Л пBп-4г+1) nBn+4r-l)\
r=0
where n is any positive integer. These three formulas are obtained from Entry
31 by setting (a, b) = {q, q), A, q\ and (q, q3), respectively.

50 16. <j-Series and Theta-Functions
Example (i). We have
q>2(q) , (p2(r) , q>2(s) , <p2(q)<p2(r)<p2(s)
<p2(-r) + cpH-s) (p2(-q)(p2(~r)(p2(-s)
<p2(q2)<p2(r2)<p2(s2)
^
Proof. By Entries 25 (vi) and (v), the right side of C1.4) can be written
2 Г <Р2(-Я)П срЧ-г)П <Р2(~*)У
Upon simplification, the expression above yields the left side of C1.4).
Example (ii). We have
1 1 1
+ <p(-q)±cp(q2)
and
+
^P?j (p(-q2)±(p(q) + (p(-q2)±(p(-qy
Proof. It is easy to see that the equation
a- = cTb + dTb CL7)
is satisfied if
2A = С + D and 2A2 - B2 = CD. C1.8)
Take A = ф(^4), В = q>(q2), С = <p(q), and D = cp(-q). By Entries 25(i), (iii),
and (vi), equalities C1.8) are seen to be satisfied for these choices of А, В, С,
and D. Hence, C1.5) follows at once from C1.7).
Second, the equality
? = E±C + ?±D CL9)
is seen to be satisfied if CD = E2. Now by Entry 25 (iii), this equality is satisfied
when С = cp(q), D = q>(- q), and E = cp(- q2). Hence, C1.6) follows from C1.9).
Example (iii). For each natural number n, the coefficient of q" in the expansion
°f [i/(l — q)№(q2) is the integer nearest to y/n.

16. <j-Series and Theta-Functions 51
Proof. By Entry 22(ii),
>¦ — Q. J=l k = \
Thus, the coefficient of q" is equal to the number of powers qk(k~1] such that
k(k — 1) < n. In other words, the coefficient of q" is the unique integer к such
that
(k -1J < k(k - 1) + 1 < n < k(k + 1) < (k + iJ.
Clearly, from the inequalities above, к is the nearest integer to y/n.
Example (iv). We have
and
(p(-q)-q>(q2)= -'<
Ш
Proof. Putting a = q,b = q3, and с = d = q2 in Entries 29(i) and (ii), we find
that
ФШ(Я2) + M-q)(p(-q2) = 2/V, q5)
and
t(q)<p(q2) ~ *(-qM-q2) = 2qf2(q, q1).
These equalities reduce to the desired identities on using the fact
ip(-qM-q2) = (p(-qmq% C1-10)
which is deducible from Entries 25(iii) and (iv).
Example (v). We have f(q, q5) = il/( —
Proof. By Entries 19, 22(ii), and 22(iv),
f(q, q5) = (~q; q6U~q
Entry 32. We have
n
<p(q)

52 16. <j-Series and Theta-Functions
ф(д) (jo( — q) Aq
and
H ^_gVj)=_4^'(-,2)
(p(q) <p( — q) <P(~q )
Proof of (i), (ii). Using first Entry 25 (iv) and then Entry 23 (ii), we find that
<p'(q) Ф'(ч) Ф'(я) ~ |A'(^2)
<p(q) Ф(я) Ф^) Ф(я2)
А ой лк Д on Л2к
А 2ЛA
_ d « (-1)*"У
dq t=i /c(l + qk)
_ » (-I)**/*-1 1 -q>4(-<
A + q"J 8q
by C3.5).
Proof of (iii). Proceeding in the same manner as above, with the help of Entry
23(i) and C3.5), we find that
in'tn\ m'l — n) d °° ' л2* л2*-1
+ ; i- = 2—
dq ^ 1B* - 1)A + e2*~i) B/c -
_0
4*J *tid+(-9ГГ
| У4(9)-1
which concludes the proof of (iii).
Proof of (iv). The desired equality is an obvious consequence of Entry 25 (iii).
Entry 33(i). If\q\<lande is real, then
[1+2 ?"=1 qk2 cos(fcfl)-| » (-1ГУ ,. m
L08L 7F?) J = 2й Щ^51)cos(fe0)-
Proof. In B1.2), set a = qew and b = де"ш, and then employ Entry 22(iii).

16. q-Series and Theta-Functions
Entry 33(ii). If\q\ < 1 and n is real, then
00
iLog
«c=i
sin n
k=l
Proof. Letting z = e2'" and using Entry 19, we find that
53
C3
C3.2)
- z~112
= 1
By letting n tend to 0 in C3.2), or by employing Entry 24(ii) along with Entry
22(iii), we see that
k=l
Thus, if Q is the quotient in large parentheses on the left side of C3.1), we have
shown that
2ing;
(e2mq;
Taking the logarithm of Q above and using B1.1), we find that
which completes the proof.
Entry 33(Hi). If\q\ < 1 and n is real, then
1 + 2 ? q*2 cos(fcn)
C3.3)
1+2

54 16. q-Series and Theta-Functions
Proof. Observe that the right side of C3.3) can be written
9 ( q >f{-zq, -
where we have employed Entry 19 and B2.4). If we now replace n by — 1 and
z by ein in the corollary to Entry 17, we find that the right side of C3.4) is equal
to
and this completes the proof.
Corollary. // \a\, \b\ < 1, then
а" +b"
Proof. Put qein = a andqe~in = bin Entry 33(iii).
Entry 3 3 (iii) essentially gives the Fourier series of the elliptic function dn u,
where и = Kn/n, and where К denotes the complete elliptic integral of the
first kind.
We now derive a useful consequence of Entry 33 (iii). Replacing n by л — n
in C3.3) and multiplying the two results together, we find that
Form the product of the two series on the right side and then integrate both
sides with respect to n over the interval [ — n, я]. Since the set of functions
{cos(kri)}, 0 < к < oo, is orthogonal on [ — я, я], we deduce that
/H2)=1 + 8S(T^' <33.5)
a result due to Jacobi [1], [2].
Formulas for q>2"{q), 1 < n < 12, similar to that found above, have been
derived by Ramamani [1].
Entry 34(i). If\q\ < 1 and n is real, then
<p2(q)
Log
1+4 cos n
„2*-l

16. ^-Series and Theta-Functions 55
Proof. Letting a = /? = q in Entry 17, we see that
(-zq2;q2U-q2/z;q2Uq3;q2)l
1 -9 у
1 + q (*=i 1 - <T
oo r_i\k-l/72*-l
oo С \\к~1п2к~1(тк~11г Л- 7-*+l/2w.l/2
+ ?( l-« w 1 - 92*
-9b + I
1 - q"" l
With 2 = e2'" and the use of Entry 22(i), we may rewrite the foregoing
conclusion in the form
2/ /W 1^л ^ (irVcosBfcl)n
(q)W)= 1 + 4 cos" S r^T^ ' C4-2)
where
F(n) =
Comparing C4.1) and C4.2), we see that it suffices to show that
F(n) л« (~\)kgksm2(kn)

56
16. q-Series and Theta-Functions
To show the equality above, it suffices to show that
-> V (-lTVcosBM
l
The proof of this is quite straightforward and much like the proofs of Entries
21 and 23, and so we omit it.
Entry 34(ii). If\q\ < 1 and n is real, then
iLog
1 +
V qk cosB/cn)
*Z4
+ q
2k
q2*-1 sin2B/c - l)n
Bfe^
C4.3)
Proof. Putting a = jS = -1 and z = e2in in Entry 17, we find that
?* cosBfcn) (-zg; д2Ы~Ф; q2)Jq2; 42)l
k=l
C4.4)
'G(O)'
where we have used Entry 22(i) and where
Comparing C4.3) and C4.4), we see that it suffices to show that
if2* cos{2Bfe -
Log G(n) = 4
Bk -
Like the calculation of Log F(n) in the previous proof, the proof of the equality
above is quite straightforward.
Ramanujan now states two "corollaries." We have not been able to discern
why the appellation "corollary" has been given to these two results. Moreover,
the "corollaries" are incorrect. We give two corrected versions of each cor-
corollary. First, we prove versions where the "right sides" are corrected; second,
we establish reformulations when the "left sides" are corrected. Most likely,
the first versions are what Ramanujan had in mind. Our first proof below uses
Watson's quintuple product identity, which has been rediscovered several
times and which appears in the literature in several guises. We provide a
thorough discussion of this identity at the end of Section 38, after giving a
proof based on one of Schroter's formulas which we develop in Section 36.

16. q-Series and Theta-Functions 57
Corollary (i) (First Version). If\q\ < 1 and z = e2'", where n is real, then
'2) sin{2C/c -
iLog
k=-oo
sinBn)
i
2, qksin2(kn)
fc=-oo
ksin2(
sin2B/cn)
Proof. By Watson's quintuple product identity C8.9),
C4.5)
k=-oo
q3k2+k(z3kq~3k -
_Зк2-2*2-3*+1 I
*=-oo
= 2iz
t=-oo
sin{2C/c -
where we obtained the penultimate equality by replacing к by к — 1 in the
previous latter summands. Thus, upon dividing both sides above by 1 — z2,
we find that
1
k=-cc
F@)
sinBn)
k=-oo
where
F(n) = (qz; q2)Jq/z; q2Uq*z2;
Now a straightforward calculation yields
Log F(n) = -2 ? fnC
so that
iLog
" qAk cosD/cn)
?'(! ~ cosBfcn)) 1 ^ qAk{\ - cosDfcn))
F@)~2)tti k{\-q2k) +2kk k{\ -
Combining C4.6) and C4.7), we readily arrive at C4.5).
C4.6)
C4.7)

58 16. ^-Series and Theta-Functions
Corollary (i) (Second Version). If \q\ < 1 and z = e2m, where n is real, then
iLog
_ »
qksin2(kn) « q4k sm2Bkn)
+
C4.8)
Proof. Let
P(z) =
(zq; q2Uq/z;
l q2)l
2l q2)
Then a straightforward, but rather lengthy, calculation shows that
C4.9)
Using the factorization
(zV;e4)« = (z92;92U-z«2;e2)«,
a similar factorization for (q4/z2; q*)x, and Entry 19, we find that
p,, _ (zg; g2)oo(gA; <z2U-z<?2; g2)J-g2/z; ga),,(ga; g2)
(Z)~ (q;q2)l(q*;q*)l
_f(-zq,-q/Z)f(z,q2/z)
(l+z)(q;q2)l(q*;q*)l
C4.10)
where we have used Entry 30(i) with a = —q/z and b = z.
Set
F(n) = ¦
\-z '
and observe that, by Entry 22(ii) and B2.2),
C4.11)
(q;
(-9; «2U«; q2)l(-q2;
Hence, C4.10) may be written
P{z) =
F(n)
C4.12)

16. q -Series and Theta-Functions
By C4.8), C4.9), and C4.12), it suffices to show that
F{n) = sec n ? {-q)kik-1)l2 cosBfc - l)n.
k=l
59
C4.13)
But,
by C4.11),
F(n) = -
z
z
.1/2 +
1
1/2 +
/2
z/2
Z'2
00
V (_д)*(*-1)/22«с
00
fc = l
from which C4.13) is apparent. This completes the proof.
Corollary (ii) (First Version). If \q\ < 1 and z - e2in, where n is real, then
Hog
sin n ? F/c + l)q11
k=-oo
« qksin2(/cn) S q*sin2B/cn)
- qk)
- q2k)
C4.14)
Proof. Applying Watson's quintuple product identity C8.9), we find that
(q; qUqz; qUl/z; qUqz2; Ч2)Лфг; q2)x
= I 4
k=-oo
X sinFfc
fc=-oo
Dividing both sides by A - 1/z), we deduce that
G(n) I «C*2+*)/2 sinFfc +
G@)
C4.15)
sin n ? Ffe
*=-oo
where
G(n) = (qz; qUq/z; qUqz2; q2Uq/z2; q2)x.
Proceeding in the same fashion as in the first proof of Corollary (i), we find that
? L°g
G(n) » qk sin2(/cn)
к
sin2B/cn)
к k(l - qk) + к fc(l - q2k)
Taking C4.15) and C4.16) together, we deduce C4.14).
C4.16)

60 16. q-Series and Theta-Functions
Corollary (ii) (Second Version). If\q\<\ and z = e2in, where n is real, then
x /4 cos и » (- 1)V cosB^ + \)n
Ж 8 Wi^Q) ^ l+2q2koos{2n) + 4
_ «> qk sin2(fcn) » q* sin2Bb)
"A fe(l-qk) +»4i fc(l+9») " l J
Proof. Let
(zq; qUq/z; qUz2q; q2Uq/z2; Я*Ш; Q2)l
Then an elementary calculation yields
Put
Using the factorization
and a similar factorization for {q2/z2; q2)^, we find that
(z2q;q2Uq/z2;q2Uq2;q2)l
G{H) {-zq2; q2U~zq; q2U-q2lz; q2U~q/z; <?V
Employing Entry 17 with z, a, and /? replaced by — z2, — 1/z, and — z,
respectively, we deduce that
+
f (-l)V{(z*+1/2
_ /1/2 , -1/2ч у I ^J g lZ
1 \^ A 2*)(l
« (-l)VcosBfc+l)n
= 4 cos n У -;—^
ki^oo 1 + 2o-
Next, by B2.4),
G(n)
C4.20)

16. q-Series and Theta-Functions
61
Putting C4.19) into C4.20) and combining the result with C4.18), we complete
the proof.
For other results in the spirit of those in Section 34, see two papers by
Rogers [1], [2].
Recall that the Bernoulli numbers Bn, 0 <, n < oo, are defined by (Grad-
shteyn and Ryzhik [1, p. 1076])
Д.
ел -
2я.
Recall also that the Euler numbers E2n, 0 <, n < oo, are given by (Gradshteyn
and Ryzhik [1, p. 1078])
secx-
2,
„=o
C5.1)
These conventions for the Bernoulli and Euler numbers are different from
those used by Ramanujan.
Entry 35(i). For each positive integer n, let
п=~2п + к ГГ^'
where В„ denotes the nth Bernoulli number. If n is any nonnegative integer, let
V (_
1 k=i
ы n+l |
Then for each positive integer n,
k=i
Proof. We shall write Entry 33(ii) in the form
L := 2 Log
C5.2)
C5.3)
and equate coefficients of like powers of Q. Expanding sinB/c — 1H in its
Maclaurin series and then inverting the order of summation, we readily find

62 16. ^-Series and Theta-Functions
that
On the other hand, using a familiar expansion for Log((sin в)/в), which, in fact,
Ramanujan derived in Chapter 5 (p. 52) (Part I [5, p. 122, Entry 16]), we find
that, for 10| <n,
1 (-iyBvBe) (-1У2 кд
2 h Bj)B/)! h B/)! »ti 1 - <j*
- ?
Using C5.4) and C5.5) in C5.3), we deduce that, for |0| < n,
Differentiating both sides with respect to 0, we find that
<» f iw) oo / iy+122j'+1 a1 i— Uk
4 U 2j i
„4 Bи - 1)! ~ U B/ - 1)! 2j i
oo
_ у / 1\n
+l у ^ r2j\i2n-2j
у
kBj-l)\Bn-2})\ ¦
Equating coefficients of 02n~1, n > 1, on both sides, we readily deduce C5,2).
In particular, if n = 1 in C5.2), we deduce that
\Qi = -4P2.
If, as usual, <r(fc) denotes the sum of the positive divisors of k, we find, by
equating coefficients of qm, that the foregoing equality is equivalent to the
arithmetical identities
24 t (_ 1Г™<2* - Ц^ - fci>) = B» - IK,
if m = n(n — l)/2, n > 1, and
otherwise. Here, <r@) - — ^j. These last two identities were first established
by Halphen [1] in 1877. Otherwise, Entry 35(i) and its general associated
arithmetical identity appear to be new.

16. <j-Series and Theta-Functions
63
Entry 35(ii). For each positive integer n, let
П — 2")B °° (-
2n '& l+qk '
where Bn denotes the nth Bernoulli number. For each nonnegative integer n,
define
(-If'1 Bk- lJ^2*-1
1-9
2«c-l
*2" 1_ « (_i)"-i^-i
4 + к 1 - q2*-1
where E2n denotes the 2nth Euler number. If n is any positive integer, then
;- 1
Proof. Write Entry 34(i) in the form
oo С 1
C5.6)
L:= -Log
sec в + 4
k=l
1-9
2*-l
C5.7)
As in the previous proof, we expand both sides in powers of в and equate
coefficients.
Using C5.1), we find that, for |0| < я/2,
,% Bn)!
.tb Bn)!
Putting n = 0 in Entry 34(i), we find that
Bn)!
2n. C5.8)
C5.9)
C5.10)
On the other hand, using a well-known expansion for Log(sec в), which,
Combining C5.7)-C5.9), we conclude that

64 16. q-Series and Theta-Functions
in fact, Ramanujan found in Chapter 5 (p. 52) (Part I [5, p. 64, Entry 17]), we
find that
» (iy«2*(l2«)B
в
-lqk v
k
= - {-iy2^Pv
Hence, putting C5.10) and C5.11) in C5.7), we deduce that
„U Bn)! P\M Bj)\
Differentiating both sides with respect to в, we find that
V (~1)2.fl2-1 = f (-1У2^+1Р2, » (-lfQ2k
h Bn - 1)! jti Bj - 1)! ktb BЛ)!
= V Г-П" V ^ J P2jQln-2j д2л-1
n4xl ' ? B/- l)!Bn - 2ЛГ '
Equating coefficients of в2"'1, n > 1, on both sides, we easily deduce C5.6).
Ramanujan (p. 202) has an erroneous factor of ( —1)"~* in the sum on the
right side of C5.6).
In order to state some arithmetical deductions from C5.6), we need to define
the divisor sums
o*{r)= Z (-lf-\2k-\y
Bfc-l)|r
and
The case n = 1 of C5.6) gives the equality Q2 = 8P2 from which we may deduce
the curious formula
*}{n) = 8 ? а№*Лп - к), C5.12)
*o
where n is any nonnegative integer, ^(O) = \ = — o?@), and ст^О) = — f. Not
only is C5.6) new, but even the arithmetically equivalent special case C5.12)

16. q-Series and Theta-Functions 65
does not appear to have been given in the literature before we mentioned it
in [6].
Define
— q
fc=i l — Я
and
oo ksnk
N = 1-504 ? A«
*=i i — Я
Ramanujan remarks that
Sn := ? (- lf+1Bfc - ip+y<*-i>A n > 1;
can always be expressed in terms of L, M, and N. Now by Entry 35(i), Sn can
be written as a polynomial in P2, P4,..., P2n, n> 1. In an epic paper, Ramanu-
Ramanujan [6], [10, pp. 136-162] proved that Р2к, к > 1, can be expressed as a
polynomial in L, M, and N. (In [6], L, M, and JV are denoted by P, Q, and R,
respectively.) Hence, Sn can be represented as a polynomial in L, M, and JV.
Examples. Let Qn, n>2,be defined as in Entry 35(i). Let L, M, and N be defined
as above. Then
@ 3Q2 = L,
5L2 - 2M
() 5Q
and
35L3 - 42ML + 16ЛГ
(i") 7Q6 =
Proof. The three desired equalities follow by putting n = 1, 2, and 3, respec-
respectively, in Entry 3 5 (i). The calculations are straightforward.
Note that Entries 36(i), (ii) below reduce to Entries 29(i), (ii), respectively,
ifp = 1.
Entry 36.1fp = ab/cd, then
(i) S := {{f{a, b)f(c, d) + f(-a, -b)f(-c, -d)}
= ? (adY*+1)l2{bc)k*-1)l2f(acpk,bd/pk)
*=-oo

66 16. q-Series and Theta-Functions
and
(ii) D := \{f(a, b)f(c, d) - f(-a, -b)f(-c, -d)}
k=-oo
Proof. We prove just (i), since the proof of (ii) is similar. Putting n — m = 2/c
and n + m = 2/, — oo < j, к < oo, we find that
m,n =—oo
m+n even
S = V /с^ут2+п2-т-п)/2дт^п_т(т-1)/2
m,n = -
m+n ev
CO
= .z_
oo oo fac\JU+l)'2
*=-oo /=-oo \P~/
which, upon the replacement of к by — k, is seen to equal the desired result.
We now state and prove some very general and useful formulas originally
due to H. Schroter in his dissertation [1]. These formulas will be employed
in Sections 37 and 38. But even more importantly, Schroter's formulas will be
utilized in proving many of Ramanujan's modular equations, especially in
Chapter 20. Ramanujan evidently never stated these general formulas in his
writings. However, from the many special cases that he clearly had proved,
he at least possessed the ideas needed to prove the general formulas.
Put
a = Aq»+\ b = q»+v/A, с = B<f~\ and d = q^/B,
where ц and v are integers such that ц > v > 0. Then
p = fl4v, abed = a4", and — = A2 IB2.
be
Entry 36(i) now takes the form
Now let к = fin + m, — oo < n < oo, 0 < m<, ц — 1. Thus,
S= Z I p qf(ABq,
m=O n = -oo \O/ \ Ab
Next, apply Entry 18(iv) with
a = ABq2»+4vm, b = q2»-Avn/AB,
and n replaced by vn. Hence,

16. ^-Series and Theta-Functions 67
oo / A\iin+m
)
m=0 я=-оо
/i-l / 4\
V I I л2*"
L. \ ч
0
In summary, we have shown that
S = \{f{Aq^\ q"
+ f(-Aq"+\ -
q J
C6.1)
We now examine Entry 36(ii) under the same substitutions as above. Thus,
letting к — цп + m, — oo<n<oo, 0 <m< fi — 1, we find that
= А У У iABY"+mq(l'+y)i2'"'+2m+1)+2>'l>'"+mJ
m=0 n=-oo
(pn+m)+2v+4.fi ~
Apply Entry 18(iv) with
c.4-v(pn+m)+2v+4.fi _-
and with n replaced vn. Therefore,
fi-l oo /,4
D = A Y У (^ т^+я'^+'Хгдя+гт+к+г^^я+тJ [ ^
0
=0 и=-оо
4vm
__-2v-4vm
= А У
0
я=— с»
"¦ 4/( + 2v+4v»i " _-2v-4vm
jq

68 16. ^-Series and Theta-Functions
In summary, we have shown that
-f(-Aq"+\ -q»+*/A)f{-Bq»-\ -
= A Y (^B)mfl<2m+1)<"+v)+2"m2
m=O
n{2fi+4m+2)(fi2-v2) 4
x f(^q4"+2y+4vm, ~q-2v-*vm\ C6.2)
We now record a couple of special cases of C6.1). Letting A = В = 1 in
C6.1), we find that
— V a2?1fLqVp+imMp2—»2) „Bд-4т)(д2-у2)\у/ 2fi+*vm _2/i-4v»i\ /-jg y,
0
m=0
Next, putting Л = q"+v and В = q"~v in C6.1) and using Entries 18(ii), (iii), we
find that
ц-l
_ у qlnmi + Zvmri B^+4m)(^2-v2)> Bд-4ж)(^2-у2)чу/ 4(j+4vm^ д-^ж\ rtg 44
m=O
Adding Entries 30(ii) and (iii) yields
f(a, b) = f(a*b, ab>) + af(^,~a*b^.
Putting a = q2™+»i2 and b = q-2vm+^2, we see that
Multiplying C6.4) by q11'2, adding the resulting equality to C6.3), and using
C6.5), we deduce that
m=0
X
/i-l
__ у 2/im2rl B/i+4»i)((j2-v2) Bfi-Am)(ft2-v2)\j-r 2vm+fi/2 -2vm+fi/2\ pg g\
m=0
Looking back at the proofs of C6.1) and C6.2), we observe that we can

16. ^-Series and Theta-Functions 69
replace m by m + )\i for any integer; and not alter the summands on the right
sides of C6.1) and C6.2). Note that C6.3) and C6.6) also remain unchanged if
m is replaced my — m. Finally, observe that, with the use of Entry 18(iv), we
may replace m by — m on the right side of C6.4) as well. These observations
are useful in simplifying these formulas somewhat.
To illustrate the remarks above, consider C6.4). Replacing q2 by q, we
deduce that
m=l
if ц is odd, and
(f,+2m)(fi2-v2) _((j-2m)(;i2-v2)\y/ 2vm _2
г дЪ-1»), C6.8)
if ц is even.
As a second illustration, set A = qM+v and В = q~<"~v> in the "difference"
formula C6.2). Replacing q2 by q, employing Entries 18(ii)-(iv), and using the
remarks above, we deduce that
v2^ Ul-2m-l)(^-v2)\
X f(q*+*+2™\ q"-v-2m), C6.9)
if ^ is odd, while
_ V _/im(m+l)y/_(/i+2m+l)(;i2-v2) q(ti-2m-l)(fi2-v2)\fr ц+v+Zvm p-v-2m\
m=0
C6.10)
if /^ is even.
The next formula will be particularly useful in Chapter 20. Let ц be an even
positive integer and suppose that со is an odd positive integer such that
in, oo) = 1 and 2ц - со2 > 0. Then
-шЧ q2"-<°2/A)f(qB, q/B)
f(-q2"-'°2A, -q2^2IA)f{-qB, -q/B)}
d-2b
* 24). C6.11)

70 16. ^-Series and Theta-Functions
The restriction (ц, со) = 1 is not strictly necessary. However, its removal
would cause complicated modifications in the formula C6.11). For brevity,
the expression q2ft~a2 will be replaced by Q when convenient.
To prove C6.11), first let a = qB, b = q/B, and n = со in Entry 31. Then
apply Entry 18(iv) with n = r. Accordingly,
ю-1
f(qB,q/B)= ? qr2Brf{qm2+2rmBm,qm2-2raB-a)
r=O
ю-1
= X 1(a> *' Г В <Ш 1)г/(^ш 2ш<ш 1)ГВШ, qa +2ш(ш 1)ГВ ш).
Now substitute this sum and the corresponding sum for /(—qB, —q/B) into
the left side of C6.11). Recall that со is odd so that (со — l)r is even. We then
apply the "sum" formula C6.1). In this application, A and fi remain unchanged,
but В is replaced by Ba>q~2<a{u'~1)T and v is replaced by ц - со2. Now C6.1) was
derived under the assumption that 0 < v < ц. However, by a similar argu-
argument, we can conclude that C6.1) is valid for all integers v such that |v| < fi.
Thus, our application is valid provided that 2fi — со2 > О. Hence,
1 ю-1
n = л Z q(a>~1Jr2B-(B>-1)r{f(QA, Q/A)f(q<o2-2<°«°-1)rB<o,,
2 r=o
+ /(-64 -Q/A)f(-q<°2-2<o('o-1)rB<o, -t
to—1 ft— 1 /a q2(o(v>—l)r\m
~ hi »V B q" \ в°>~
/(CO2 D(OBfl-(O2)
A
s/ f\ A Dco-,2u+4mu—4co-zm—2co(co— l)r _2u-4mu+4co2m+2co(co—l)r
= "У* V1* 04!ют+<ю-1)г/2!2д-2{ЮЯ1+(ю-1)'1/2}
X f
where in this last step we applied Entry 18(iv) with n = m.
In each expression on the right side above, m and r occur only in the
expression com + j{co — l)r. For each pair (m, r), apply the division algorithm
to deduce that
com + |(co — l)r = qn + n, 0 < n < ц — 1,
for a certain nonnegative integer q. Suppose r is fixed and m varies from 0 to

16. ^-Series and Theta-Functions 71
ц — 1. We claim that n then assumes all values from 0 to ^ — 1 in some order.
To that end, let n and ri correspond to the distinct values m and m'. Then
n — ri = co(m — m')(mod ц).
But \m — т'\<ц and (со, fi) = 1. Thus, ц\{п — ri); that is, n ф ri, and our
claim is established.
On the other hand, suppose we consider two different values of r, say r and
r', and select values of m which yield the same n. This is possible by the
conclusion of the preceding paragraph. Let q and q' be the corresponding
quotients. Then
\{co - l)(r - r') = n(q - q')(mod со).
Now \(co — 1) and со cannot have a factor in common, and 0 < \r — r'\ < со.
Thus, co\(q — q'). In other words, q and q' cannot differ by a multiple of со.
Hence, as r assumes the values from 0 to со — 1, the со values of q differ from
multiples of со by the numbers 0, 1, 2,..., со — 1 in some order.
Now each term of the last double sum above remains unaltered if a
multiple of цсо, say l/xco, is added to the argument com + \(co — \)r. To see this,
apply Entry 18(iv) with n = I and n = co2l, respectively, to the pair of theta-
functions in each summand. Consequently, when we effect the substitution
com + %(co — \)r = n + q\i, n runs from 0 to ц — 1, and it can be assumed that
q runs from 0 to со — 1. Thus, replacing the index q by s, to avoid a conflict
in notation, we have proved that
ш—1 ft—I
Q _ у у _4(n+S(*J2j-2(n+S0)
s=0 n=0
= "f
\2fi<D2+4<D(n+sn)
X V (
? Bi2"-'a2)s
(А
where we have applied Entry 18(iv) with n = cos. Lastly, apply Entry 31 with
n = со,
g2fi-a2 ^ '

72 16. ^-Series and Theta-Functions
and
132ft —a>2
b = - Q2fl-4m.
We find that the inner sum above reduces to
( A<° 2 +4m В2ц~ш2
This then completes the proof of C6.11).
We now record some special instances of C6.11) that will be useful in
establishing certain modular equations in Chapter 20. In each case, of course,
ц is even, at is odd, and {fi, со) = 1.
First, if Л = B= 1, then
_ у _4m2r/ Bд-ю2)B/1+4т) _Bд-ю2)B0-4»!)\/Y 2fi-4-<om „2ц+4ют\ pg
Second, let A = q2*-™2 and В = q°>. By Entry 18(iv) with n = {co- l)/2,
f(q»+\ q-°>+1) = q-^-^fiq2, 1) = 2q-^-
Hence,
H
_ V д4т2+д/2-2шту/ Bц-аг)Bц+4т)^ {2ц-а>*%2р-4т)\г1 Ар-Ашт д4ют\
C6.13)
Adding C6.12) and C6.13), we find that
^ + cp(-q2""°2M-q)} + b^-^-^
_ у _4ту(дBд-ю2)B/1+4»1) B/i-ra2)B;i-4m)\ f у-/ 2ц-4<от 2р+Аа>т\
_|_ _/1/2-2ю»1/-/_4д-4ю»1 _4ют\|
= У д4т2Дд<2^-ю2)B^+4т)^ Bд-ю2)Bд-4т)чд д/2-2сат^ ф+2ют\
т=0
C6.14)
where we have utilized Entry 31 with a = q"/2-2com, b = q"i2+2<am, and n = 2.
An indicated above, C6.1) and C6.2) are due to Schroter [1] who did not
publish his proofs outside his thesis. However, he did write three short papers
[2], [3], [4] in which he took special cases of his general formulas to establish
certain modular equations. Proofs of Schroter's formulas may be found in the
books of Tannery and Molk [1, pp. 163-167] and Enneper [1, pp. 470ff]. A
more recent proof has been given by T. Kondo and T. Tasaka [1].

16. ^-Series and Theta-Functions 73
An elegant generalization of Schroter's work has been discovered by
R. Blecksmith, J. Brillhart, and I. Gerst [2, Theorem 2]. We translate their
formula into Ramanujan's notation. It will be convenient, however, to put
fo(a,b) = f(a,b) and Л (a, b) = f(-a, -b).
Theorem. Let a, b, c, and d denote complex numbers such that \ab\, \cd\ < 1.
Suppose that there exist positive integers a, /?, and m such that
{abf = (cd)"(M-"w.
Let Eu?2e{0, 1}. Then
ftl(a,b)fjc,d)
+l ~2r)l2 h(rdV**+1 +2r>'2
, f(a/bf(cdf (b/a)(cdy\ „ ч
x Js2 у -^а^ф . ^^ф J. C6.15)
where R is a complete residue system (mod m),
[0, ifel + ae2iseven,
<5i = i,
[1, if ?x + a.e2 is odd,
and
[0, ifetl
u.
Letting a = /? = 1 and m = 2 in C6.15), we obtain an equality that is also
achieved by adding Entries 29 (i) and (ii).
Blecksmith, Brillhart, and Gerst [1], [2] have employed theta-functions in
proving theorems on the parity of partition functions while also obtaining
some elegant new identities for theta-functions. Using Ramanujan's theory of
theta-functions, Bhargava, Adiga, and Somashekara [3] and Blecksmith,
Brillhart, and Gerst [3] have given alternate proofs of these theta-function
identities.
Entry 37. We have
<p(-a)<p(-b)} = cp(ab) + 2 ? (abff(ab
о
(i) Ш<*)<РФ) + <p(-a)<p(-b)} = cp(ab) + 2 ? (abff(ab^k,
\ о
^
(ii) НФ)Ф) - <p(-a)<p(-b)} = 2 t
and
(iii) ф(а)ф(Ь) = ф(аЬ) + t

74 16. q -Series and Theta-Functions
Proof of (i). In Entry 36(i), put b = a and d = c, and then replace с by b.
Proof of (ii). In Entry 3.6(ii), put b = a and d = c, and then replace с by b.
Replacing к by fc — 1, we find that
,37.,)
In the first sum on the right side, replace к by 1 — к and apply Entry 18(iv)
with и = 1. Hence,
О /Mk-l „2К-1
lc=-oo
а,
k=l
oo „2*-l / „2*-l h2k-l
Simplifying the right side of C7.2) and then substituting it into C7.1), we
complete the proof.
Proof of (iii). In Entry 36(i), let а = с = 1 and then replace d by a. Using
Entries 18(ii), 18 (iii), and 22(ii), we deduce that
Replacing к by — к in the former sum on the right side above and employing
Entry 18(iv) with n = 1, we readily complete the proof in the same fashion as
in the proof of Entry 37(ii).
Corollary. We have
(i) ф(Я3)ф(я13) - ф(-Ч3)ф(-Ч13) = Яъ 3
(ii) ifrtfWiq11) - i/f(-q5m-qn) = q5
and
(iii)
Proof of (i). In C6.8), set ^ - 8 and v = 5 to obtain the equality
з
i V1 Sm2-5mft 312 + 78m _312-78m\ r/fl10m 16-10m\
m=l

16. ^-Series and Theta-Functions 75
Replace q by — q and subtract the two formulas to get
= 2q3f(q390, q23A)f(qX0, q6) + 2q51f(q5*6, qis)f(q30, <f14). C7.3)
Now apply Entry 18(iv) with a = q14, b = q2, and n = 1 to find that
f(ql\q2) = ql*f(q30,q-1A).
Employing this in C7.3) yields
= 2q3f{q23\ q39O)f(q6, q10) + 2q43/(«78, q^)Rq2, qlA\ C7.4)
Now from Corollary (ii) in Section 31, we see that
ФШ) =f(q6,ql0)±qf(q2,qi*)
and
ФШ39) = f(q23*, q390) ± q39f(q1&, q546)-
Using these equalities in C7.4), we conclude that
Ф(-q)}{Ф(q39) + ф{-qг9)}
- ф(-д)} {ф(Ч39) - ф(-Ч39)}),
from which the desired result readily follows.
Proof of (ii). Letting fi = 8 and v = 3 in C6.8), we find that
з
у 8m2-3m/7 <U0+110m _440-l 10m\ Мдбт _16-бтл
m=l
8m2-3m/7 <U0+110m _440-l 10m\ Мдбт _16-бтл
Changing the sign of q and subtracting the two equations, we find that
= 2qSf{q5™, q33o)f(q6, q10) + 2q63ftf™, qllo)f(qls, q~2)
= 2q5f{q330, qS5O)f(q6, q10) + 2q61f(q^0, qlio)f{q2, q"), C7.5)
where we have applied Entry 18(iv) with a = q2, b = q1*, and n = 1. By
Corollary (ii) in Section 31,
Ф(±q) = f(q6,q10)±qf(q2,q1Л)
and
ФШ55) = f(q330, q550) ± q55f(qi10, q110\
Using these equalities in C7.5), we arrive at the desired result.

76 16. ^-Series and Theta-Functions
Proof of (iii). In C6.8), let fi = 8 and v = 1. Replacing q by — q and subtracting
the two equalities, we find that
= 2<?W78, q63°)f(q2, q1*) + 2q69f(q126, qSS2)f(q6, q10). C7.6)
By Corollary (ii) in Section 31,
H±q)=f(q6,q10)±qfi^,q1A)
and
Using these equalities in C7.6), we readily complete the proof.
Example. We have
q>{-q*)q>{-q120) 15 6 120
q xi-q2M-q*°) q q q
Proof. In C6.7), let ц = 3 and v = 2 to find that
Replacing q by — q and subtracting the two formulas, we deduce that
ФШ(Ч5)-Ф(-Ч)Ф(-Ч5)
= ф^6){(р^15) — (p(—q15)} + qf(q2, q*)
Applying Entries 25(ii), 19, and 30(ii), we obtain
-2q(-q2;<
2q{-q2;c
X I „120. л120ч • C7.7)
(qi20;q120l
(~q120;q12X
By B2.4),
,_ , {q;q)a:
9 q ~(-qiq)
and by Entry 22(iv) and B2.3),
Using each of these equalities twice in C7.7), we achieve the desired result.

16. ^-Series and Theta-Functions 77
Entries 38(i), (ii). For \q\ < 1,
f(-q5) « qf
f(-q5) S qk(k+1)
f(-q\-q3) *=o («)» ¦
Entries 38(i) and (ii) constitute the famous Rogers-Ramanujan identities.
For the early history of these fascinating identities, consult Ramanujan's
Collected Papers [10, pp. 344-346], Hardy's book [3, pp. 90-99], or Andrews'
book [9, Chap. 7]. Briefly, for several years after Ramanujan initially dis-
discovered these identities, he was unable to supply proofs. In fact, he [3]
submitted the identities to the problem section of the Journal of the Indian
Mathematical Society. In 1916, while browsing through past issues of the
Proceedings of the London Mathematical Society, Ramanujan discovered a
proof of the identities in a paper by Rogers [1]. Stimulated by the renewed
interest in his work, Rogers [4] published another proof. Ramanujan soon
found his own proof, and Rogers devised still another proof. Hardy thereupon
arranged for these two proofs to be published together (Ramanujan [8], [10,
pp. 214-215]; Rogers [5]). At about the same time that Ramanujan unearthed
Rogers' work, I. J. Schur also independently discovered Entries 38(i), (ii) and
published two proofs [1], [3, Vol. 2, pp. 117-136] as well as proofs of similar
results [2], [3, Vol. 3, pp. 43-50]. Watson's paper [2] gave still another proof
of the Rogers-Ramanujan identities.
We would now like to point out that although Ramanujan discovered the
Rogers-Ramanujan identities in India and it took several years before he
found a proof, these identities are limiting cases of Entry 7. In fact, the proof
is not much different from Watson's proof. We are grateful to R. A. Askey
for informing us that the Rogers-Ramanujan identities are deducible from
Entry 7.
If we let c, e, and / tend to oo in G.3), we find that
f (-lf(a)k(l-ag2k)a2kq^2-k'2= » akqk2
h Ш1 - a) mU &0 (q)k '
Letting a = 1 yields
k=o (q)k k=i
If we now apply the Jacobi triple product identity (Entry 19) to f(—q2, -q3),
we obtain the first Rogers-Ramanujan identity, Entry 38(i). Similarly, letting

78 16. ^-Series and Theta-Functions
a = q above, we deduce that
oo пЩ+1) oo
(«)- I -т^= I (-i)*(i-42k+V
k=0 (q)k k=0
= f(-q,-q*)-
To obtain this last equality, apply the distributive law in the penultimate line
and replace к by к — 1 in the second sum. Applying Entry 19 to f(—q, —q*),
we obtain the second Rogers-Ramanujan identity, Entry 38(ii).
Today, there exist many proofs of the Rogers-Ramanujan identities as well
as considerable generalizations. It would be impossible here to list all of these
proofs and generalizations, and so we describe only a selected sampling of
papers and conclude with a description of sources where more complete
bibliographies may be found.
After the work of Rogers and Ramanujan, some of the most important
earlier papers were written by Slater [1], Alder [1], and Gordon [1]. Other
generalizations have been found by Andrews [5], [15], Bressoud [1], Denis
[4], Milne [5], Paule [1], Verma [1], and Verma and Jain [1], [2]. Using
primarily the g-binomial theorem, Bressoud [2] has developed an especially
elegant and simple proof of the Rogers-Ramanujan identities, which has been
reproduced by J. and P. Borwein in their book [2]. Emphasizing operators
and explicit solutions of functional equations, Ehrenpreis [1] has developed
a new approach to Rogers-Ramanujan identities. His paper also contains a
discussion of several other proofs. For an enlightening exposition of Rogers'
first proof of the Rogers-Ramanujan identities and their connections with
certain ^-orthogonal polynomials, see Askey's paper [7]. Andrews [14] has
also discussed the importance of Rogers' work and has reproduced two
fascinating letters of Rogers. To see how computer algebra can be an aid in
proving the Rogers-Ramanujan identities, see Andrews' paper [20]. Proofs
of the Rogers-Ramanujan identities employ the Jacobi triple product identity
at some stage, except for one proof found by Andrews [19]. In the 1980s,
R. J. Baxter discovered the Rogers-Ramanujan identities and several beauti-
beautiful analogues in his work on the hard hexagon model. For descriptions of this
work, see Baxter's book [1] as well as papers by Baxter [2], Andrews [11],
and Andrews, Baxter, and Forrester [1]. An interesting proof of the Rogers-
Ramanujan identities motivated by their discovery and occurrence in the
solution of the hard hexagon model has been given by Andrews and Baxter
[1]. A. K. Agarwal and Andrews [1] have proved some Rogers-Ramanujan
identities for certain partitions with "n copies of n," which also have applica-
applications to the hard hexagon model. For many years a purely combinatorial
proof of the Rogers-Ramanujan identities was sought. Finally, in 1981, a
bijective proof of the identities was devised by Garsia and Milne [1]. Relying
heavily on the work of Garsia and Milne, Bressoud and Zeilberger [1] found
a simpler bijective proof. Lepowsky and Wilson [1] have proved the Rogers-
Ramanujan identities within the setting of Euclidean Lie algebras.

16. q -Series and Theta-Functions 79
We have listed only a minority of the proofs and generalizations of the
Rogers-Ramanujan identities found by Andrews. His survery papers [4]-[6],
[8], [21], book [9], and monograph [18] provide references for many original
papers on this subject, including his own important contributions. Particu-
Particularly recommended is Andrews' paper [21], which provides a classification
and discussion of almost all of the known proofs of the Rogers-Ramanujan
identities. Askey's paper [7] also offers many references. Another survey paper
has been written by Verma [2].
Entry 38(iii). For \q\ < 1,
f(-Q2,-q3) 1 + 1+1 + 1 +¦¦¦'
Proof. Set a = 1 in the corollary to Entry 15. Using Entries 38(i), (ii), we
complete the proof.
Entry 38(iii) is another famous theorem of Ramanujan and is generally
known as "Ramanujan's continued fraction" or as the "Rogers-Ramanujan
continued fraction." As pointed out in Section 15, the first proof of Entry 38(iii)
was given by Rogers [1]. Ramanujan's proof is found in his paper [8], [10,
pp. 214-215]. Shortly thereafter, Rogers [6] gave another proof. Although
the continued fraction was mentioned in Ramanujan's [10, p. xxviii] first letter
to Hardy, the equality of Entry 38(iii) was not. Ramanujan eventually found
several generalizations and ramifications of his continued fraction which he
recorded in his "lost notebook" [11], in the unorganized pages of his second
notebook, and in his third notebook. For an account of some of these develop-
developments, see two papers by Andrews [10], [13], several papers by Ramanathan
[1]> [2], [4]-[6], [9], and a paper by Andrews, Berndt, Jacobsen, and Lam-
phere [1]. After the work of Rogers and Ramanujan, no significant generaliza-
generalizations were found until Selberg [1], [2, pp. 1-23] published his first paper in
1936. In addition to papers cited in Sections 15 and 16 and in our discussion
of the Rogers-Ramanujan identities above, further generalizations and re-
related work may be found in papers by Carlitz [1], Carlitz and Scoville [1],
Gordon [2], Hirschhorn [3], [6], Al-Salam and Ismail [1], Bhargava and
Adiga [1], [2], Bhargava, Adiga, and Somashekara [1], [2], Bhargava [1],
Churchhouse [1], Denis [l]-[3], Verma, Denis, and Rao [1], Singh [1], and
Hovstad [1].
The Rogers-Ramanujan continued fraction has combinatorial interpreta-
interpretations, a fact first recognized by G. Szekeres [1]. We mention one such combi-
combinatorial interpretation, discussed by A. M. Odlyzko and H. S. Wilf [1].
An (n, k) fountain is an arrangement of n coins in rows such that there are
exactly к coins in the bottom row, and such that each coin in a higher
row touches exactly two coins in the next lower row. Let f(n, k) denote
the number of (n, k) fountains, and put f(ri) = X*=i/(n> k). Thus, /A) = 1,

80 16. ^-Series and Theta-Functions
/B) = 1, /C) = 2, /D) = 3, /E) = 5, /F) = 9, /G) = 15, and so on. Then
{/(n)} has the generating function
1 + ? f(n)x" = T T T T
n=l 1 — 1 — 1 — 1 —
Similar interpretations have been examined by Glasser, Privman, and Svrakic
[1] and Privman and Svrakic [1].
For further combinatorial interpretations of the Rogers-Ramanujan con-
continued fraction, see papers by Flajolet [1] and Andrews [13].
Entry 38(iv). For \q\ < 1,
f\-q\ -q*) - q2'5f2(-q, -q*)=f(-
C8.1)
We establish the following formulation of the quintuple product identity
from which we deduce Entry 38 (iv).
Theorem (Quintuple Product Identity). For \q\ < 1,
f(B\ q*/B3) - B2f(q/B\ B>q>) = /(-c?2)/(~*/ ~^f]. C8.2)
J(Bq, q/tt)
Proof. In C6.1), set ц = 3 and v = 1 and replace q2 by q. Recalling also the
remarks made after C6.6), we find that
, q2/A)f(Bq, q/B) + f(-Aq2, -q*/A)f(-Bq, -q/B)}
AB
C8.3)
Make the same substitutions in C6.2) but also replace A by I/A. Accordingly,
-Xf(q2/A,Aq2)f(Bq,q/B)-f(-q2/A, -Aq2)f(-Bq, -q/B)}
i /в\т /В* А2
— Д-l у [_) „3m2+4m+2/-f " „32+16т п „16-16m
,7 + 2m
C8.4)
Note that
l'А2 В* \ /В4 Аг
f [ „д24+16ж _д24-1бт|_ ft _„32+16т п -1б-1бт
when A = B2q2. Giving A this value and subtracting C8.4) from C8.3), we

16. <j-Series and Theta-Functions 81
deduce that
f(-B2q*,-l/B2)f(-Bq, -q/B)
= /D28, q2°){f(B3q5, q/B3) - A/B2)f(qs/B3, B3q)}
+ f(q*\ qA)q5{Bf{B3q\ \/B3q) - (l/B^fiq1/B3, B3/q)}. C8.5)
We now apply Entry 18(iv) twice. First, letting a = 1/qB3, b = B3qn, and
n = 1, we deduce that
f(B3q\ l/B3q) = ^/(qS/B3, B3q).
Second, putting a = B3/q, b = q1 IB3, and n = 1 yields
Using the two foregoing equalities in C8.5), we find that
f(-B2q*,-l/B2)f(-Bq,-q/B)
= {f(q3*, q20) ~ qW\ qA)} {/B»V, q/B3) - A/B2)f(q*/B3, B3q)}
= /(-«"*, -q8){f(B3q5, q/B3) - (l/B2)f(q5/B3, B3q)}, C8.6)
where we have applied Entry 31 with a = —q*,b= -qs, and n = 2.
Next, replace В by \/B in C8.6) and successively employ Entries 30(iv),
30(i), and 24(iii) to conclude that
f(q$/B3, B3q) - B2f(q/B3, B3qs)
_f(-B2,-qVB2)f(-Bq,-q/B)
K~qA)
_f(-B2, -q*IB2)f(-B2q2, -q2/B2)cp(-q2)
f(-q*)f(Bq,q/B)
_/(-В2,-д2/В2)ф(д2)<р(-д2)
f(-q*)f(Bq,q/B)
_A-q2)f(-B2,-q2/B2)
f(Bq, q/B)
which completes the proof.
Proof of Entry 38(iv). In Entry 31, let a = —q,b= —q2, and n = 5. Using
also Entry 18(iv) three times, we find that
f(-q) = -qf(-q25) + {f(~q35, -<t°) ~ q'bf{-q~'°, ~q85)}
-q2{f(-q20, ~q55) ~ q5f(-q\ "V0)}- C8.7)

82 16. ^-Series and Theta-Functions
We now apply C8.2) twice. In each case, replace q by q25'2, and then let
В = ~q1512 and — q5'2, respectively. Hence, C8.7) becomes
Replace g by q1/5 and multiply both sides by f{ — q). Using the corollary to
Entry 28, we deduce that
-в2,-V)
which is precisely C8.1).
As Ramanathan [8] has pointed out, the quintuple product identity can
be found in Ramanujan's "lost notebook" [11] in the form
Л"*2> -Дх)(GЯх!> = /(-AV, -Ax6) + x/(-A, -AV>). C8.8)
f(-x, -Ax2)
To see that C8.2) and C8.8) are equivalent, set Ax3 = q2 and x = - q/B. Then
C8.8) takes the form
=/(в W/i?3) -
by an application of Entry 18(iv).
Next, we put C8.8) in a form that is perhaps more common and that
legitimizes the designation "quintuple product identity." Let Ax3 = q. By
Entry 22(iii) and the Jacobi triple product identity, Entry 19, the left side of
C8.8) equals
(fr q)Jx2; g)Jglx2\ q)oo = (q; q)Jx2; q^jq/x2; q)J-x; gU-g/x; д)ю
(x; q)x(q/x; e), (x2; g2Ue2/x2; «2)«>
= (e; q)Jx2q; ч2)ЛФг; q2)J-x; q)ao{-q/x; q)x.
On the other hand, from the definition of f(a, b), the right side of C8.8) is
readily seen to equal
CO
? {-I)nq*3n-1)l2x3n(l+xqn).
п—~ со
Lastly, replacing x by — 1/z, we summarize our calculations above with a more
familiar form of the quintuple product identity,
C8.9)

16. ^-Series and Theta-Functions 83
The quintuple product identity has a long history, and it is difficult to assign
priority to it. In one form, it was probably known to Weierstrass, for in H. A.
Schwarz's book [1, p. 47], published in 1893, the quintuple product identity
is written in terms of Weierstrass sigma functions. In R. Fricke's book [2,
pp. 432-433], the quintuple product identity is presented in terms of theta-
functions. Watson's name is associated with the quintuple product identity
because in 1929 he [3] proved it en route to establishing C9.1) below. W. N.
Bailey [1], who was familiar with Watson's work, found a proof in 1951.
Shortly thereafter in 1952, D. B. Sears [1] showed that the quintuple product
identity followed easily from some work he had done a year earlier. In the
course of proving a conjecture of Dyson, A. O. L. Atkin and P. Swinnerton-
Dyer [1] established the quintuple product identity in 1954 without realizing
its prior occurrence in the literature. The identity was rediscovered in 1961 by
B. Gordon [1]. L. J. Mordell [2], attributing the result to Gordon, gave
another proof shortly thereafter. In 1970, M. V. Subbarao and M. Vidyasagar
[1] found a proof. In 1972, L. Carlitz [3] discovered two proofs and, in the
same year, in collaboration with Subbarao, published still another proof [1].
Andrews [7] showed that the quintuple product identity is a consequence of
Bailey's summation of a well poised 6ф6. In 1988, employing the Jacobi triple
product identity, M. Hirschhorn [5] established a significant generalization
of the quintuple product identity. A year later, Blecksmith, Brillhart, and Gerst
[2, p. 307] pointed out that the quintuple product identity is a special case of
their theorem, which we related in Section 36. Lastly, in 1990, R. J. Evens [1]
used complex function theory to give a short, elegant proof of the quintuple
product identity that is completely unlike previous proofs.
Entry 39. // a, )8 > 0 and aj8 = n2, then
. 1 О~2Ф о-г* _-4«
2
" /5 + 1
+
2 1 + 1 + 1 +
s
and
- 1 в'*'5
1 - 1 + 1 -•
S — 1 р~Ыъ <>-$ p~2V
~~ + ~Г - T" + T~ -

84 16. <j-Series and Theta-Functions
Jacobsen [1, p. 435] has shown that, in fact, Entries 39(i), (ii) are valid for
all complex numbers a and )8 with a/? = n2, Re a > 0, and Re /? > 0.
Formula (i) was communicated by Ramanujan [10, p. xxviii] in his first
letter to Hardy and was first proved in print by Watson [4]. Ramanathan [1],
[4] has proved both (i) and (ii) and has established additional theorems of this
type. We shall give below a proof of (ii) which is different from the proofs of
both Watson and Ramanathan but which possesses features of both proofs.
It seems likely that our proof is close to that found by Ramanujan. Our proof
of (i) is very similar, and we give only a brief sketch of it.
Proof of (ii). Let
е~ф е~' е'2" е'3*
v ' 1-1+1-1 +•¦¦¦
Then employing Entries 38(iii), (iv) and the corollary to Entry 28, we find that
»+>
(-e", e3a)
f(e~*, -ea)/(-e-2a,e-3a)
/(e-a)/(ea)
= e-~~s~. C9.1)
Thus,
In Entry 27(iv), replace a by a/5 and /? by 5)8 to deduce that
C9.2)
where a/? = n2. Using this equality and a similar equality with the roles of a
and )8 reversed, we find from C9.2) that
- - F(O + lj{—L ~ F[e-') + 1 j = 5. C9.3)
For brevity, set A = F(e~") and В = F(e~0). Then C9.3) takes the form
(A2 - A - l)(B2 -B-l) = 5AB,

16. ^-Series and Theta-Functions 85
or, after a brief calculation,
{AB - %(A + B) - I}2 = ЦА + BJ. C9.4)
Suppose that
AB - \(A + B) - 1 > 0. C9.5)
Then, from C9.4),
AB - \{A + B) - 1 = ~-{A + B),
since А, В > 0. After some elementary manipulation, we find that
BA-y/l- 1)BB - ф - 1) = 10 + 2^5.
By C9.1), А, В < (y/l + l)/2, and so, since А, В > 0, the left side above is no
greater than
(,/5 + IJ = 6 + 2^5 < 10 + 2^5.
Since this is an obvious contradiction, our assumption C9.5) is incorrect, and
we must conclude that
AB-%{A + B)-\= -^—{A + B).
After some elementary algebraic manipulation, the foregoing equality may be
written in the form
Ц2А + J5- Щ2В
which is equality (ii).
Proof of (i). Define
Пе~*) = -г- . -г
1 +1+1 + 1 +
Then, proceeding as above, we can show that
Putting A = F(e~2x) and В = F(e~2fi), we find, with the use of Entry 27(iii),
that
(A2 + A - 1)(B2 + В - 1) = 5AB,
which is equivalent to
{AB + \{A + B)- I}2 = f(A + BJ.
The remainder of the proof is parallel to that of (ii).

86
16. ^-Series and Theta-Functions
Corollary. We have
(i)
and
(ii)
1 - 1 + 1 -¦
1 + 1 + 1 +
Proof. Let x denote the continued fraction on the left side of (i). Putting
a = p = л; in Entry 39 (ii), we observe, after simplification, that x satisfies the
equation
x2 + (,/5 - l)x - 1 = 0.
Solving this equation and observing that x > 0, we easily obtain the desired
result.
In a similar fashion, Corollary (ii) follows from Entry 39(i).
Corollaries (i) and (ii) are both found in Ramanujan's [10, p. xxvii] first
letter to Hardy. Ramanathan [1], [2], [4], [5], [9] has not only proved
Corollaries (i) and (ii) but has established several additional beautiful results
of this sort.
Some of the proofs in this chapter appear in the doctoral dissertation of
C. Adiga [1] at the University of Mysore.

CHAPTER 17
Fundamental Properties of Elliptic Functions
Chapter 17 is almost entirely devoted to the theory of elliptic functions. The
groundwork was prepared in the sections on theta-functions in Chapter 16.
In the present chapter, Ramanujan introduces Jacobian elliptic functions and
elliptic integrals. It is interesting that Ramanujan does not use the classical
notation and terminology from the theory of elliptic functions and integrals.
In Section 6, we identify the functions and parameters employed by Ramanu-
Ramanujan with the more familiar notations in the theory of elliptic functions.
Much of Chapter 17 concentrates on various types of infinite series that
can be evaluated in terms of parameters that arise frequently and naturally in
the theory of elliptic functions and integrals. Many of Ramanujan's identities
involving infinite series may be derived from theorems found in Jacobi's
Fundamenta Nova [1], [2]. In particular, the Fourier series of the Jacobian
elliptic functions can be utilized to establish many of Ramanujan's findings.
Generally, however, we prefer to employ theta-functions, as did Ramanujan.
It is difficult to assess how many results in this chapter are original with
Ramanujan. Perhaps a majority of the formulas in Chapter 17 cannot be found
in print. However, if they are not in the literature, most can be derived without
too much difficulty from published results. In particular, as indicated above,
many of the results in Chapter 17 can be deduced from Jacobi's Fundamenta
Nova [1], [2].
We conclude this introduction with a few remarks about notation. As usual,
put
where a is any complex number and к is a nonnegative integer. The generalized
hypergeometric series pFq is defined by

88 17. Fundamental Properties of Elliptic Functions
Гя1,Я2,...,Яр 1 » («i)t(«2)t"-(«,)tX*
p «|_Л, &,•••, ft' J *tb (/Ш/У* • • • (ft), k\'
where p and q are nonnegative integers and a1,a2,...,ap, j81;j82,...,ft are
complex numbers. If the number of parameters is "small," we use the notation
pi^oti, <x2,..., ap; j8i, j82,..., ft; x) instead of that on the left side of @.2). If
x = 1, we omit the argument. In this chapter, p = q + I, and so pFq converges
for |x| < 1 always, and if Re^ + a2 + • • • + a,+1) < Re(ft + j82 + • • • + ft),
q+iFq converges for x = 1 as well.
As is customary in the theory of q-series, we also utilize the notations
(a)n:=(a;q)n:="f\(l-aqk) @.3)
and
(a). := (a; q)x := f[ A - aqk),
k=0
where \q\ < 1 here and throughout the sequel. The notations @.1) and @.3)
evidently conflict. However, the context will immediately make it clear whe-
whether @.1) or @.3) is being used.
Finally, if i/>(z) = T'(z)/T(z), recall that (e.g., see Whittaker and Watson's
text [1, p. 247])
where у denotes Euler's constant. Formula @.4) will frequently be used in the
sequel, often without comment.
Entry 1. Let n and x be real numbers with 0 < x < 1. Then
Г
J
* cos{(l - 2n) sin-* ф sin ф)} ^ = n _ ^ n, 1; x)
2
о у/1 — х sin2 q> *¦
Proof. By Entry 35(iii) in Chapter 11 of Ramanujan's second notebook (Part
II [9, p. 99]),
A — x2)-1'2 cosBn sin x) = 1F1(\ + n, \ — n; \; x2),
where n is arbitrary and 0 < x < 1. Replace In by \ — 2n and x by 4/jc sin q>
and then integrate both sides over 0 < (p < я/2. Accordingly, upon inverting
the order of integration and summation, we find that
I
11/2 cos{(l - In) sin (^x sin q>)}
— x sin2
ZU П)к\П)ъ. t, 1 . ->t
u X S1D Ф
k=0 (l)*^' JO
dcp

17. Fundamental Properties of Elliptic Functions 89
1 « A - п)к(п)кГ(±)Г(к + i) .
from which the desired conclusion readily follows.
We now transcribe Entry 1 into perhaps more familiar forms. Put
sin в = »/x sin (p. Then a brief calculation gives
f
Jo
A.1)
- sin2
By a result of Murphy, which can be found in Whittaker and Watson's
treatise [1, p. 312], 2^A - n, n; 1; x) = Р_„A - 2х), where Pn denotes the nth
Legendre function. Thus, A.1) may be written in the form
i:
- sin2 в
In particular, setting n = \, we deduce that
f
Jo
/x - sin2 в 2
=?P.1/2(l-2x).
Thus, we have a representation of the Legendre function P_1/2 in terms of an
elliptic integral of the first kind. This result is due to Kleiber [1, p. 10]. Lastly,
if we replace 1 — 2x by cos а, и by — и, and в by q>/2 in A.2), we further find that
COS(n ¦, z/-r , D, ч
/ ^<P = ^.(cos a),
jo ^/cos cp — cos a
which is known as the Mehler-Dirichlet integral (Whittaker and Watson
Corollary (i). For any real number n,
Г
Proof. Letting x = j in Entry 1, we find that the left side of A.3) is equal to
G1/2J^A — n, n; 1; j). Recall now the following theorem of Gauss, which
may be found in Bailey's tract [4, p. 11] and which was rediscovered by
Ramanujan in Chapter 10, Entry 34 (Part II [9, p. 42]). If a and b are arbitrary,
then

90 17. Fundamental Properties of Elliptic Functions
In particular,
Y(-)
¦41-"'i'-i
and so the result follows.
Corollary (ii). For 0 < x < 1, let
*'2 cos{(l - In) sin (,/x sin q>)}
Г
=
J
о J\ — х sin2 cp
d(p.
Then, for 0 < x < 1 and nonintegral n,
exP ( -
SinGin) Ux
x {1 + {In2 -2n+ l)x + A - §(n - n2) + ^{n - n2J)x2 + •••},
where, as usual, ф(х) = Г'(х)/Г(х) and у denotes Euler's constant.
Proof. We turn to Corollary 1 of Entry 25 in Chapter 11 (Part II [9, p. 77]).
In that corollary, let n = 0 and then replace a and b by n — 1 and — n,
respectively. Using Entry 1 as well, we find that, for 0 < x < 1,
sinGin) * x
= X —k (u\\2—* {^(" + ^) + Ф(^ — и + fc) — 2i/>(fe + 1) + Log x}x*
2
= — Mx{Log x + ф(п) + фA — n) — 2ф(\)}
n
00 (уЛ (^ уЛ
-2ф(к+ 1) + 2^A)}х*.
Since фA) = — у, this may be written in the form
SinGCH) Ux
+ ^— Ё /i 142 {^(" + fc) - <A(") + 1^A - И + fe)
2ux t=i (к!)
- ^A - и) - 2i/^(fe + 1) - 2y}x*. A.5)
Exponentiating the equality above, we find that

17. Fundamental Properties of Elliptic Functions 91
= x ац#(п) + фA - n) + 2y)e», A.6)
where, with the use of @.4),
fl 1
«1 -n)<- + -
1 n(n +1)A - n)(l - n) fl 1 1 1
2}x + - < ++ +
J 4
1 + n(l - n)x + ¦ ¦ •
_ (In2 -2n + l)x - jC»4 - 6n3 + n2 + In - 2)x2 + ¦¦¦
1 + (n - n2)x +~7-
= Bn2 - In + l)x + Цп4 - fn3 + ^n2 - §n + i)x2 + •••.
Expanding exp w and putting the result in A.6), we complete the proof.
For 0 < x < 1, let
We can extend this definition to x = 0 and x = 1, because it is easily seen that
lim f(x) = 0 B.2)
and
lim F(x) = 1. B.3)
The function F(x) was briefly examined in Section 27 of Chapter 11.
Entry 2(i). IfO < x < 1, then
у
O2 j^l 2/B/- 1)'
Entry 2(i) was stated as a corollary and proved in Section 26 of Chapter
11 (Part II [9, pp. 78-79]).
The next result is very characteristic of Ramanujan, and we quote him
exactly. (We need to assume that x > 0 below.)
Entry 2(ii).
F(l — e *) = ¦ very nearly. B.4)
10 + л/36 + x2
Proof. By B.2) and B.3) it is readily seen that the left- and right-hand sides
of B.4) agree at x = 0 and x = oo.

92 17. Fundamental Properties of Elliptic Functions
Calculating the Maclaurin series of the right side of B.4), we find that, for
x
3
280x5
On the other hand, replacing x by x/8 in Entry 2(vii) below, we deduce that,
for x > 0,
y y3 279x5
Т + M
Comparing B.5) and B.6), we complete the proof of Ramanujan's excellent
approximation for x > 0 and x small.
Observe, from B.5) and B.6), that the coefficients of x5 differ by only
l/B19 • 33 ¦ 5). Thus, Ramanujan's approximation is uncannily accurate.
H. Waadeland has communicated to us a very plausible explanation for
Ramaunjan's approximation. Replacing x2 by (in B.6), we arrive at
53
29-3-
t
16
t
16
1
5
t2
3072 '
t
192 :
+ 1 +
1
93t2
219-32
53t
29 ¦ 3 • 5
1
144.905660...'
i
•5 '
+ ••¦
Now,
Ramanujan liked highly composite numbers. Thus, replacing 144.905660 by
144 and replacing all subsequent numerators in the continued fraction above
by f/144, we find that
t t t t t
_xs 16 192 144 144 144
xF(l — e ) x —
V ; 1+1 + 1 + 1 + 1 +•••
?/16
io + V36 +1
Entry 2(iii). For 0 < x < 1,
Log F(x) Log FA - x) = л2.
Proof. This result follows at once from the definition of F in B.1).

17. Fundamental Properties of Elliptic Functions 93
Entry 2(iv). We have
F(l - x) + F{\ - 1/x) = 0.
Proof. Using Entries 30 and 32(ii) in Chapter 11 (Part II [9, pp. 87, 92]), we
find that
dxy
1
¦ + -:
*(\, |; 1; 1 - x)x(x -
1 1
2FAh i; 1; 1 — x)x(x - 1) 2^2(|, i; 1; 1 — x)x(x - 1)
= 0.
Thus, for some constant c,
F(l-x)
C
By using Entry 2(i), we easily see that
Hence, с = — 1, and the proof is complete.
Entry 2(v). We
Proof. In our proof of Entry 32(iii) in Chapter 11 [9, p. 93], we showed that
^ ( i, i; 1; jj—p ) = (I + x) 2FX (i \; I; x2). B.7)
Replacing x by A — x)/(l + x), we find that
J \ \ л л "% ~ __ / Л \ Л Л ^Tjb
B.8)
Dividing B.8) by B.7), we arrive at
' . . 4x
4x

94 17. Fundamental Properties of Elliptic Functions
Multiplying the equality above by — ж and exponentiating, we deduce the
desired result.
In a note following Entry 2(v), Ramanujan describes a very ingenious
algorithm for calculating the power series expansion of fBx/(l + x)), which
we now relate in detail.
First, by Entry 2(iv),
- x/ V 1-х/ V l +
Thus, since FBx/A + x)) is an odd function of x, we can write
% пкХ2к~1 + 0{х2я+1)
in a neighborhood of the origin (in fact, for |x| < 3). Setting y2 = 2x/(l + x),
we find that, in a neighborhood of у = О,
= I hy2k + o(y^2),
k=X
where b2m_, and b2m are expressible in terms of at, a2, ..., am for each m,
1 < m < n. Hence, by Entry 2(v), in some neighborhood of у = О,
(In \ 1/2
LWy2k+
= I скугк~1 + O(/"+1), B.10)
k=l
where cm is expressible in terms of bx,..., bm, 1 < m < In. Hence, c2m^ and
c2m are expressible in terms of ax, ..., am for each m, 1 < m <, n. Next, set
x = 2y/{i + y2). Then
x 2x Ay
у = Т+у/Г^ ап ГТ^ = (i + yf
Hence, for |x| sufficiently small, B.10) becomes
X ' 0(x4"+1)
0(x4n+1), B.11)
where each dm, 1 < m < 2n, is expressible in terms of cu ..., cm. Thus, since
d2m-i is expressible in terms of cu..., c2m-i and d2m is expressible in terms of
cls..., c2m, we deduce that ^2m-i and ^2m are expressible in terms of alf...,

17. Fundamental Properties of Elliptic Functions 95
am. But, comparing B.9) and B.11), we see that ak = dk,k> 1. In conclusion,
we have therefore shown that alm-i and a2m can be determined from a1,...,am.
Entry 2(vi) illustrates the algorithm described above.
Entry 2(vi).//|x| <%,then
/ 2x \ 1 5 з 369 . 4097 . 1594895 .
fITT^) = Vх + Vх + W^x + ~i*rx + -?27 x +¦¦¦¦
Proof. From Entry 2(i), it is clear that a1 = %, in the notation B.9). Thus, we
write, for |x| < j,
Setting y2 = 2x/(l + x), we find that
Hence, by Entry 2(v),
4v \ f 1 , 1 A A1'2
V 32J
1 ,
Setting x = 2y/(l + y2), we find from the equality above that
2x\ 1 x / Ч
+O(x5) B.12)
for |x| < \. Now, by Corollary 1, Section 14 of Chapter 3 (Part I [5, p. 71]),
2 i , v
= 1 + na + n X
where n is any real number and \a\ <\. Hence, for |x| < 1,
2 x2 x4 5x6 7x8
and
8 3x2 9x4 7x6
4 16 16
Putting B.14) and B.15) in B.12), we arrive at
B-15)

96 17. Fundamental Properties of Elliptic Functions
Repeating the procedure above, but with n = 2 in the algorithm, we have
F(v2) = — v2 + — V4 + -^-V6 + -^-y8 + 0(v10)
l' ' 16' 32' 1024' 2048' 1У '
and
4' ' 16' ' 512' ' 2048-
By B.13),
32 . 5x2 5x4
A + J\ — x2M 4 4
and
128 . 7x2
B-16)
A + J\ - x2O 4
Thus,fromB.14)-B.17),
2x \ 1 5 , 369 , 4097 .
We repeat this procedure once more, but with n = 3. Accordingly, we find
that
F (У2) = ~y2 + ~y* + ~y6 + ~y8 + 6^S-y10
from which we deduce that
4y \ 1 1 з 17 , 45 _ 4239 „
TW) = y + y+y+y+y
Finally, using B.14)-B.17), we find that the coefficient of x9 in the power series
expansion of FBx/A + x)) is 1594895/227. This completes the proof.
Entry 2(vii). For x > 0,
-8X4 l l 3 31 5 661 7 219677 Я
F{i~e } = X-X +Х -Х +Х+-
Proof. We shall apply Entry 2(vi) with x replaced by tanhDx). Since
2 tanhDx) _ _8x
1 + tanhDx) ~ ~e '
we find that
FA — e~8x) = - tanhDx) + —— tanh3 Dx) + -^ tanh5 Dx)
tanh7 Dx) + -—2=— tanh9 Dx) + • • •. B.18)

17. Fundamental Properties of Elliptic Functions 97
Now, for |x| < л/2 (Gradshteyn and Ryzhik [1, p. 35]),
tanhDx)=E
1931
^7x9 + -' B-19)
where В„, 0 < n < oo, denotes the nth Bernoulli number. Substituting
B.19) into B.18), we achieve the desired expansion after a rather tedious
computation.
Example 1. We have
F @) = 0, F(i) = e'\ F(l) = 1,
F((Jl - IJ) = e~' A and i%/2 - IL) = e*.
Proof. The values for F@) and F(l) were previously observed in B.2) and
B.3), respectively. The value for F(^) follows immediately from B.1).
In B.7), put x = yjl - 1 to find that
2*"i(i, i; 1; 1 - (v/2 - IJ) = 2Fi(i i; 1; 2(^ - 1))
The proposed value for F((sf2 — IJ) follows immediately from B.1) and the
extremal equality above.
Lastly, set x = {^/l - IJ in Entry 2(v) to get
Squaring the extremal equality above finishes the proof.
Example 2. // 0 < x < 1, then
1 1 3 31 , 37 . 5981 .
2ХПХ + 240X + 2520X + 145l520X +-
Proof. In Entry 2(vii), replace x by x/(l — x2) to find that, for 0 < x < 1,
/ ( *x\\l x1 x3 31 x5
V
2 1-х2 6A- х2K 240 A - х2M
_ 661 х7 219677 х9
~ 5040 A - х2O + 1451520 A - х2)9 + "''
Expanding A — х2)",0 < и < 4, in Maclaurin series and collecting coeffi-
coefficients of like powers, we complete the proof.

98 17. Fundamental Properties of Elliptic Functions
For|g|< 1, let
<?(<?)= I q"\ C.1)
one of the classical theta-functions studied by Ramaoujan in Chapter 16.
Lemma. For \q\ < 1,
Proof. By Entry 32(iii) in Chapter 11 (Part II [9, p. 93]),
2^1 BЛ; i; i - (j——) ) = (! + x) 2*1 (i ъ U *2)- C.2)
Now if
ьПГ~^р {3'3)
then
by the corollary in Section 25 of Chapter 16. An elementary calculation now
shows that
_
By Entry 25(iii) in Chapter 16,
<p(qM-q)=cp2(-q2). C.5)
Hence,
i + x = ^!M C6)
(p2(q2)'
Substituting C.3), C.4), and C.6) into C.2), we complete the proof.
Entry 3.1f\q\ < I, then
Proof. Iterate the identity of the foregoing lemma a total of m times. If n = 2m,

17. Fundamental Properties of Elliptic Functions 99
we then find that
Now let n tend to oo. Since q>(—q") and (p(qn) tend to 1, the desired result
follows.
The proof of the lemma and Entry 3 are very briefly sketched by Ramanu-
Ramanujan (p. 206). Proofs in the latter half of the second notebook are very rare
indeed.
In his sketch, Ramanujan seems to claim that
= j(l + x) 2i"i(i, h l; x2).
Since the right side is analytic at x = 0 while the left side is not, the proposed
identity is false. Fortunately, there is no evidence that Ramanujan actually
used this claim.
Lemma. For \q\ < 1,
p (i i. i. <P4(~9)\ _ (P2(q) p Л i. , фЧ — q
Proof. We shall employ B.7) with x = (p2(-q)/(p2(q). By Entry 25(vi) in
Chapter 16,
<P2(q) + V2(-q) = 2cp2(q2). D.1)
Using C.5) and D.1), we readily find that
4x q>\-q2)
A + xJ <p V)
and
1 , x_V(g2)
Substituting D.2) and D.3) into B.7), we complete the proof.
Entry 4(i). // m is a nonnegative integer and n = 2m, then
D.2)
D.3)
<f>4q) J \ <f>4qn)
Proof. Iterate the identity in the previous lemma m times to obtain the
equality
1 1. 1. У V 41 \ _ У УЧ> г 1 I.i.
2' 2> l, „4/_\ / ~ „,.2/_n\2rl \ 2, 2' l>

100 17. Fundamental Properties of Elliptic Functions
Combine this equality with C.7) to deduce that
,4.4,
Multiplying both sides by — ж and exponentiating, we complete the proof.
Entry 4(ii). Ifn = 2m, as above, then
Proof. This follows immediately from D.4).
The proofs of the results in Section 4 were also sketched by Ramanujan. It
was doubtless the importance of the inversion formula in Section 5 below
which led Ramanujan to include sketches of the results in Sections 3 and 4 in
his notebooks.
Entry 5 (Inversion Formula). For \q\ < 1,
Proof. We shall let n tend to oo in Entry 4(ii).
As x tends to 0, by Entry 2(i), F(x) ~ x/16. Thus, if ?„ = (p4(-qn)/(p4(qn),
lim yF(l — ?,„) = lim
Let
another classical theta-function studied by Ramanujan in Chapter 16. By
Entry 25(vii) in Chapter 16,
= Log q + lim -
= Log q.

17. Fundamental Properties of Elliptic Functions 101
Thus,
lim JFA - U = q,
n-*oo
and applying Entry 4(ii), we complete the proof.
Entry 6. In the notations B.1) and C.1),
<P2(F(x)) = 2F1(H;l;x). F.1)
// furthermore
z = 2f,(ii;i;*) F-2)
and
then
<p(e->) = yfz. F.4)
We remark that the notations F.2)-F.4) will be used extensively in the
sequel.
Proof. By Entries 3 and 5, respectively,
(p2(F(x)) = 2i4(U; 1; 1 - u) F.5)
and
F{\ - u) = F(x),
where и = u(x) = (p4(-F(x))/(p4(F(x)). Thus,
с П 1. i. i ^\ = i7/l 1. i. ..\ • F-6)
From F.6) we would like to deduce that
2iMii;i;i-«) = 2fi(ii;i;x) F.7)
and thus deduce F.1) from F.5) and F.7).
Suppose that F.7) is not true. Then there are values x0 and u0 = u(x0) such
that
2Fj(l - u0) := 2F,(i i; 1; 1 — u0) Ф 2F1(i, i; 1; x0) =: 2^(х0). F.8)
Assume, without loss of generality, that 2fi(l — u0) < 2Ft(x0). Then, by F.6),
2Fl(u0) < 2Fi(l — x0). Now 2^(х) is increasing on @,1). Thus, 1 — u0 < x0
and м0 < 1 — x0. These two inequalities are incompatible. Hence, F.8) is
invalid, and so F.7) is established to complete the proof of F.1).
From the definitions B.1) and F.3), F(x) = e~y. Using this fact, we easily
see that F.1) and F.4) are equivalent.

102 17. Fundamental Properties of Elliptic Functions
At this juncture, we should identify the quantities x, y, and z with the
customary parameters in the theory of elliptic functions. The complete elliptic
integral of the first kind is defined by (Whittaker and Watson [1, pp. 499,500])
ft/2 J
К := K(k) := -|_—- = 12Ft(i, \; 1; fc2) = frcp2(q). F.9)
Jo y/1 — к sin (p
Here k, 0 < к < 1, is the modulus of K. To obtain the second representation
for K, expand the integrand in a binomial series and integrate termwise. (See
Part II [9, p. 79].) The last equality is one of the most fundamental results in
the theory of elliptic functions and follows from F.2) and F.4). Ramanujan
does not use the universal notation к and sets x = fc2. Later, when deriving
modular equations, Ramanujan puts a = k2. The complementary modulus fc'
is defined by fc' = *Jl — k2. From Entry 3, F.9), and the monotonicity of <p2(q)
for 0 < q < 1, we see that
F.10)
From F.2) and F.9),
z--K. F.11)
71
Also from F.3) (Whittaker and Watson [1, p. 486]),
q = e-» = e'nK'IK. F.12)
Ramanujan uses the notation x instead of q, which is universally employed
today, and so we use q as well.
The following corollary is the famous inversion formula for the theta-
function (p. This formula is also found in Section 7 of Chapter 14, p. 169, and
in Entry 27(i) of Chapter 16, p. 199. The following, perhaps new and novel,
proof is obviously the one which Ramanujan found at this point and is
different from either of his two previous proofs.
Corollary. Let a, ]B > 0 with a/} = n. Then
Proof. Let у = a2. Since F(x) = e'\
F{\ - x) = e''2!' = е-"*1'2 = е'О2. F.13)
From F.1) and F.3),
<p2{F(l -x)) у а
<p2(F(x)) n /Г
Using F.13) in F.14), we complete the proof.
F.14)

17. Fundamental Properties of Elliptic Functions 103
n1/4
Example (i). ф(в-«) =
1 Ш
Proof. By Example 1 in Section 2, F(j) = e~*. Therefore by Entry 6,
<P2(e"'t) = 2JF1(i,i;l;i).
But, by A.4),
and the desired result follows.
Example (ii). <p(e-
Proof. Recall from Example 1 in Section 2 that F({^/l - IJ) = e~K A Thus,
by Entry 6,
V - IJ)- F-16)
In order to evaluate г? x{\, \; 1; (^fl — IJ), we invoke Entry 33(iv) of Chapter
11 (Part II [9, p. 95]),
jFi (*•h 1;
Letting x = (yfl — IJ and using A.4), the duplication theorem, and the
reflection principle, we find that
= a-FtCi, f; i; i)
тг23'4Г(|)
_Г2(|)у/4-2ч/2Г(В
Г2(|) 2^71:
Substituting this into F.16), we complete the proof.

104 17. Fundamental Properties of Elliptic Functions
Ramanujan (p. 207) inadvertently omitted the factor >/Г(з)/21/* in his
formulation.
Example (iii). cp(e-2") =
Proof. By Example 1 in Section 2, F{(y/2 - IL) = e~2n. Thus, by Entry 6,
lL). F.17)
To evaluate 2^1B. к U (л/2 - IL), we shall employ C.2) in which we set
x = {y/2- IJ. Thus,
4 - V2
4 Г2Ш'
by A.4). Substituting this in F.17), we complete the proof.
Example (iv). ? {k2n - \)е~пкг = i
Proof. In the corollary above, differentiate both sides with respect to a to get
Letting a = /? = yjn and simplifying, we obtain the desired result.
Section 7 consists of a large collection of results on elliptic integrals. As we
shall see, some are quite elementary, others are less elementary but known,
and perhaps a couple may be new. Throughout Section 7, we tacitly assume
that the parameter x is chosen so that the integrals exist, and that all upper
limits on integrals do not exceed я/2 so that all changes of variables are valid.
Proofs of some of the results in Section 7 have also been given by Thiruven-
katachar and Venkatachaliengar [1].
Entry 7(i). // sin a = y/x sin /}, then
Cx dq> _ f" dq>
Jo y/x — sin2 (p Jo yj\ — x sin2 (p
Proof. In the former integral, make the change of variable sin q> = y/x sin в.

17. Fundamental Properties of Elliptic Functions
105
Since
dtp Vxcosfl /
-jn= /-— . , and Vx ~
«У ,/1 - x sm2 0
sin <P = Vх cos
the desired result follows at once.
Entry 7(ii). // tan a = y/l-x tan P, then
Jo -v/l — x cos2 (p Jc
— x sin2 (p
Proof. In the former integral, make the change of variable tan <p =
^Г^ос tan в. Then elementary calculations give
2 1-х dcp
1-х cos cp = n-^ and
'1-х
1-х sin2
The result now follows.
Entry 7(iii). // tan a = ¦sf\—b tan jB, then
d<p _ л—ft f
a — fe . , Jo
Г
Jo
'1-
1-b
sin
— a sin2 </>)(! — Ь sin2 <p)
Proof. In the former integral, put tan (p = ,/l — b tan в. Then
a — b . 2 I — a sin2 0 , d(p ,Vl — b
l d
l___sm «,, = ____
The sought result now follows.
Entry 7(iv). // tan a = ^/1 + x tan /?, the
Jo y/l + x cos 2<p Jc
, d(p
and _e
— x2 sin4 (p
Proof. In the former integral, put tan (p = ^/l + x tan 0. Then elementary
calculations give
A + x)(l — x sin2 в) da>
I + x cos 2q> = r-r^ and — =
1 + x sin2 в dd
'l+x
The desired result immediately follows.
The next result is a degenerate form of the addition theorem.

106 17. Fundamental Properties of Elliptic Functions
Entry 7(v). // cot a cot 0 = ^1 - x, then
f' dq> [' dcp n x
Jo ^y 1 — x sin <p Jо ,y 1 — x sin <jo z
Proof. Noting F.11), we see that the proposed formula may be written
C" dcp _ C"'2 dcp
Jo J\ — x sin2 cp jp y/l — x sin2 cp
In the former integral, make the change of variable cot cp = 4/T — x tan в.
The equality above follows very easily from calculations similar to those in
the foregoing entries.
Entry 7(vi). If cot a tan(]B/2) = ,/1 - x sin2 a, r/ien
о -yjl — х sin2 cp Jo yj\ — x sin2 cp
Proof. Although a proof may be given along the same lines as the previous
proofs, we alternatively observe that Entry 7(vi) is a special case of the
converse of Entry 7(viii), (a) below. (In fact, the conditions (a)-(d) in Entry
7(viii) are both necessary and sufficient.) To see this, replace 0 and у by a and
0, respectively, in this converse theorem.
In fact, Entry 7(vi) is the classical duplication formula. The next result is
known as Jacobi's imaginary transformation. See Cayley's text [1, p. 68].
Entry 7(vii). // a = Log(tanGi/4 + 0/2)), then
i
dcp _ . f' dcp
о y/\ — x sin2 cp Jo y/l — A — x)sin2 cp
Proof. On the left side, let sin cp = i tan в, or cp = — i Log((l — sin 0)/cos в).
Elementary calculations give
1-х sin2 cp = 1 + x tan2 в and —- = i sec в.
du
Upon substitution in the integral on the left side, we see that it remains to
show that the given hypothesis is equivalent to a = Log(cos 0/A — sin 0)).
This is an elementary exercise with trigonometric identities.
Entry 7(viii) offers the addition theorem under four different sets of hy-
hypotheses. Let

17. Fundamental Properties of Elliptic Functions 107
dcp
f d9 Г'
Jо J1 — x sin q> Jo
— x sin2
and w =
— x sin2
Then the addition theorem 'states that
u+v = w. G.1)
In Entry 7(viii), Ramanujan assumes G.1) and derives four implications. As
intimated in the proof of Entry 7(vi), the steps in the proofs are reversible.
Thus, each of the four conditions below implies G.1). We remark that formula-
formulation (c) below is Legendre's canonical form of the addition theorem.
It will be convenient to use the theory of the Jacobian elliptic functions as
set forth, for example, in the texts of Whittaker and Watson [1, Chap. 22] or
Cayley [1, Chap. 4]. In particular, heavy use will be made of the many
identities connecting the Jacobian functions sn, en, and dn.
Entry 7(viii). // G.1) holds, then
.sin ou/l — x sin2 в + sin 6,/l — x sin2 a
(a) tan f у = ^ • -^ ,
cos a + cos p
(b) у = tan (tan cc^/l - x sin2 j8) + tan (tan p\/l - x sin2 a),
cos у
coi a coi p = —
Si
and
(c) cot a cot /? = :—- + J\ — x sin2 y,
sin a sin /? v
^/sin s sin(s — a) sin(s — /S) sin(s — y)
sin a sin /S sin у 2 '
where 2s = a + /? + y.
Proof of (a). From the theory of elliptic functions, it suffices to show that
sn w sn u dn i; + sn i; dn u
1 + en w en и + en i;
Setting w = и + v, employing the addition theorems for sn(u + i;) and cn(u + i;)
(see Cayley [1, p. 63]), cross-multiplying, and simplifying, we can establish the
required identity.
Proof of (b). The proposed identity may be put in the form
tan <x4/l — x sin2 в + tan /L/l — x sin2 a
tan у = , v
1 - tan a tan p\/(l - x sin2 a)(l - x sin2 0

108 17. Fundamental Properties of Elliptic Functions
Thus, in terms of elliptic functions, we must show that
snu , sn i;
dn v -\ dn и
sn(u + v) en u en i;
cn(u + v) „ sn м sn » ,
1 dn и dn v
en и en v
This identity is an immediate consequence of the addition theorems for
sn(u + v) and cn(u + v).
Proof of (c). The third equality is equivalent to
en и en v cn(u + v)
sn u sn v sn и sn v
dn(u + v).
Using the addition theorems for cn(u + v) and dn(u + v) (Cayley [1, p. 63]),
we easily complete the proof.
Proof of (d). To the identity of part (c),
cos a cos /} = cos у + sin a sin /3^/1 — x sin2 y,
add ± sin a sin /? to both sides to obtain, after some manipulation,
— 2 sin j(tx + P + y) sin j(tx + (} — y) = sin a sin /?(+1 + J\ — x sin2 y).
Multiply these two equalities together and use the definition of s. The pro-
proposed identity readily follows.
Entry 7(ix). // |x| < 1, then
n Г*/2 dq> _ Г*/2 cos^ sin2 q>) dep
2 Jo ,/l + x sin <p Jo yj\ — x2 sin4 q>
We shall provide two proofs, neither of which is completely satisfactory,
because they are in the nature of verifications.
First Proof. Expanding A + x sin <p)~1/2 in a binomial series, we find that
Jo y/l + x sin <p z fc=o
~ 4 kk fcimfc +1) ¦ ( j
Next, set
cos 1 и
We want to find the Maclaurin series for y. An elementary calculation shows

17. Fundamental Properties of Elliptic Functions 109
that у satisfies the initial value problem
A - u2)y' - uy + 1 = 0, y@) = я/2.
Solving this problem by customary power series methods, we find that
cos и л » Bk)\u2k s 22k(fc!)V+1
= У = ~~ iL ~2k 2 ~~ Zj > V-3)
where |u| < 1. Hence,
**/2 cos (x sin2 <p)
1
о ^/l — x2 sin*
я ^ Bk)\x2k C*>2 . u - 22*(fc!Jx2k+1 C
= я Е ^71^2 sin4* cpdcp-Y. - nk,u,
Z k=0 Z (K!j Jo fc=o (•iK + Ij! Jo
sin
K ]
Comparing G.2) with G.4), we see that it suffices to show that
, fc>0.
~22kW
and
Bfc+l)!
Both equalities are immediate consequences of the duplication theorem, and
the proof is complete.
Second Proof. Expanding A + x sin в sin2 cpY1 in a geometric series, we
readily find that
Jo Jo l+xsin0sm2<p 4*tt)
Thus, from G.2), we have shown that
dddcp
n C dQ _ f1 C*'2 _
2j0 J\ + x sin в Jo Jo 1
+ x sin 0 sin2 <p'
Comparing this with Entry 7(ix), we observe that it suffices to show that
dQ cos и
I
0 1 + и sin в
\u\ < 1. G.5)
By expanding the left side of G.5) in powers of и and comparing the result
with G.3), we may deduce the evaluation G.5).

110 17. Fundamental Properties of Elliptic Functions
Glasser [1] has constructed tables of elliptic integrals from which Entry
7(ix) can be deduced from Table 1, formula A0).
Entry 7(х)Л/|х| < I, then
M2 Г*'2 dO_dcp / Г*/2 dcp у
Jo Jo y/{l — x sin2 0)A — x sin2 в sin2 cp) VJo y/l — x sin4 cp)
Proof. Expanding the integrand in a binomial series and integrating termwise,
we find that
= ?Е,т!**™»«. G-6)
o у/1-х sin2 в sin2 <p
Proceeding in a similar manner and using the calculation above, we further
find that
IT
Jo Jo
dQ dcp
- x sin2 0)A - x sin2 в sin2
¦2
^Jtl2M2)
4 k% h (k\Jj\(j + k)\
_n2 f (f %№n-k Ш .
4 U V*tb (fc!J( fc)lj !
4 „tb \*=o (fc!J(n - k)\) n\
In the same fashion,
Hence,
;o y/\ - Jsin4 cp) = T „?o So k\Bk)\(n - k)\Bn - 2k)!*"' (?'8)
Comparing G.7) and G.8), we see that it remains to show that
у (l)fcB72*(l)n-fcB72n-2)t
kh к!Bк)!(и - к)!Bи - 2k)!
n! ktb(fc!J(n-fc)!' l ' ;
Using the elementary relations
(fl),-t = (_(~^} and (flJt = 22k(ia)t(ia + i)fc, G.10)
we find, after some calculation and simplification, that G.9) is equivalent to
the identity

17. Fundamental Properties of Elliptic Functions 111
U-n-n| ЩAJ [II-n
In order to prove G.11), we shall combine some results from Chapter 11
with a formula connecting two terminating 4F3's, one of which is Saalschiit-
zian. First, from our book (Berndt [9, p. 98, lines 13, 17]), we deduce that
G12)
From Bailey's tract [4, p. 56],
x, y, z, -n~| (v- z)n(w -z)n [ u-x,u-y,z,-n
J |
4 3| u,v,w
G.13)
where u + v + w = x + y + z — n+l. Let x = —\ — n,y = z = ^,u — w =
| - n, and v = \. Then, from G.12) and G.13), we find that
hl-n,-n
G14)
Using G.10), we can easily show that G.11) and G.14) are equivalent, and so
the proof is complete.
Entry 7(xi).//|x| < 1, then
x sin q> d6 dtp
Г*/2 Гп/2
Jo Jo
Ля/2 Л sin
Jo Jo
- x1 sin2 <p)(l - x1 sin2 в sin2 q>)
dOdcp
— x1 sin2 q> — sin2 в cos2 q>
dq> V j/f"'2 dq>
21 I / , "I
о V1 ~ г(! + x) sin2 <P^ VJ° V 1 - i(l ~ x) sin2 <P
Proof. The first equality is elementary, while the second is somewhat more
recondite.
On the right side of the first equality, let sin в = x cos x/z/y/l — x2 sin2 ф.
The limits 0 = 0, sin x are sent into ф = я/2, 0, respectively. Elementary
calculations show that
d0 Xy/l — x2 sin ф
dx// 1-х2 sin2 ф
and
1-Х2 Sin2 fl) — Sin2 в COS2 fl) =
1-х2 sin2

112 17. Fundamental Properties of Elliptic Functions
Using these equalities in the integral on the right side of the first equality in
Entry 7(xi), we complete the proof.
To prove the second equality in Entry 7(xi), we first expand by using G.6)
and then expand again via the binomial series to find that
, ,**/2 x sin cp d6 dcp
П ,
o Jo y/(l — x2 sin2 cp)(l — x2 sin2 в sin2 cp)
oo /14 V2j Ли/2
I^M sin2^-v^
j=0 J\ Jo
у. 2J+2k+1
where we have used G.10). Employing two results from Chapter 11 of our
book (Berndt [9, p. 98, line 17; p. 97, Entry 34(iii)]), we deduce that
00
2n + l
_ _ , \X
' л=0
= у {2fi2(i h i; i(i + *)) - 2*1i2(i i; 1; i(i - x))}.
Using F.11), we finish the proof.
Our proofs of Entries 7(x) and 7(xi) are undoubtedly not those found by
Ramanujan. However, our proofs do depend on results from Chapter 11, and
so possibly Ramanujan might have started with these theorems on hypergeo-
metric series and then was led to elliptic integrals.
Entry 7(xii). // A + x sin2 a) sin /? = A + x) sin a, then
dcp
C" dcp [*"
Jo y/l — x2 sin2 cp Jo Г
Ax . 2
sin cp
A + X)
Proof. In the integrand on the right side, make the substitution sin cp =
A + x) sin 0/A + x sin2 в). Then elementary calculations yield
dcp _ A + x)(l - x sin2 в)
M ~ 7l - x2 sin2 61 A + x sin2 в)
and

17. Fundamental Properties of Elliptic Functions 113
4x . 2 1-х sin2
'I* ' 1 -1- V Cln2 fi
The desired result now follows.
Entry 7(xii) is known as Gauss' transformation, while Entry 7(xiii), which
is very similar in appearance, is called Landen's transformation [1], [2].
Entry 7(xiii). // x sin a = sinB/? — a), then
9 dq>
J\ — x2 sin2 q> Jo I. 4x . ,
/ 1 - T, о sin <P
V A + X?
Proof. In the latter integral, let x sin в = sinB<p — 0), or
i(sin-x(x sin 0) + 0). Then
and
and
A
V
the proof is
dd
4x
A+x)
complete.
xcos 0
J
+
1
X
-X2S
COS 0
- X
in'
+
2 sin2 0
4
7l - x2 sin2 0
1 +x
In his "lost notebook" [11], Ramanujan recorded many deep results on
elliptic integrals. See a paper by S. Raghavan and S. S. Rangachari [1] for
proofs of several of the these beautiful theorems.
Much, of course, has been written about elliptic integrals. The most com-
complete tables have been compiled by Byrd and Friedman [1]. Other sources are
tables of Gradshteyn and Ryzhik [1] and Glasser [1].
In the sequel, we shall be rearranging the terms of absolutely convergent
double series. To describe the different rearrangements, we employ the termi-
terminology of MacMahon [1, pp. 26-32]. Let us set forth the terms of a double
series Xm,»=i amn in an array
a21 a22 a23 a24
%i <*32 a33 a34
«41 «42 «43 «44
The two most common methods of summation are by rows and by columns.
It sometimes will be convenient firstly to sum the first row, secondly to sum

114 17. Fundamental Properties of Elliptic Functions
the remainder of the first column, thirdly to sum the remainder of the second
row, then to sum the remainder of the second column, and so on. This is called
the row-column method of summation. Similarly, we may want first to sum
the first column, then the remainder of the first row, then the remainder of the
second column, and so on. This is called the column-row method of summa-
summation. It is occasionally convenient, especially when amn = anm, 1 < m, n < oo,
to sum first all elements in the first row and first column, next to sum the
remaining elements in the second row and second column, and so on. We
designate this procedure the Clausen transformation.
In Chapter 16, Ramanujan studied the general theta-function
f{a,b) = ? а^Щ*"-1, \ab\<l.
k=-x
Recall from Entry 22 of Chapter 16 that q>(q) = f(q, q), \j>(q) = f(q, q3), and
f(-q)=f(-q,-q2)- (8-1)
We shall quote extensively from Chapter 16 in the sequel.
Entry 8. We have
(i)
(ii)
(iii)
*'<*)= 1+4
•tt>-l + 8
q>(q)(p{q2) =1-2
*=1 1 + ( — <?)
q
oo / iV4"l|/in2l-l
*=i 1-q
where (fc/3) denotes the Legendre symbol,
00
(vi) \l/{q)(p{q2) = ? (-
со 1 j_ л2*+1
(vii) ^2(
k=o У — q
со fc/7* со 1 i «ft
(viii) У -i= У (-l)*+1g*(k+1)/2^
(ix) <p2(-q)f(-q)= Z Ffc

17. Fundamental Properties of Elliptic Functions 115
00
(x) ^{q2)f2{-q)= Y, Cfc + i)<?3*2+2*,
*=-00
(xi) f{-q)f{-q2) = q>i-q)*l>{q),
and
f{-q) = cp(-q2)
f(-q*) i//{q)
Proof of (i). In Entry 33(iii) of Chapter 16, let n = я/2 and replace q2 by — q.
We immediately find that
V2(q) = 1+ 4 ? r^-H. (8.2)
*=i 1 + q2le
Expand 1/A + q2k), 1 < к < oo, in a geometric series so that we obtain a
double series above. Summing this double series by columns, we find that
& 1 + q2k *=i 1 - q2"'1 '
This completes the proof of (i), which is due to Jacobi [1], [2].
Bhargava and Adiga [5] have used Entry 17 in Chapter 16 to give simple
proofs of Entries 8(i), (ii).
Proof of (ii). In C3.5) of Chapter 16, replace q2 by -q to find that
oo „к
= 1 + 8
&(l+(-qW
Writing the series on the right side as a double series and summing by columns,
we complete the proof of Entry 8(ii), which again is due to Jacobi [1], [2].
Proof of (iii). Applying the corollary in Section 33, Chapter 16 with a = q
and b = q3, we find that
Rq, q3) 2( 4v , , » f, qk + q3k (t.,
Using Entries 25 (iii) and 24(i) in Chapter 16, we deduce that
(8.4)
Writing the series on the right side of (8.3) as a double series, we find that it
is represented by the array

116 17. Fundamental Properties of Elliptic Functions
q
q2
q3
q3
q6
q9
~q5
-qi5
-q1
~q1A
~q21
q9
q18
q21
q11
q22
q™
Summing by columns and using (8.4) also in (8.3), we complete the proof.
Proof of (iv). Putting a = q and b = — q2 in the corollary of Section 33 of
Chapter 16, we find that
тУ)Л* ~q2) = 1 + 2 f ?+l-ft
*(q)f(-q,q2) + к l+(-qKk-
Using (8.1) and Entries 30(iv) and 24(iii) in Chapter 16, we have
2{ з/te-i2) f2(q)v2(q3)
Summing by columns, we deduce that
k\ qk
к i + (-q
The desired result now follows.
Proof of (v). From (8.2),
f (-q)k
к 1 + q
2k-
Summing by the column-row method, we complete the proof of this result,
originally discovered by Jacobi [1], [2, p. 187].
Proofs of (vi), (vii). From the same corollary in Section 33 of Chapter 16,
Reversing the roles of a and b and subtracting the two equalities yields
-b) f(-a,b)\= a a» - b2*-1
(-a,*) f(a,-b)i к l-iabJ"-1 •
Applying Entries 30(iv) and (vi) in Chapter 16, we deduce that
" к \l - a2kb2k~2 1 - a2k~2b2k)' 1 J
where we have summed by columns.

17. Fundamental Properties of Elliptic Functions 117
Letting b = a3 in (8.5) and employing Entry 25(iv) of Chapter 16, we arrive
at
Ф2(а*) = ср{а*)ф(а8) = ? ° ^ ~ S T~8k+6- (8-6)
t=o 1 — а к=о У ~ а
Replacing a by ia, we get
2 4 = <а*Ша8) = V __a4"
(8.7)
1=0 1 + fl k=0
Adding the latter two equalities and replacing a* by q, we arrive at
oo „к oo
Zo r^T Zo r
Applying Clausen's transformation to each series on the right side, we con-
conclude that
1 i /,**+3
Bk+l)Bk+2) l ^ Ч
n2k+1
4
which is Entry 8(vii).
Next, replace q by + y/q in (8.7) and add the two equalities to get, by Entry
,
1 + q$ г h \l - q2k+3/Z 1 + q2k+3'2
i + ^+
2 »4 Ь -
2 ttb ji - qAk+312 +1 + qAk+3l2i 2 h li - <z"+7/2 r+T^
q"
k=o l — q k=o i — 1 k=o 1 — ^ k=o 1 — q
1 J. Д8* + 3 a, , j_ ,,8^+7
„Bк+1)Dк+1)^_ЛЛ Y
»=b I-?-" »tb* l-q8t+7
t 1
2*+l

118 17. Fundamental Properties of Elliptic Functions
where, in the penultimate equality, we transformed the series by Clausen's
method. Thus, (vi) is established.
Proof of (viii). Consider the series
2q2k~1 _ Bk - I)?2*
- q2"'1J Г- q2"'1
where we have transformed the series on the left side by the row-column
method. On the other hand, summing by columns, we get
у 4 = у kq
&=1 и ~ q ) fc=i >¦ ~ q
Hence, combining these equalities, we deduce that
oo 1 j_ „k
( _ n*+lfl*(*+l)/2_
k=l
-LK ' Ч (\-qkJ
kqk », Bk -
q2k \ v{2k - l)q2k~1
к
,2k
4
к oo Ъ„к
-qk l~q2k к
= S 2kqk _ « _kqk
»=i i - ? *=i i —
S kqk
qk
k~i i-q~
which completes the proof.
Proof of (ix). From the proof of the first version of Corollary (ii), Section 34
of Chapter 16,
fc=^oo sin n
where z = expBm). Letting n tend to 0, we find that
00
I Fk + l)q^2^'2 = (q; q)l(q; q2Jx = 9>(-q)f(-q),
fc=-00
by Entry 22(iii) and B2.4) in Chapter 16.
Proof of (x). From the proof of the first version of Corollary (i), Section 34
of Chapter 16,

17. Fundamental Properties of Elliptic Functions 119
(q2; q2Uqz; q^Uq/z; qXtfz2; q*Uq*/z2; «*)и
V 3*2 + 2* ^^±
* До sin Bи)
where z = expBin). Letting и tend to 0 yields
? Ck + l)q3k2+2k = (q2; «?2Ш q2JJq*; <?4J
k=-oo
= <l>(q2)f2(-q),
by Entries 22 (ii), (iii) in Chapter 16.
Proof of (xi). This result is contained in Entry 24(iii) of Chapter 16.
Proof of (xii). Using first Entry 8(xi) above and then Entries 25(iii), (iv) in
Chapter 16, we find that
f(~q) Я>(
<p(-q2№(q2)
_<Р2(-д2Жд2)_<р(-д2)
Example.
/2(-«*) = <p2{-qS)f(-q8).
We give two proofs. The first and shorter proof is probably the one that
Ramanujan had. The second proof shows that the result is not as deep as the
first proof indicates.
First Proof. By Entry 8(x),
00 00
? {Зк + \)q3k2+2k + 2q ? Cfc + l)q12k2+8k
k=-oo k=-x
: ? Ffc + 1)^12*2+4* + t (-6fc-2)q12"
k=— oo k= —oo
2q ? Cfc+l)«
fc=-oo
00
X Fk + l)q12k2+"
= <p2(-q8)f(-q8),
by Entry 8(ix).

120 17. Fundamental Properties of Elliptic Functions
Second Proof. By C.1) and E.1), we first observe that
00 00
<p(-q) + 2#(«?8) = 1 + 2 X (- 1)V2 + 2 ? qBk+1J
= 1+2 f; q^ = <p(q4).
Multiplying both sides by f2(—q*) then gives us
<p(-<?)/2(-<?4) + 2qi/,(qs)f2(-q*) = cp(q4)f2(-q4). (8.8)
By using the product formulas from Entry 22 of Chapter 16, we can easily
show that
and
Substituting these equalities into (8.8), we finish the second proof.
Entry 9. Recall that x, y, and z are related by B.1) and F.2)-F.4). Then
dy 1
(i)
(ii)
dx x(l-x)z2'
dz Jo z dx
dx = 4x(l - x)'
where n > 0, and
(iv) l-24f ^A~ = (l-2x)z2 ^
Proof of (i). This formula is the special case и = ^ of the corollary in Section
30 of Chapter 11 (Part II [9, p. 88]).
Proof of (ii). By L'Hospital's rule,
Also, z'@) = |. Thus, in order to prove that (ii) holds, it suffices to show that
the derivatives of both sides of (ii) are equal. Multiplying both sides of (ii) by
4x(l — x) and then differentiating the resulting equality, we find that
4A - x)z' - 4xz' + 4x(l - x)z" = z. (9.1)
But (see Bailey's tract [4, p. 1]), this is precisely the hypergeometric differential
equation satisfied by z = 2F1 (j, \; 1; x), and so the proof is complete. (In Entry

17. Fundamental Properties of Elliptic Functions 121
31(i) of Chapter 11, Ramanujan states an equivalent form of the hypergeo-
metric differential equation.)
Proof of (iii). This result is a special case of Entry 31 (ii) in Chapter 11 (Part
II [9, p. 88]). In particular, set a = /? = \ and у = 5 = 1 and replace n by n + 1
to obtain the present result.
Proof of (iv). In the derivation below, we employ the following results from
Chapter 16: Entry 22(iii), namely, /(- q) = (q; q)x, Entry 24(iv), Entry 25(iv),
and Entry 25 (vii). We also use F.10) and F.4). Accordingly, we find that
= 1-П~ Log f[ A -
ay k=i
dy
j Log{i(l
dy
= -xz2
which completes the proof. Note that in the penultimate line we employed
Entry 9(i).
In the notation of Section 9 of Chapter 15,
L{e-2')=\-2Af-J-^. (9.2)

122 17. Fundamental Properties of Elliptic Functions
Thus, we have shown that
Це~2у) = A - 2x)z2 + 6x(l -x)zj~- (9-3)
Example. For у > О,
|?И„-11У _ „ЦП I iiv
iiilx + ililii
This example is more properly placed in Section 2. It is not clear why
Ramanujan was led to examine exp(— lly).
Proof. Replacing x by x/{2 — x) in Entry 2(vi), we find that, for |x| < 1,
or
16F(x) = x + \x2 + fix3 + -^x4 + O(x5).
Using B.1) and raising each side to the 11th power, we arrive at the desired
formula after a moderate amount of calculation.
Although the results in Entries 10-12 are easy to prove, their importance
cannot be overestimated, for we shall utilize them many times in proving
Ramanujan's modular equations in Chapters 19-21.
Entry 10. // x, y, and z are related by F.2)-F.4), then
(i) <p(e~y) = yfz,
(ii) <p(-e->')
(Hi) <р(-е-^)
(iv) cp(e-2')
(v) (p(e-Al>)
(vi) vie-*2)
(vii) <Р(-е-^2)
(viii) <p(e~yl*) = yfz(\ + x1'4),
and
(ix) <p(-e->"*)
Proof. Part (i) repeats F.4).
Part (ii) follows from F.10) and part (i).

17. Fundamental Properties of Elliptic Functions 123
For (iii)-(v), we employ the identities
and
<p2(q) + (p2(-q) = 2cp2(q2), A0.2)
found in Chapter 16, Entries 25 (hi) and (vi), respectively. (These identities were
also established by Jacobi [1, 2, Section 37].)
Part (iii) follows at once from A0.1) and parts (i) and (ii).
Part (iv) is an immediate consequence of A0.2) and parts (i) and (ii).
Part (v) arises from A0.2) and parts (iii) and (iv).
Using A0.1) and A0.2), we readily can show that
V(q) ± <p{~q) = yfi{<p\q2) ± <p2(-q2)}112.
Hence,
cp(±q) = -UfoV) + ср2(-Я2)У'2 ± {q>\q2) - cp^-q2)}^2), (Ю.З)
2
which will be used to establish parts (vi)-(ix).
To prove both (vi) and (vii), we use A0.3) along with parts (i) and (ii).
Lastly, parts (viii) and (ix) follow from A0.3) with the help of parts (vi) and
(vii).
Entry 11. Recall that i/t(q) is defined by E.1). Then
(i) ф(е-')
(ii) ф(-е-")
(iii) ф(е-2")
(iv) ф[е-*)
(v) ф(е-8')
(vi) ф{е-*2)
(vii) ф(-е->)
(viii) ф{е-^)
and
(ix) Ф(-е-у1А)
Proof. Our proofs depend on employing the following formulas from Entry
25 of Chapter 16:
Ф^2) = к~1/4(И(<?) - (p4-q)V'\ (ИЛ)
^<р(-ч)), (Н.2)

124 17. Fundamental Properties of Elliptic Functions
and
' , (П.З)
in conjunction with Entry 10.
Part (i) follows from A1.1) and Entries 10(vi), (vii).
To prove (iii), employ A1.1) along with Entries 10(i), (ii).
Part (ii) follows on using A1.3), Entry 10(ii), and Entry ll(iii).
Using A1.2) along with Entries 10(vi), (vii), we may deduce (iv).
Use A1.2) and Entries 10(i), (ii) to easily deduce (v).
To establish (vi), use A1.3), Entry 10(vi), and Entry ll(i).
The proof of (vii) is identical with that of (vi), except that Entry 10(vii) is
used instead of Entry 10(vi).
To prove (viii), employ A1.3) along with Entry 10(viii) and Entry 11 (vi).
The proof of (ix) is identical with that of (viii), except that Entry 10(ix) is
used instead of Entry 10(viii).
Entry 12. Let f be defined by (8.1) and recall from Entry 22 in Chapter 16 the
definition
(-qiq2)x- A2.1)
Then
(i) f(e->) = чД2'б{хA - х)е»У'2\
(ii) /(- e~y) = yfz2~ll6(l - x)ll6(xeyI12*,
(iii) f(-e-2») ^ 2
(iv) Я-е-*')
(v) x(e-y)
(vi) x(-e~y)
and
(vii) x(-e~2y)
Proof. We employ the relations
4 2 A2.2)
A2.3)
and
which are contained in Entries 24(ii), (iv), and (iii), respectively, in Chapter 16.

17. Fundamental Properties of Elliptic Functions 125
The proof of (i) uses A2.2) and Entries 10(i) and 11 (ii).
To prove (ii), use A2.2) and Entries 10(ii) nad ll(i).
Employ A2.3) and Entries 10(ii) and ll(i) to prove (iii).
To prove (iv), use A2.3) and Entries 10(iii) and 11 (iii).
Use A2.4) and Entries 10(i) and 12(i) to establish (v).
Part (vi) follows from A2.4) and Entries 10(ii) and 12(ii).
To prove (vii), employ A2.4) and Entries 10(iii) and 12(iii).
Before proceeding further, we describe three procedures in the theory of
elliptic functions by which "new" formulas can be produced from "old"
formulas.
Consider a formula of the form
Cl(x, e->, z) = 0, A3.1)
and suppose that x', y', and z' is another set of parameters such that
Cl(x', e-"', z') = 0
and
A +
Solving for x', we find that
\ * J \1 + y/l - xj
From Entry 2(v),
ey —
that is, y' = 2y. From B.7),
/ 4 fx'
V
/
— P /1 1.1. -Л — F 1 l.i.
- 2^l(l. 2. 2' X> ~ 2*1 I 2. 2. 4 J
\ A +
= A + у/хJРЛ\, ъ U x') = A + /
Solving for z', with the aid of A3.2), we find that
z' = ±z(l + y/l - x).
Hence, given the formula A3.1), we can deduce the formula
+ y/l - x) ?)
This process is called obtaining a formula by duplication.

126 17. Fundamental Properties of Elliptic Functions
By reversing the transformation, we obtain the formula
We designate this process as obtaining a formula by dimidiation. These two
processes are equivalent to Landen's transformation.
Next, let
x X
X = - ОГ X =
x-l x'-l
Replacing x by 1 — x in Entry 2(iv), we may deduce that
F(x) + F(x') = 0.
Hence,
e~y = F{x)= -F(x')= -e~y'.
By Entry 32(ii) in Chapter 11 (Part II [9, p. 92]),
\, \; 1; x) = J\ - x z.
In conclusion, given A3.1), we can deduce the formula
This process is called obtaining a formula by change of sign and is due to Jacobi.
We have previously defined the function L in (9.2). Now define M and N by
M(q) = 1 + 240
kti 1 - qk
and
N(q) = 1 - 504 ? t^e,
t=i 1 - 9
where |g| < 1. These two Eisenstein series along with L were extensively
studied by Ramanujan in Chapter 15 and in his paper [6], [10, pp. 136-162].
Results akin to those in the next few sections have been used by Rama-
nathan [3] in proving some results of Ramanujan in his first notebook, "lost
notebook" [11], and letters to Hardy.
Entry 13. Let L, M, and N be defined as above. Then
(i) M{e~2y) = z4(l - x + x2),
(ii) N{e~2y) = z6(l + x)(l - ?x)(l - 2x),

17. Fundamental Properties of Elliptic Functions
127
(iii) M{e~y) = z4(l + 14x + x2),
(iv) N(e~') = z6(l + x)(l - 34x + x2),
(v) M(e-*') = z*(l - x + ±x2),
(vi) N(e~*') = z6(l - |x)(l - x - ?x2),
(vii) "if x is changed to (A — y/\ — x)/(l + y/l — x)J then у is changed to
(viii) 2L(e-2') - L(e->) = 1 + 24
f
= z2(l +
(ix) 2L(e-*») - L(e-2^) = 1+24 J,
(x) 2M(e-2jI) - M(e-") = 1 - 240
(xi) 2N{e-2') - N(e~') = 1 + 504
(xii) 2M(e-*>) - Mie'2") = 1 - 240
T
= z4(l - 16x + x2),
f
and
(xiii)
A e2k* + 1
2N(e~*y) - N(e~2y) =1 + 504
= z6(l + x)(l + 29x + x2),
= z4(l - x - |д
& e2k> + 1
= z6(l - |x)(l - x + fix2).
Proof of (i). From Section 13 in Chapter 15 (Part II [9, p. 330]),
dL(t)_L2(t)-M(t)
dt
12
Thus, by the chain rule,
dy
Moreover, by Entry 9(i),
Hence,
dx
M(e-2*) - L2{e'2y)
1 dL(e~2*)
- x)z2 dy '
-x(l - x)z
2dL(e-2y)
dx 6
Thus we see that we can determine M(e~2y) from (9.3) and A3.3).
A3.3)

128 17. Fundamental Properties of Elliptic Functions
Using (9.3) and the hypergeometric differential equation (9.1), we find, upon
a direct calculation, that
Thus, from (9.3), A3.3), and A3.4),
x)zS
Upon simplifying with the use of the hypergeometric equation (9.1), we reach
the desired conclusion.
Proof of (ii). The proof is similar to that of (i). From Section 13 of Chapter
15 (Part II [9, p. 330]),
dM _ LM{t) - N(t)
l~dT 3 •
By the chain rule and Entry 9(i), this equality may be written in the form
-3x(l - x)z2—1—? = 2N(e-2>) - 2LM(e-2").
Solving for N(e~2y) and using (9.3) and Entry 13(i), we readily deduce part (ii).
Proof of (iii). Apply the process of dimidiation to Entry 13(i).
Proof of (iv). Apply the process of dimidiation to Entry 13 (ii).
Proof of (v). Apply the process of duplication to Entry 13(i).
Proof of (vi). Apply the process of duplication to Entry 13 (ii).
Proof of (vii). This is just Ramanujan's statement of the principle of duplica-
duplication.
Proof of (viii). An elementary calculation shows that
2L(e-2') - Це-') =1+24 § -J-. A3.5)
k=l & +1
Since we know the value of L(e~2y) from (9.3), it remains to determine L(e~y).
Our proof is similar to that of Entry 9(iv).
Using Entries 24(ii), 25(iv), and 25(vii) from Chapter 16 and F.4) and F.10),
or Entries 10(i), (ii), we find that

17. Fundamental Properties of Elliptic Functions 129
Y Log{e->l8<p2(-e-y)<pll2(e-1')\l/1/2{e-2>>)}
= x(l - x)z2^- Log{(l - x)V2x}
= A -5x)z2 + 12x(l -x)z-^.
ax
Using this last equality along with (9.3) in A3.5), we complete the proof.
Proof of (ix). Apply the principle of duplication to Entry 13(viii).
Proof of (x). An elementary calculation shows that
Using parts (i) and (iii), we finish the proof.
Proof of (xi). An elementary calculation gives
1 + 504 Z пЛ-г = 2N(<T2>) - N(e-»). A3.7)
t=i r + 1
Now use parts (ii) and (iv).
Proof of (xii). Use A3.6) with у replaced by 2y and then use parts (i) and (v).
Proof of (xiii). Employ A3.7) with у replaced by 2y and then use parts (ii) and
(vi).
Entry 14. We have

130 17. Fundamental Properties of Elliptic Functions
1
? (~
(iv) 17 + 32 ? l ' = z8(l - x2)A7 - 32x + 17x2),
k-i e +1
(vi) 1 + 8 | ^^ = ^6A - x)(l - x2),
- xJA7 - 2x + 17x2),
-46x-
(vii)
(viii)
(ix)
17-
31
1-
¦32
+ 8
-16
к
oo f_11t
i + 8 Ех -^hr = z6A - x)A -2X)'
17 - 32 J; {^S
and
(xii) "jJ x is changed to — x/(l — x), then e~y is changed to —e~y."
Proof of (i). An elementary calculation gives
Apply Entries 13 (viii), (ix) to complete the proof.
Proof of (ii). By an elementary calculation,
Now use Entries 13(x), (xii) to complete the proof.

17. Fundamental Properties of Elliptic Functions 131
Proof of (iii). An elementary calculation yields
Using Entries 13(xi), (xiii), we finish the proof.
Proof of (iv). From Entry 12(ii) in Chapter 15 (Part II [9, p. 326]),
M2(e'2y) = 1 + 480 ? -щ—r. A4.1)
An easy calculation gives
2M2{e-2>) - M2(e->) = 1 - 480 У -^—.
*=i « + 1
Thus,
= 256{2M V4') - M2(e-2jI)} - {2M2(e-2*) - М2(е^)}.
Employing Entries 13(i), (iii), and (v), we complete the proof.
Ramanujan (p. 212) inadvertently wrote 17 — 32x + x2 instead of
17 - 32x + 17x2 on the right side of (iv).
Proof of (v). A routine calculation yields
2 ) = 15 1-16 X Ц5Ц
*=i e — J
Using Entries 13(i), (iii), we reach the desired conclusion.
Proof of (vi). A simple calculation gives
64JV(e-2)I) - N(e^) = 63A + 8 У *~^ *
Now use Entries 13 (ii), (iv).
Proof of (vii). By A4.1),
256M2{e~2y) - М2{е~У) = 15 A7 - 32 У (~? ' ).
\ t=i e y — 1 /
Applying Entries 13(i), (iii), we finish the proof.
Proof of (viii). By Entry 12(iii) of Chapter 15 (Part II [9, p. 326]),

132 17. Fundamental Properties of Elliptic Functions
M{e-2>)N{e-2') = 1 - 264 ? 1Jf
k=i e
and so
Using Entries 13(i)-(iv), we complete the proof.
Proofs of (ix), (x), (xi). Apply the principle of duplication to Entries (v), (vi),
and (vii) to deduce (ix), (x), and (xi), respectively.
Proof of (xii). This is an enunciation of the principle of change of sign.
Entry 15. We have
k=lS
00
00
00
»-i si:
00
*=1 SI
& si
2k
sinhB
smhB
linl
i
iinl
i
iin:
i
iinl
h
nh
h
nh
h
nh
A
nh
+
fe-
fe-
h(ky)
b(ky)
h(ky)
h(ky)
Bky)
Bky)
Bky)
,9
Bky)
1
f l)y
h i)y

17. Fundamental Properties of Elliptic Functions 133
oo 2k + 1
(ХШ) J
(xv) 1*ьъ№ +
and
Proof of (i). Observe that
Now use Entries 13(i), (iii).
Proof of (ii). The sum to be evaluated is equal to -2b{N(e~y) ~ N(e~2y)}-
Employ Entries 13 (ii), (iv) to complete the proof.
Proof of (iii). The sum to be determined is equal to 24o{M2(e~y) — M2(e~2y)}.
Now use Entries 13(i), (iii).
Proof of (iv). The sum to be evaluated is equal to — ^г{М(e~y)N(e~y) —
M{e~2y)N{e~2y)}. Apply Entries 13(i)-(iv) to complete the proof.
Proofs of (v)-(viii). Apply the process of duplication to (i)—(iv) to obtain
(v)-(viii), respectively.
Proof of (ix). By a straightforward calculation,
= ЫЦе~2у) ~ W~y)} ~ ЫЦе~4у) ~ Ue~2y)}
= ^{2Щ-2у) - L(e~y)} -
Use Entries 13 (viii), (ix) to complete the proof.
Proofs of (x)-(xii). Trivially,
- Bk + IK » k3 » BkK
h sinh{iBfe + \)y) ~ h sinh(fej;) h sinhBfej)'
Use Entries 15(ii), (vi) to complete the proof of (x).
The proofs of (xi) and (xii) are similar.

134
17. Fundamental Properties of Elliptic Functions
Proofs of (xii)-(xvi). Apply the principle of dimidiation to (ix)-(xii) in order
to obtain (xiii)-(xvi), respectively.
Entry 16. We have
00 (
(i)
(-l)*Bfc + IK
(iii)
(iv)
(v)
(vi)
(vii)
k% cosh{|Bfe + \)y)
=iz8A'2x){l ~136xA ~x)
-2Z {1 1232x(l~x>
+ 7936x2(l - xJ} УхA - x),
- H072x(l - x)
+ 176896x2(l - xJ}

= i sin
t=o
(viii) ? (-1)" tan
k=0
= i tan x1'4,
(ix)
1
,cosh{iBfc + l)>>}
Bk + IJ
(xiii)
148
A) cosh{iBfc + l)y} г" li
The appearance here of formulas 16(v) and (vi) is rather mysterious, because

17. Fundamental Properties of Elliptic Functions 135
the intermediate results needed to prove (v) and (vi) are not given by Ramanu-
jan. In particular, series with summands involving eleventh powers in them
have not heretofore been considered by Ramanujan in this chapter. Of course,
we could establish (v) and (vi) by first deriving the aforementioned ancillary
formulas. However, we proceed in an entirely different fashion and use the
Fourier series of the Jacobian elliptic function sn. In fact, most of the results
in Sections 13-17 could similarly be established by employing the Fourier
series of the appropriate elliptic function.
Entry 16(vii) is originally due to Jacobi [1], [2, p. 164] who remarked "quae
inter formulas elegantissimas censeri debet." Likewise, Entry 16(viii) is an
elegant, beautiful formula.
Proofs of (i)-(iv). We obtain the sought formulas by the process of change of
sign in Entries 15(xiii)-(xvi), respectively. Observe that, under this procedure,
_ e-Bk+l)y/2 ' ?o
where, here, n = 1, 3, 5, 7. The four desired formulas follow without difficulty
from A6.1).
Proofs of (v), (vi). We use the notations F.9) and F.10).
By a theorem of Hermite, which may be found in Cayley's book [1, p. 56],
for \u\ < K',
sn и = и - A + fe2)^ + A + 14fe2 + fc4)^
+ A + 1228fc2 + 5478fc4 + 1228fe6 + fc8)^
- A + U069fe2 + 165826fc4 + 165826fe6 + 11069fc8 + k10)^- + ....
A6.2)
On the other hand, by a result of Jacobi [1], [2, p. 165], which may be found
in Whittaker and Watson's treatise [1, p. 510], for и = 2Kt/n and \u\ < K',
In » д<2я+1>/2 sinBn
sn « = Yi ? 1—^r+i
In » (-iyt2J+i » {2n
fib B/+1)! U 1 - q2n+1
Kkfib B/+1)! nU 1 - q
n - (-mnu/2KJJ+i - Bh
Kk jk B/ + 1)! „U sinh{iBn + l)y}"
( ' '

136 17. Fundamental Properties of Elliptic Functions
Equating coefficients of u9 and u11 in A6.2) and A6.3), we deduce that,
respectively,
S Bn+ If
„=o sinh{^Bn -
= iz10Vx(l + 1228x + 5478x2 + 1228x3 + x4) A6.4)
and
? Bи + II1
+ 11069x + 165826*2 + 165826*3 + 11069x4 + x5).
A6.5)
Formulas (v) and (vi) are now obtained from A6.4) and A6.5), respectively, by
the process of change of sign.
Proof of (vii). Integrate Entry 16(i) over [0, y]. On the left side, we find that
? cosh{iBfc
= 4 ? (-1)*+1 tan e-<»+1^. A6.6)
)i=O
Using Entry 9(i) and making two changes of variables, we find that on the
right side we get
2=-sin"V^ A6.7)
Jo y/l — u2
Combining A6.6) and A6.7), we complete the proof.
Proof of (viii). Apply the principle of dimidiation to Entry 16(vii). Examining
the right side of (viii), we see that we are required to show that
4 sin-1 ( = tan x1'4.
1+JxJ
We leave the verification of this equality as a straightforward exercise for the
reader.
Proof of (ix). Apply the process of duplication to Entry 17(i) to find that

17. Fundamental Properties of Elliptic Functions
137
1
Subtract this formula from Entry 17(i) to get
= iz(l - JT^xy
k% coshBfe + l)y
Applying the principle of dimidiation, we obtain the desired formula.
Proof of (x). Apply the process of duplication to Entry 17(ii) to deduce that
A k2
Subtract this formula from Entry 17(ii) and find that
ttb coshBfe + l)y
Employing the principle of dimidiation, we complete the proof of (x).
Proof of (xi). Applying the process of duplication to Entry 17(iii), we find that
oo lA
16 I"
k=l О
Subtract the formula above from Entry 17(iii) and obtain
? Bk + IL
k% coshBfc + l)y
x (|x2 - 2x + 2 - Jl - x - A - xK/2).
Using the principle of dimidiation, we achieve the proposed formula.
Proof of (xii). The proof is similar to the foregoing proofs. We first apply the
principle of duplication to Entry 17(iv). The formula so obtained and Entry
17(iv) then give
f Bk + lN _ t
»:=o coshBfe + l)y 2
+ 44x2(l - Jl - xJ + A - Jl - xN}.
The proposed formula now follows by dimidiation.
Proof of (xiii). The proof is like previous proofs. Applying the principle of
duplication to Entry 17(v) and combining the result with Entry 17(v), we
deduce that

138
17. Fundamental Properties of Elliptic Functions
Entry 17. We have
+ 2 ?
ti
? —j^ = z,
t=i cosh(fcy)
oo 1.2
t=i cosh(fcy)
+ л/1 - xJ9F4x2(l + ,/
+ 408x2(l - У*7*L + A - V1
Using dimidiation, we obtain the desired formula.
=z (
5 + 4
and
(я) 61 - 4
(— UkGk
*=o
- x),
= z7(l - x)F1 - 46x + x2).
Proofs of (i)-(v). From Cayley's book [1, p. 57], for |u| < K',
dn u = 1 - k2— + fe2D + fe2)^- - fc2A6 + 44fe2 + fe4)|-
+ fe2F4 + 912k2 + 408fe4 + fe6) — + •¦•.
A7.1)
(We have corrected a slight misprint; Cayley has written и instead of 1 for the
first term on the right side.) Also, from Whittaker and Watson's text [1,
p. 511], for и = IKt/n and \u\ < K',

17. Fundamental Properties of Elliptic Functions 139
n 2% « qk cosBfct)
U = — + — ? \
1 2Д 1 « (-iy/2b\2j
2 + z ,4 cosh(ky) A Bj)! \ z )
1 2 °° f— lV'/2u\2'/ °° k2j
/)! V z
where the notations F.9)-F.12) have been utilized. (In fact, this Fourier series
was essentially derived by Ramanujan in Entry 33(iii) of Chapter 16.) Equat-
Equating coefficients of и2", 0 ^ n < 4, in A7.1) and A7.2), we arrive at (i)-(v),
respectively.
Proof of (vi). This result is identical with Entry 8(i), since z = <p2(e~y).
Proofs of (vii)-(ix). These results are obtained from Entry 35(ii) in Chapter
16 by setting n = 1,2, and 3 there in turn. In the notation of that theorem, we
need to determine P2, P4, and P6. But these are constant multiples of the series
in Entries 14(i)-(iii), respectively.
Examples. Recall that cp(q) and \j/(q) are defined by C.1) and E.1), respectively.
Then
@ <P8(q) = 1 + 16 ? кУ ..,
k=l 1 ~\~Q)
and
oo 1-9 * / oo JLS_*
I ^ -^W(l + 504 I jit,
Proof of (i). From Entry 10(ii),
<p8(-g) = z4(l-xJ.
The desired result now follows from Entry 14(v) after replacing q by — q.
Proof of (ii). From Entry ll(i),

140 17. Fundamental Properties of Elliptic Functions
е->ф*(е-у) = ^z4x. A7.3)
Using Entry 15(i), we complete the proof.
Proof of (iii). By Entry 11 (iii),
The desired result now follows from Entry 15(ix).
Proof of (iv). The proposed formula follows from the equality immediately
above and Entry 16(ix).
Proof of (v). By Entry 16(x), F.4), F.10), A1.1), and A1.3),
k=O
Proof of (vi). The desired formula follows at once from Entries 13 (xi) and
15(iv) along with A7.3).
Entry 18. // n is a positive integer, then
(i) * <"*
*=o Bk + 1) cosh{\Bk + 1)тц/3} 24'
? (-1)" = 5k
*=o Bk + 1) cosh{iBfe + 1)^/^3} 24'
and
oo / I^^Ot -1- 1 ^6n—1
(ill) > 1=- = 0
*=o cosh{iBfe + 1O1^/3}
= z
=o cosh{iBfe + 1
Formula (i) and the first part of (iii) were, in fact, first established by Cauchy
[3, p. 317]. Rao and Ayyar [1] and Riesel [1] each rediscovered the first part
of (iii). Proofs of (i) and the first part of (iii) can also be found in Berndt's paper
[4, Corollaries 7.2, 7.6]. Zucker [2] has derived both (i) and (ii), while Ling
[3] has proved both parts of (iii). Zucker [1] has also established the first part
of (iii) for и = 1.
Ramanujan probably established Entry 18 via partial fractions. For exam-
example, if oj is a primitive cube root of unity, then, for each nonnegative integer n,

17. Fundamental Properties of Elliptic Functions 141
U6n
cos и cos(cou) cos(<y2u)
*=o {(jfc + i)V - и6} cosh{(fc + %
after a somewhat lengthy, but routine, calculation. Letting и tend to 0, we
deduce (i) for n = 0 and the first part of (iii) for n > 0.
Entry 18 may seem a bit out of place in relation to the remainder of this
chapter. However, Ramanujan probably chose to place these formulas at this
juncture because of their obvious connection with Entry 16(iii). Ramanujan
returns to these sums in Chapter 18; in particular, see Section 10.
In Chapter 14, Ramanujan established several additional results in the
same spirit as Entry 18. For many other results of this type and for numerous
references to the literature, see Berndt's papers [3], [4] and book [9].
Examples. Recall that x(q) is defined by A2.1).
CM/
h 1 + q2k+i
then
X{q) = 2vW2* or
(ii) V
i-i 1 , -2(c+l
then
X(q) = 21/4{A54 ± 6^645L}1/24.
(iii) //
+ l)^
q2k+1 ~
then
4 or 21'4DqI/24 or 21/4B764gI'24.
Proof of (i). Consider Entry 16(iv). If the left-hand side is equal to 0, then,
since 0 < x < 1, either x = \ or x(l — x) = jj^. The offered conclusion now
follows from Entry 12(v).
Proof of (ii). Turn to Entry 16(v). We see that the left side vanishes when
{x(l - x)} = 616 ± 24^645. By Entry 12(v), we may complete the proof.

142 17. Fundamental Properties of Elliptic Functions
Proof of (Hi). The left side of Entry 16(vi) vanishes if and only if x = \ or
{x(l - x)} = 16 or 11056. The desired results now follow from Entry 12(v).
We do not know Ramanujan's motivation in deriving the previous
examples.
In conclusion, we remark that Ling [l]-[3], Zucker [1], [2], and Schois-
sengeier [1] have evaluated several series like those found in the latter sections
of Chapter 17 in terms of parameters in the theory of elliptic functions.
Many of the results in Chapter 17 were independently proved by S. Bharga-
va and С Adiga and can be found in Adiga's thesis [1].

CHAPTER 18
The Jacobian Elliptic Functions
In Chapter 18, Ramanujan continues his development of the theory of elliptic
functions begun in Chapter 16 with the theory of theta-functions and con-
continued in Chapter 17 with an introduction to elliptic integrals and the compila-
compilation of a large catalog of series that can be evaluated in terms of elliptic
function parameters. This chapter contains further series identities depending
on the theory of elliptic functions. Such results are considerably fewer in
number here than in Chapter 17 and generally are more difficult to prove. In
particular, see Sections 4-7.
Chapter 18 also contains Ramanujan's introduction to the Jacobian elliptic
functions sn, en, and dn, although we have already used knowledge of these
functions to prove some of Ramanujan's results on elliptic integrals in Chapter
17. In contrast to Jacobi [1], [2] and other writers, Ramanujan introduces
these functions in Section 14 via their Fourier series. He derives only a handful
of the basic facts about Jacobian elliptic functions and terminates his develop-
development rather early, but he also obtains some results apparently not in the
literature. One of Ramanujan's most interesting results is the very unusual
identity
2ГЧ1)
i + 2 У cosWj , Ii , 2
к hMj +{
j 2 | =
coshMj +{ h coshMj к '
which, at first appearance, does not seem to have any connection with Jaco-
Jacobian elliptic functions. However, this highly intriguing formula arises from
elementary Jacobian elliptic function identities. As in other aspects of his
development of the theory of elliptic functions, Ramanujan does not use any
of the historical or standard notations for Jacobian elliptic functions.
Three sections A2,13, and 22) are concerned with continued fractions that

144 18. The Jacobian Elliptic Functions
arise in the theory of elliptic functions. Most of these results are connected
with the work of T. J. Stieltjes and L. J. Rogers. Undoubtedly, the most
interesting result is a corollary in Section 12. If F(<x, /?) is a certain continued
fraction, then
F(i(« + /?), yfifi) = ±{F(a, P) + F(P, a)},
or, in words, F determined at the arithmetic and geometric means of a and /?
is equal to the arithmetic mean of F(a, /?) and F{P, a).
Two sections, 23 and 24, contain beautiful new theorems on theta-functions.
As in Chapter 17, the parameters x, y, and z designate the principal
parameters in Ramanujan's study. See Entry 6 of Chapter 17 for the meanings
of x, y, and z and the latter part of Section 6 for the relationships of x, y, and
z with the more standard notations in the theory of elliptic functions. Because
Chapter 18 also contains material not particularly related to elliptic functions,
we list here those sections that are entirely devoted to elliptic functions: 1, 2,
4-7, 11-18, and 22. In these sections, x, y, and z always have the meanings
indicated above. In other sections, x, y, and z denote generic variables. There
should be no cause for confusion.
Evidently, Ramanujan was greatly intrigued with the problem of approx-
approximating the perimeter of an ellipse. Two long sections, 3 and 19, are primarily
devoted to this topic. Ramanujan's approximations are very accurate, and
those in Section 19 have a rather unusual character. Ramanujan also finds
several approximations to к, some arising out of geometrical considerations.
Approximations to к and some geometrical problems are considered in Sec-
Sections 3, 20, and 24.
Sections 8-10 and 21 are devoted to partial fraction expansions that
superficially resemble series connected with elliptic functions found in Chapter
17 and elsewhere in Chapter 18. In these sections, we proceed in the standard
manner via the residue calculus. We calculate the principal parts of a certain
meromorphic function / and conclude by the Mittag-Leffler theorem that /
is equal to the sum of its principal parts plus an entire function g. In every
instance in this chapter, it is easily shown that g(z) = 0 by letting z tend to oo.
This aspect of the proof is always tacitly assumed in the sequel. Lastly, Я2о
denotes the residue of/at a pole z0.
Entry 1. Recall from Section 9 of Chapter 15 the definition
Then
iFii-ъ -\\ 1; x) = z(l - x) + z dx
Jo

18. The Jacobian Elliptic Functions 145
Proof. Recall from Entry 6 of Chapter 17 that
г = л(й;1;4 (i-i)
Elementary calculations then show that
Z[l XI == 1 T" 7. ~, rv2~\4 П)Х {*-•¦?•)
»=i (и!J
and
Гх oo (X.\2
zdx=Y-?>^x»- A-3)
Adding A.2) and A.3), we readily deduce the first equality of Entry 1.
By Entries 9(ii) and 9(iv) in Chapter 17,
z(l - x) + ГX z dx = z(l - x) + 4x(l - x)~ A.4)
Jo "x
and
^L(e-2JI) = f(l - 2x)z + 4x(l - x)^. A.5)
Substituting A.5) in A.4) and simplifying, we easily achieve the second equality
in Entry 1.
Entry 2. In the same notation as Entry 1,
1 Cx
zdx
1 Cx
1 Jo
Proof. The proofs of these two equalities follow precisely along the same lines
as the proofs of the corresponding two equalities in Entry 1.
The formulas in Entries 1 and 2 were presumably suggested by the formulas
for the perimeter of an ellipse in Entry 3.
Entry 3. Consider the ellipse
+
with eccentricity e = (I/a) *Ja2 — b2. If L = L(a, b) denotes the perimeter of
this ellipse, then

146 18. The Jacobian Elliptic Functions
Fl(l-hl;e2) C.1)
= n(a + bJFl{-l -\; 1; t) C.2)
= п{Ца + Ь)- J{a + 3b)Ca + b)} nearly C.3)
= n(a + b)\l -\ 7 > very nearly, C.4)
(. 10 + V4 - 3tJ
where, in C.2) and C.4), t = {{a- b)/(a + b)J.
The appendants "nearly" and "very nearly" are quoted from the second
notebook (p. 217).
Formula C.1) is due to Maclaurin [1] in 1742.
The second formula, C.2), is obtainable from C.1) by using Landen's
transformation. We have not been able to find C.2) explicitly in the work of
Landen. In fact, it appears that C.2) is originally due to J. Ivory [1] in 1796.
Ivory's paper is rather unusual in that it begins with a letter to the editor, John
Playfair. Ivory informs us in his letter that he was led to this theorem by the
study of mutual disturbances of planets. Evidently then the editor considered
it fair play to print Ivory's letter. Ivory's proof of C.2) is quite ingenious and
since it is unlikely to be known to many, we give it below.
The approximation C.3) is due to Ramanujan and was rediscovered by
Fergestad in 1951. (See papers by Selmer [1] and Stubban [1].) Both C.3) and
C.4) are stated without proof in even more precise forms near the end of
Ramanujan's paper [2], [10, p. 39], where he indicates that the formulas were
discovered empirically.
However, Jacobsen and Waadeland [1], [2] have offered a very plausible
explanation of Ramanujan's approximation C.4). Wirte
Then
1 + 1 +•••]¦
3A+1+
If each numerator is replaced by — 3t/16, then we obtain the approximation
which immediately yields the approximation C.4). Since Ramanujan's ability
to represent analytic functions as continued fractions is unparalleled in mathe-
mathematical history, it seems likely that Ramanujan's formula C.4) had its source
here. Jacobsen and Waadeland [1], [2] have found a similar argument for
C.3).
Many approximations to Ца, b) have appeared in the literature. The
approximations

18. The Jacobian Elliptic Functions 147
Ца, b) « n{a + b)
and
Ца, b) ж Ьсу/iA,
given by Kepler [1, pp. 401, 402] in 1609, are perhaps the first to appear in
the literature. As might be expected, the relative sizes of a and b determine the
nature of the estimates. Most approximations, including Ramanujan's, are
best when a and b are somewhat close in size. Almkvist [1] and Almkvist and
Berndt [1] have described several such approximations when t is "small" and
discussed their accuracy. Two of the approximations that combine simplicity
and accuracy are
/a312 -J- Ь3/Л2/3
given by Muir [1] in 1883, and
published by NyvoU [1] in 1978. Other approximations of this sort have been
found by Euler [3], [6, pp. 357-370], Peano [1], Sipos (see a paper by
Woyciechowsky [1]), and Selmer [1].
The perimeter of an ellipse is intimately connected with the arithmetic-
geometric mean. See the papers by Almkvist [1] and Almkvist and Berndt [1]
for this relationship. These authors also describe some of Gauss' beautiful
contributions and how they relate to the modern day calculation of %.
Proof of C.1). Parameterizing the given ellipse by x = a cos <p,y = b sin <p,
0 < (p < 2%, we find from elementary calculus that
= 4 | (a2 sin2 <p + b2 cos2 epI/2 dq>
Jo
•я/2
= 4a Г
J(
A - e2 cos2 epI/2 d(p
о
_i\ Г «/
—г^е2п \
и! Jo
«/2
^г \ cos2" <pdq>
n=o и! J
= 2%a2Fx{\,-\;\;e2\ C.7)
First Proof of C.2). Take a special case
и)'2) = A + uY\F,(-^ -\; 1; u2)
of Landen's transformation (see Erdelyi's compendium [1, p. 111, formula E)])
and set и = (a — b)/(a + b). After simplification, we deduce C.2).
Ivory's Proof of C.2). Using C.7), we find that

148 18. The Jacobian Elliptic Functions
f * f a2 — ft2 ]1/2
L = 2a \ <1 j— A - cosBo>))> d(p
Jo I 2a2 J
= f" {2(a2 + ft2) + 2(a2 - ft2) cos^)}1'2 dq>
Jo
Proof of C.3). For brevity, define the coefficients аи, п > 0, by
= E «.«"• C.8)
O
E
n=O
Next, after some rearrangement, define the coefficients /?„, n > 0, by
(. + Ц - > + 36)Ca + W . „
= (а + Ь) ? Д,(", C.9)
л=О
where \t\ < 4. Comparing C.8) and C.9), we see that а„ = fin, 0 < n < 2,
a3 = 2/?3, and a4 = 5^4. Thus, the approximation C.3) differs from Ь/{к(а + ft)}
by only about t3/29. It furthermore appears that а„ > Д, for n > 3. We prove
this in the next theorem.

18. The Jacobian Elliptic Functions 149
Theorem 1. For n > 3, Д, < а„/2"~2.
Proof. From C.8) and C.9), for n > 1,
ал+1 Bв - IJ
а„ Bn + 2J Д,
Thus,
П+l I
2Bn-l)-2'
if n > 2. Proceeding by induction, we deduce that
а 2 а 2"'
ал+1 z ая z
for n > 2, and the proof is completed.
Proof of C.4). Define the coefficients yn, n > 0, by
3t ,1 1 , 1 , 25 . 95
= E И,*", (ЗЛО)
n=O
where \t\ < |. Comparing C.8) and C.10), we find that а„ = у„ for и < 4, while
>'s = If as- Thus, Ramanujan's approximation is amazingly accurate, with the
error being about 3t5/217.
We are very grateful to G. Almkvist and R. A. Askey who each provided
the following proof of an analogue of Theorem 1.
Theorem 2. Let а„ and yn, n>0,be defined by C.8) and C.10), respectively. Then
for n>5,yn< а„.
Proof. We have, for |tj < §,
where, for n > 0,

150
18. The Jacobian Elliptic Functions
The terms comprising dn alternate in sign for к > 1 and are increasing in
absolute value as к increases. For n > 3, it is easily seen that
It follows that for n > 4,
1 1
Since
it follows that
16 C2)"-1
1
b-2
i
n-1
B4)'
,n-2
B4)"
-1
In
(-*« _ _
' A)B2 24"B«-lJV"
yn ^ nBn - 1K
n>0,
2n
6Bn - 3)
for n > 4. It is easily seen that un is a decreasing function of n for n > 4. A
short calculation shows that u6 = §5. Hence, since we have already shown
that y5 < a5, the proof of Theorem 2 is complete.
Suppose that we let A(t) be an approximation for L(a, b)/{n(a + b)}, where
2
(З.И)
The first nonzero term in the power series of the right side of C.11) gives an
indication of the accuracy of the approximation A(t). Let us say that A(t) is
of order n if the leading power in C.11) is the nth. Thus, C.8) and C.9) show
that, for C.3), A(t) is of order 3, while for C.4), we see, from C.8) and C.10),
that A(t) is of order 5. The aforementioned approximations of Kepler, Euler,
Sipos, Peano, and Muir are of orders 1,1,2,2, and 2, respectively. Selmer [1]
found an approximation of order 3 and two of order 4. See the paper by
Almkvist and Berndt [1] for more details.
M. B. Villarino [1] has examined Ramanujan's second approximation in
closer detail and has proved that
-@.000512272.. .)t5 = - (- - ^ ) t5
\n 11 /
A(t) -
- ~ts = -@.000022888.. .)t\

18. The Jacobian Elliptic Functions 151
Example.
(i) я = 3.14159 26535 89793 23846 26434,
(ii) Log 10 = 2.30258 50929 94045 684018,
(Hi) e"« = 0.0432139182 63772 25,
(iv) e*12 = 4.81047 73809 6535165547 3.
It is not clear whether Ramanujan recorded these values on finding them
in books or calculated the values himself. Ramanujan was acquainted with
few books in India. We have examined those of which we know he had
knowledge, and we are unable to find any of these decimal expansions, which,
indeed, are correct.
Gauss [2, p. 427] recorded я to 100 decimal places and Log 10 to 50 decimal
places but doubtless took his values from Wolfram's tables [1]. However,
Gauss himself calculated e~n to 50 decimal places [2, p. 428] and ея/2 to 34
places [2, p. 431]. Abramowitz and Stegun [1, pp. 2,3] give these four decimal
expansions, although they record less digits than Ramanujan for (i) and (iv).
The calculations of Gauss, Ramanujan, and Abramowitz and Stegun are in
agreement.
Corollary. According to Ramanujan,
_ 355/ 0.0003\
Я~Ш\ 3533 /
and
я = (97^ — tjI/4 nearly.
In fact,
hi/ nnnm\
1 = 3.14159265358979432.
A comparison of this expansion with that of n given above shows that this
approximation is greater than n by about 10~15. Ramanujan also gives this
approximation in his paper [2] and says that he found it by taking the
reciprocal of 1 - ИЗя/355 [10, p. 35].
The second approximation
(W 2 - и)Щ = 3.14159265262
is less than n by about 10~9. Ramanujan [2], [10, p. 35] informs us that he
empirically discovered this approximation. However, N. D. Mermin [1] has
suggested how Ramanujan might have discovered this approximation and why
it is so accurate. The simple continued fraction expansion for я4 is given by
1111 1 1
= 97 +
2 + 2 + 3 + 1 + 16,539 + 1 +

152 18. The Jacobian Elliptic Functions
If we truncate the continued fraction after the fourth partial quotient, we
obtain the approximation я4 as 97^.
Entry 4. Let x and у be as in Section 6 of Chapter 17. Then
о Д\2 „л со 1 i _-Bn+l)ji/2
Proof. First observe that
1 + g-Bn+l))>/2
I 00 on
-u
The second series on the far right side is evaluated in Entry 17(vii) of Chapter
17, while the determination of the first series can be gotten from the second
by the process of dimidiation. Thus,
= \ j " A -
S = \ j A - z3(l - x)(l -y/i) + z3(l - x) - 1) dy
Employing Entry 9(i) in Chapter 17 to make a change of variable and then
using A.1), we deduce that
4jox/^ 4BU(n!JJ0
s
and the desired result follows.
Entry 5. Let x and у be as given in Entry 4. Then
1A oo (l.\2 v-n oo
+ 1) Log(l - е~Bл
Proof. First we observe that
T := 4 ? (- l)"Bn + 1) Log(l - е-<2я+1»)
n=0
¦^.
v n% e<2»+1»-l

18. The Jacobian Elliptic Functions 153
Just as in the last proof, we employ Entries 17(vii) and 9(i) in Chapter 17 and
A.1) to find that
•-Г
Г = (z3(l-x)-l)d>;
)y
dx
Cx d Cxz-l
—{y + Log x) dx + dx
Jo dx Jo x
oo AJ Гх
= у + Log x - lim (y + Log x) + Y -^ x" dx.
x->o+ n=i (n!) Jo
Now from Entry 2(i) in Chapter 17, we may easily deduce that
у + Log x = Log 16 + o(l)
as x tends to 0 + . Using this in E.1), we readily deduce the desired formula.
Entry 6. With x, y, and z as in Section 6 of Chapter 17,
„^o Bл + IJ cosh{iBn + l)y} 2z 3 2V ' ' ' 2' 2' ;'
Proof. An elementary calculation yields
d2 ( 1 \ Bb + IJ e<2B+1))I + e-<2"+1» - 6
^ eBn+l))./2 + e-an+l)y/2l 4 ^Bn+l))./2 + g-Bn+l))./2\3 •
Thus,
00 1
Bn + IJ cosh{|Bn
» вB"+1)у + e-<2"+1» - 6
If и = exp(—\y), the series in the integrand above can be written in the form
ц5Bп+1)
2j (л , „4я+2\3 '
n=o (i + w ;
We now expand A + и4в+2) in a binomial series. The resulting double series
can be represented by the array
w-6«3 + u5 -3w2(w-6w3 + w5) 6w4(w - 6w3 + w5)
w3-6w9 + w15 -3«6(«3-6u9 + «15) 6w12(w3 - 6w9 + w15) •¦¦
w5-6w15 + u25 -3wlo(w5-6w15 + w25) 6w2O(w5 - 6w15 + w25) ¦••.

154 18. The Jacobian Elliptic Functions
Arranging this array in ascending powers yields
и -9и3 25u5 -49u7
u3 -9«9 25m15 -49m21 -
w5 -9u15 25u25 -49u35
Summing the new array by columns, we obtain the sum
и _ 3V 52u5 7V
(-Щ2П+1)
2
Hence, using this in F.1), we find that
(-l)"Bn+lJ
rr
2 jy J, „U
The sum in the integrand appeared in our proof of Entry 4, and so using our
calculations therefrom, we find that
F.2)
Now Entry 9(iii) in Chapter 17 can be written in the form
z
Г°° Г" хп
x"(l - x)z3 dy dy = -j3F2(n + i n + \, 1; n + 1, n + 1; x),
Jy Jy и
where n > 0. Setting n = | and substituting the result in F.2), we complete the
proof.
Before stating Entry 7, we offer a remark about transforming a formula of
the sort
п(х, y,z) = 0
into a "new" formula. Suppose that we replace x by 1 - x. Then by F.3) and
F.2) in Chapter 17, у and z are transformed into ж2/у and yz/n, respectively.
We therefore obtain a new formula
Q(l - x, n2/y, yz/n) = 0.
This process is equivalent to JacobVs imaginary transformation.
Entry 7. Recall that Catalan's constant С is defined by

Then
f. (-1)"
18. The Jacobian Elliptic Functions 155
(-1)"
„U Bb + lJ(e<2»+1» + 1)
Proof. Apply Jacobi's imaginary transformation to Entry 6 to discover that
^\~XK 3f2(l,l,l;!,!; 1-х)
00
= Jo
?0 Bb + IJ cosh{iBn + \)п2/уУ
Next, we expand
tan w
/(w) =
w coshGiw/.y)
into partial fractions. The function / has simple poles at 0, iy(n + j), and
(n + j)n, — oo < n < oo. First, we find that
for each integer n. An elementary calculation then shows that the sum of
the two principal parts corresponding to the poles w = iy(n + ^) and
w = — iy(n + ^), 0 < n < 00, is equal to
2(-l)"ytanh(n + |)y
*0>2(n + iJ + w2) '
Next,
for each integer n. Another brief calculation shows that the sum of the two
principal parts associated with the poles (n + \)n and — (n + \)n, 0 < n < oo,
is equal to
. 142-2 ...2,- G-3)
COSh{7r2(n +
Hence, from G.2) and G.3), we deduce that

156 18. The Jacobian Elliptic Functions
„=o cosh{я2 (п + j)/y}((n + jJn2 — w2)'
Letting w tend to 0, we find that
o Bb + IJ cosh{;i2(n
я2 7i » (-1)" tanh(n + ^)
1- 2
8 у „% {In + IJ I eBn+1» + 1
= n^_n 2«- (-If
8 j j ?<, Bл + 1J(еBи+1» + 1)' l '
Substituting G.4) into G.1), we complete the proof.
Although the following example appears in Section 7, it does not appear
to be closely connected with Entry 7.
Example. Let x and у be as above. Then
00 1
у =
„tb {In + 1) sinh{^Bn + l)y}
Proof. By an elementary calculation,
.. S 1
"I I
Ъ Bn + 1) sinh{|Bn
со а, еBп+1))>/2 ц. е-Bи+1))>/2
We wish to transform the series in the integrand. If и = exp(—jy), this series
can be written
oo u2n+1
„U (l -«4"+2J = L ^(m+1H«
Bn+l)Bm+l) . Bn+l)Bm+3)\
n~0 m=0
S m+l - m
Л u-<2m+1» - u2m+1 + „4b u"<2m+1> - u2m+1
2m+ 1
Zj л „:+
Putting this representation into G.5) and utilizing Entries 15(xiii) and 9(i) in

18. The Jacobian Elliptic Functions 157
Chapter 17, we find that
2jy „U sinh{|Bn + l)y} У
1 Г"
- J
1
- x)
1 С' d . /1 +
= - — Log I
Jo \i
= i Log У
J + y/X.
This completes the proof.
Entry 8. Let 0 be real. If\d\< n, then
» ircosBn +1H + 2 cos{^Bn + 1H} cosh{|Bn + 1)^/39} _ ж
«=o(~ Bb + 1) cosh{|Bn + 1O1^/3} 8'
andif\9\ <7t/2, then
V n*cosBn + lH(cosBn + 1H + cosh{Bn + 1)^/30}) _ ж
1111 / I II y^. — ,
в=о Bn + 1) cosh{jBn + 1O1^/3} 12
lH(cosBn + 1H —
n% ч ' Bв + IL cosh{iBn
and
„cosBn + lH(cosBn + 1H + cosh{Bn +
a ^„sinBn + lH(cosBn + 1H — cosh(Bn + iK/ JVU «, л,
| 111 1 \ I 1 \™ ^j V *_ /J-J
»=o Г2и+ П4сояЬШ2и+ Ik. /3\ 12 '
у 1)B
Bb + IO cosh{iBn + \)Пу/Ъ}
n1 пв6
11520 180"
These beautiful series evaluations apparently have not been given previous-
previously in the literature. As the proofs below make clear, even more general
theorems undoubtedly exist.
Proof of (i). Let to = expBni/3) and define
_ cosB0z) + cosBco0z) + cosBco20z)
cos(tiz) cos(fta)z) cos(nca2z)
We expand /(z) into partial fractions.

158 18. The Jacobian Elliptic Functions
First, /(z) has simple poles at z = n + }, — oo < n < oo. After an ele-
elementary calculation and simplification, we find that
where
cosBn +1H + 2 cos{iBn + 1H} cosh{iBn +
J\n, v) =
The sum of the two principal parts corresponding to the poles n + \ and
—(n + |), 0 < n < oo, is thus equal to
(8.1)
Second, / has simple poles at z = co(n + j), — oo < n < oo, with
The sum of the two principal parts corresponding to the poles a>(n + %) and
—a>(n + 5), 0 < n < 00, is equal to
(8.2)
z2 - co2(n + ^J
Lastly, / has simple poles at z = a>2(n + 5), — 00 < n < 00, with
The sum of the two principal parts associated with the simple poles ca2(n +
and — co2(n + j), 0 < n < 00, is equal to
Hence, from (8.1)-(8.3), we deduce that
12 » (-l)"+1(n + jM(cosBn+1H + 2 cos{jBn+1H} cosh{jBn + \\/Ъв})
2~л h
Letting z = 0 above, we deduce Entry 8(i).
Proof of (ii). Since the proofs of (ii) and (iii) are similar to the proof of (i), we
are brief in our details.
Consider
1 + cosD0z) + cosDco0z) + cosDco20z)
COS(Tlz) COSG1COZ) COSGlC02z)

18. The Jacobian Elliptic Functions 159
Of course, g has the same simple poles as / in the proof of (i). Thus, by
calculations similar to those above, we deduce the partial fraction expansion
24 » (- l)"+1(n+j)s cosBn + lH(cosBn + lH+cosh{Bn +1)^/30})
* »-o cosh{%Bn + l)nJ3}(z6-(n+%N)
(8.4)
Putting z = 0 yields the desired result.
Proof of (iii). We calculate the partial fraction decomposition of
._ smDfe) + sinDco0z) + sinDco20z)
Z3 COS(Tlz) COSG1COZ) COSGt(O2z)
The function h has the same simple poles as / and g in the proofs above. By
calculations like those above, we find that
24 » (- l)"+1(n+jJ sinBn+ lH(cosBn +1H-cosh{Bn +1)^/30})
я о oosh{|B + l)/3}F(+iN)
»=о
Letting z tend to 0 above, we deduce Entry 8 (iii).
Proof of (iv). Expanding both sides of (8.4) in powers of z, we find that, for
_ 24 » (-1)" cosBn + lH(cosBn + 1H + cosh{Bn + 1)^/30})
~ % -=o (R + i) cosh{iBn + ПлТз}
z
4iBn
,6
Equating coefficients of z6 on both sides, we deduce the desired result.
Entry 9. If z ф ±a)JBn + 1), 0 <j < 2, 0 < n < oo, where со = ехрBя»/3),
then
cosh{iBn + 1)Я>/З}(Bв + IN - z6)
71 1
12

160 18. The Jacobian Elliptic Functions
Proof. In A8.1) of Chapter 17, let n = 0 and и = \nz. After some elementary
manipulation we achieve the desired result.
Example. For each complex number z,
(i) | cos(iz)(cos(iz) + cosh(iy3z)) = 1 + I t Ц^
-ДО-
„=i ypny.
Bn + lN7i6
and
(ii) i sin(iz)(cos(|z) - cosh^z)) =~l
T1
t
1 11=0
Proof. First we verify the elementary identity
= ^A + cos z + cos(coz) + cos(co2z)).
The first equality in (i) now easily follows. To prove the second equality of (i),
use the elementary identity
\ cos(|z)(cos(^z)
= cos(|z) c
along with the familiar infinite product representation for cos z.
Similarly, part (ii) follows easily from the elementary identities
|(sin z + sin(coz) + sin(co2z))
/ = — sin(|z) sin(i
Entry 10. If 2ф ±оз'{2п + 1), 0 < ; < 2, 0 < п < оо, where со = ехрBш/3),
then
nz \\
— 3
п=о cosh{iBn + 1)л/у/3}(Bп + IN - z6)
, / %z \( ,. . .
4 cosh —= cosf^Tiz) + cosh
_ ж \2у/ъ)\
12 cos(|rcz)(cos(^7tz) + cosh(^
Proof. For each nonnegative integer m, we expand

18. The Jacobian Elliptic Functions 161
iuo2z
M ¦=
\
( l ( %z \ ь. fncoz\ (
cosh —j= 1 + cosh —j= + cosh
V уз/ V^y V
cos(^rcz) cosijiiwz) cos(j7ta>2z)
( ( nz \ ( I nz \\
6m' 4cosh(—= )( cos(Itcz) + coshl—= I) - 3
into partial fractions. The function /(z) has simple poles at z = coJBn + 1),
j = 0, 1, 2, — oo < n < oo. After a somewhat lengthy but elementary calcula-
calculation, we find that
/(z)=i2 ?(-№;1Nm+5(юл)
i B=o cosh{iBn + 1)яЛ/3} (Bв + IN - z6)
Setting m = 0, we complete the proof.
Clearly, A0.1) is an analogue of A8.1) in Chapter 17.
Example. Under the same hypotheses as Entry 10,
з
з
Z
Bи + IJ
6 - z6
„U Bn + IN - z
71
=
12
Proof. In a more symmetric form, this example may be written
(^tccoz) ta.n(jna>2z) » {In + IJ
12? =n4Bn+lN-z6'
We expand the left side of A0.2) into partial fractions. We first find that
1K tan(iBn + lOico) tan(iBn + \)n(o2),
where; = 0,1,2, — oo < n < oo. Straightforward calculations then give A0.2).
S (-1)" n1
•-o Bn + IO cosh{|Bn + 1O1^} 23040
Example.
Proof. From A8.1) of Chapter 17,
1 =24*' ? (-Щ2П+1M
COS Z COS(O)Z) COS(OJZ) „e0 COSh{iBn+ 1O1^/3} (Bn + lN7C6-BzN)

162 18. The Jacobian Elliptic Functions
where a> = expB;ri/3). Expand both sides in powers of z and equate coeffi-
coefficients of z6 on both sides to achieve the proposed formula. (The calculations
are very similar to those needed in the proof of Entry 8(iv).)
The last example is found in Ramanujan's [10, p. 350] first letter to Hardy
and was first established in print by Watson [1] who employed contour
integration in his proof.
Entry 11.
(i) J/|Im 9\ < 7i, then
, „ 2, cosHL»
+ 2 j ++2f j =
„=i cosh(n7r)J I „=i cosh(n7i)J %
(ii) Let x', y', and z' denote the parameters associated with the complementary
modulus k'. If \1тв\ < у/2, then
a cosBn + 1H » coshBn + 1)9 _ д , г—-
% cosh{iBn + l)y} ?b cosh{iBn
Proof of (i). Using the Fourier series of the Jacobian elliptic function dn
(Whittaker and Watson [1, p. 511]),
K9\ я Л .a cos(nfl) \
dn — =4 1 + 21^Г1 Ь ИтвКя, A1.1)
\nj 2K\ n=icosh{ny)J
we may rewrite Entry ll(i) in the form
where y = n,x = %, and by F.15) in Chapter 17, К = ^тг3/2/Г2(|). Thus, by
A1.2), it suffices to prove that
Using Jacobi's imaginary transformation (Whittaker and Watson [ 1, p. 505])
dn(iu,k) = dc(u,k) = ^y
cn(u, k)
since k' = k= l/y/2 here, we find that
n J dn2{K9/n)
_ 2 - sn2 {K9/n) _ 2 dn2 (К9/л)
dn2 (Хв/я) dn2(X6»/7i) ~ '
where we have employed elementary identities for the Jacobian elliptic func-

18. The Jacobian Elliptic Functions 163
tions (Whittaker and Watson [1, p. 493]). (See also Section 14 below.) Thus,
A1.3) is established to complete the proof of Entry ll(i).
Entry ll(i) is a fascinating identity even though it is not particularly deep,
as the proof shows. One wonders how Ramanujan ever discovered this most
unusual and beautiful formula.
Proof of (ii). Using the Fourier series for en,
n » cosBn + 1N»
and Jacobi's imaginary transformation (Whittaker and Watson [1, pp. 511,
505]), we find that the left side of Entry 11 (ii) may be written
Kk BКв\К'к' BКЧв\ KK'kk' , , r—
n ) n \ n ) т *v
77 / YY
- — л?>?> \J AA .
Entry 12. Ifn>0, then
1 » sech(j» _ z (nzJx BnzJ CnzJx DnzJ
W 2+ ]k 1 + (jnJ ~ 2 + 2 + 2 + 2 + 2
and
(ji) „ ? sech{|B/
"& l+Bj+lJn2
= zVx ^ BnzJ^ CnzJ DигJх
~~Т" + ПГ+ 1+1+1 +¦•¦
Proof of (i). Using A1.1) and integrating termwise, we find that
i
"J
1 2 °° f°° 2/u
e-2«/(.«) dn и dw = - + — У sech(;>) e-2"/<»z) Cos — du
о 2 nz?i Jo z
1 . * sech(;>) A21)
On the other hand, by a theorem of Stieltjes [1], [2, pp. 184-200] that was
rediscovered by Rogers [3],
If
_ z
e •2"'tnz' dn u da = -
о 2
A2.2)
Combining A2.1) and A2.2), we complete the proof.
The continued fraction (ii) appears in Ramanujan's [10, p. 350] first letter
to Hardy and was first established in print by Preece [1]. The following

164 18. The Jacobian Elliptic Functions
corollary is found in Ramanujan's [10, pp. xxix] second letter to Hardy and
was proved by Preece [2]. The proof that we give is much different from that
of Preece and is undoubtedly similar to the one Ramanujan must have found.
Corollary. For n > 0, Re a > 0, and Re P > 0, define
F(«P)
n+ n + n + n + n +
Then
, ^p\ = i{F(a, P) + F(p, a)}. A2.3)
What a marvelous theorem! In words, the continued fraction F evaluated
at the arithmetic and geometric means of a and /? is equal to the arithmetic
mean of F(a, P) and F(P, a).
Proof. First, let /? > 1 and choose a. such that 0 < a. < /? and
Thus, in Entry 12(ii), we set x = a2//?2 and z = p. Also replace n by 1/n. We
then find that
2 - 8есЬЩ2/ -. *,„ _ F,
14/ 12 — РУа> I
In Entry 12(i), make the same substitutions for a and /?, but replace n by 2/n.
Accordingly, we discover that
2/1 » sech(;>) \
nB + ,§rT(wJ = ^'a)- AZ4)
Hence,
Now the left side above appears, by A2.4), to be F(P, a), except that у is
replaced by \y and n by 2n. Thus, we apply the process of dimidiation
described in Section 13 of Chapter 17. Since x is transformed into
4y/x/(l + yfxJ, we see that a2//?2 is replaced by 4аД/(а + PJ. Also, z is
transformed into A + y/x)z, and so /? is replaced by а + /?. Combining these
two changes, we see that a is replaced by 2у/оф. Thus, from A2.4), after a little
manipulation,
1/1 j> sech(-i/,)\_
2 '
Combining this with A2.5), we complete the proof for 0 < а < /? < 1.

18. The Jacobian Elliptic Functions 165
By symmetry, we see that A2.3) also holds for 0 < /? < a < 1. Now each of
the three continued fractions in A2.3) converges to an analytic function of a
and /? for Re a > 0 and Re /? > 0. Thus, by analytic continuation, A2.3) is valid
for Re a. > 0 and Re f} > 0.
Entry 13. Ifn>0, then
- x)(nzJ xBnzJ A - x)CnzJ
1 + 1 - 1 + 1 -
(-1УB/+1)8есЬЩ2/+1)з>}
l+B/+l)V
- x) 22B2 - l)x(l - x)(nzL
~ 1 + (nzJ(l - 2x) + 1 + CnzJ(l - 2x)
42D2 - l)x(l - x)(nzL
+ 1 + EnzJ(l - 2x) +¦¦¦'
and
,..., , f Bj + 1) csch{jBj
- l)x(nzf 42D2 -
1 + (nzJ(l + x) - 1 + CnzJ(l + x) - 1 + EnzJ(l + x) '
where, in (i) and (ii), 0 < x < 1Д/2.
Proof of (i). Recall the Fourier series of the Jacobian elliptic function cd и
(Whittaker and Watson [1, p. 511])
l Г A, \ V (-1У«)8B/+1)и |T . !
jZy/x cd(zw) = 2j • unn- ' ^m "I < гУ-
It follows that
z^/x e~u/n cd(zw) du
Jo
= 2 E ¦ . fL- , n , e-"'n cosB; + 1)« du
fib sinh{|B; + l)y} Jo
= 2n » (-1Ус8сЬЩ2/+1)з>}
"Д 1 + B; + lJn2 '
Next, from Jacobi's Fundamenta Nova [1], [2, p. 147],
cd и = cn(k'u, i/c'/k).

18. The Jacobian Elliptic Functions
166
Thus,
Now Stieltjes [1], [2] and Rogers [3] have shown that (see also Perron's book
[1, P- 220])
x Г e-uln cd(zu) du = А Г
Jo fc Jo
') cn(M> jfe/fe') dUi A3.2)
к Г"
e-«Hnzk')
_ к (nzk' {nzk'J Bnzk'J(ik/k')
~ к~\~Г + i + i
{3nzkf Dnzkf(ik/k'J
+ 1 + 1 +•/
nzk {nzk'J {Inzkf Cnzkf DnzkJ
. A3.3)
1 + 1 - 1 + 1 - 1 +
To ensure the convergence of this continued fraction, by a theorem in Perron's
book [1, p. 53, Satz 2.16], we must require that k/k' < 1 or x < 1/^2. Taking
A3.1)—A3.3) together, we complete the proof.
Proof of (ii). We employ the Fourier expansion of the Jacobian elliptic
function sd и (Whittaker and Watson [1, p. 511]),
(-l)'sinB; + 1)и
^j sd(zu) = 2
Differentiating with respect to u, we find that
- x) - sd(zu) = 2 Д coshm+l)y} ¦
Hence, integrating by parts and integrating termwise, we arrive at
zjxjl-
[1 - x) Г»
n Jo
e-"ln sd(zw) du = zJx\\
Jo
о-»!" .
du
sd(zw) du
oo /¦ 1V"C2i + 1) C°°
= 2% ,/,.—ppr e"u/" cosB; + 1) и,
^(-iyBj + l)sech{iBj+l)y}
= 2n A 1 + Bj + \Jn2 '
From Jacobi's work [1], [2, p. 147],
sd и = sn(fe'«, ik/k').
A3.4)
Thus,
zJx(\ ~
Г1 — x) C™ 1 / X f °
e "'" sd(zw) du = -
n Jo n V 1 —x Jo
A3.5)

18. The Jacobian Elliptic Functions 167
By another result of Stieltjes [1], [2], and Rogers [3],
Jo e
1 22B2 - l)(ik/k'J
(nzkT2 + A + (ik/kf) - (nzk'T2 + 32A + (ik/k1J)
42D2 - l)(ik/k?
-~(nzk'J
n2z\\ - x) 22B2 - l)nVx(l - x)
+ (nzJ(l - 2x) + 1 + CnzJ(l - 2x)
42D2 - 1)mVxA - x)
A3.6)
+ 1 + EnzJ(l - 2x) +¦¦¦'
The assurance that this continued fraction converges is guaranteed by a
theorem in Perron's text [1, p. 47, Satz 2.11]. Equalities A3.4)-A3.6) now
imply the sought result.
Proof of (iii). Using the Fourier series (Whittaker and Watson [1, p. 511])
we find upon an integration by parts and integrating termwise that
' г f °° - &
e "'" sn(zu) du = Zy/x e y- sn(z«) du
00 Ъ 4- 1 Г0
^osinh{iB/ + l)^}jo
On the other hand, by the same result of Stieltjes and Rogers that we used
in A3.6),
1 22B2 - l)x
1°
Jo
cr, и Аи =
42D2 -
- {nzY2 + 52A + xj
(nzJ 22B2 -
1 + (nzJ(l + x) - 1 + CnzJ(l + x)
42D2 -
A3.9)
The desired result now follows from A3.8) and A3.9).

168 18. The Jacobian Elliptic Functions
For additional proofs of the continued fractions of Stieltjes and Rogers
employed in the proofs above, see the paper by Flajolet and Francon [1],
where combinatorial applications are given. Further work on combinatorial
implications of continued fractions of the Jacobian elliptic functions can be
found in Flajolet's paper [1]. Generalizations of some of the Stieltjes-Rogers
continued fractions have been discovered by D. V. and G. V. Chudnovsky [1,
pp. 30-31].
Corollary. Ifn>0, then
1 + B/ + 1)V
1 (\i2 1 • 3M4 6 • 10(и/44 15-21(и/гL
~ 4 \T + I + i + i +
where ц = ,/я/Г2C/4).
Proof. Set у = n in Entry 13 (ii). Then x = \ and, by F.15) in Chapter 17,
z = ц. The corollary now easily follows.
In Section 14, Ramanujan defines three functions S, C, and Cj, for real в, by
and
с = с@) = + |
Now, in fact, Ramanujan has replaced в by 26 on each right side above. But
in all subsequent work after Section 14, Ramanujan employs the definitions
that we have given. We have actually already encountered these functions in
A3.7), A1.4), and A1.1), respectively. More precisely,
S = \Zsfx sn(z0), С = {zjx cn(z0), and Ct = \z dn(z0). A4.1)
Entry 14. // C, S, and Ct are defined as above, then
C2 + S2 = \xz2, A4.2)
C? + S2 = \z\ A4.3)
2CS + -^ = 0, A4.5)

18. The Jacobian Elliptic Functions 169
2QS + -^ = 0, A4.6)
and
2CC, = ^. A4.7)
Furthermore, define <p, 0 < <p < 2n, by
С = \zyfx cos q> and S = jz^/x sin (p. A4.8)
Then
d = iz^/1 - x sin2 ф, A4.9)
/ ^— d sin w d<p
zcoscp^/l -xsin2q> = —^— = cosq>—, A4.10)
da do
and
The formulas A4.2) and A4.3) are simply translations of the fundamental
formulas (Whittaker and Watson [1, p. 493])
en2 и + sn2 и = 1 and dn2 и + к2 sn2 и = 1,
respectively. These formulas are generally proved by utilizing representations
for en м, sn м, and dn и in terms of theta-functions. However, Ramanujan
probably used Fourier series. Thus, we proceed by finding the Fourier series
for C2, S2, and C2. In fact, Jacobi [1], [2, p. 196] found the Fourier series for
C2 and S2 but not for C2. More elegant derivations of these Fourier series as
well as the Fourier series for C2 have been found by Glaisher [2], and since
Glaisher's work is not particularly well known, we present it.
Kiper [1] has derived the Fourier series for higher powers of the Jacobian
elliptic functions, while Langebartel [1] has developed Fourier expansions for
several rational functions of Jacobian elliptic functions.
It should be remarked that the two formulas in A4.8) are compatible
because of A4.2).
Equality A4.11) represents Ramanujan's form of the inversion of the elliptic
integral of the first kind.
Proofs. From the definition of S, we see that
CO CO ^3 Ци1 ^" ¦! ^" X fv
_4\2= V V
m— -co n= —a
\)y) sinh((n + \)y)'
The coefficient of expBin0), 1 < n < oo, in this double series is seen to be

170 18. The Jacobian Elliptic Functions
1
I =:
„?±oo sinh((m + \)y) sinh((n — m —
1
sinh(n>>) mi^oo
1
{coth((m + |)y) - ooth((m - и
\ Г М -М-2
. . ,—- lim < У coth((m + ?) v) — У coth((m + т)
Sinh^) M-oo U=J^»+l m=-M-n-l
In
sinh(ny)
A similar calculation shows that the coefficient of exp( — 2in6), 1 < n < oo, is
equal to the coefficient of ехрBш0). Hence,
The derivation for C2 is similar. From the definition of C,
CO CO
4C2= у у I
m^oo „i^oo cosh((m + \)y) cosh((n + \)y)
The coefficient of expBin0), 1 < n < oo, in this double series is equal to
„i^oo cosh((m + %)y) cosh((n — m —
1
sinh(ny) mi^oo
In
{tanh((m + \)y) - tanh((m - n +
by the same type of reasoning as that used above. The coefficient of
ехр(-2ш0), 1 < n < oo, is the same as that for expBin0). Hence,
Third,
oo oo „2i(m+n)d
m^oo »=-oo cosh(m);) cosh(ny)'
The coefficient of expBm0), 1 < n < oo, in this double series is found to be

18. The Jacobian Elliptic Functions 171
1
nt±m cosb{my) cosh(n — m)y
1
sinh(ny) mt±a
In
{tanh(m>>) — tanh(m — n)y)
The coefficient of exp( — 2in6) is the same as that for ехрBш0), 1 < n < oo.
Therefore,
4 2m^i M smh(ny)
It is evident from A4.12)-A4.14) that C2 + S2 and C2 + S2 are independent
of в. Now S@) = 0, and by Entries 16(ix) and 17(i) in Chapter 17, respectively,
C{d) = \zs/x and C!@) = -|z. The formulas A4.2) and A4.3) now follow
immediately.
We next prove A4.4). First, observe that
oo oo „2!(т+я+1)в
~ m^oo »=-oo cosh((m + \)y) sinh((n + \)y)'
The coefficient of expBm0), 1 < n < oo, in this double series is equal to
00 1
у t
nt±m cosh((m + j)y) sinh((n — m — j
1
{coth((n - m - \)y) + tanh((m + \)y}
cosh(n>0 mtLa
{coth((n — m — j)y) + tanh((m
M-.00 m=-M-l
In the last sum, all of the hyperbolic tangent terms cancel as well as all but
2и of the hyperbolic cotangent terms. Each of these 2и surviving terms tends
to 1 as M tends to oo. Hence, the coefficient of ехрBш0) is equal to 2и sechfay).
It is now not difficult to see that the coefficient of exp(—2in6) is equal to
— 2n sech(ny). The Fourier series A4.4) is now immediately evident.
Equality A4.5) is a direct consequence of A4.4) and the definition of Cx.
Upon differentiating A4.2) and A4.3) with respect to в, we deduce that
dC _ dC,
Cle~Cl d%-
Thus, A4.6) follows from A4.5) and the equality above.
Again, from A4.2),
dS _ CdC
de~ ~~S~d6'
Using the equality above and A4.6), we deduce A4.7).

172 18. The Jacobian Elliptic Functions
Equality A4.9) follows immediately from A4.3) and the representation for
S in A4.8).
From A4.7)-A4.9),
d sin a)
1 -> г r.r5„ ^ ^S
\z2Jx cos (py/l-x sin2 (jo = 2CQ = — = izJx
do do
The equalities in A4.10) are now obvious.
The important result A4.11) follows readily from A4.10) and the fact that
0 and <p vanish together.
Since S@) = 0, we may deduce from A4.12) that
and
= 2 а и sin2 И)
„=! sinh(ny)
Equality A4.15) is rather curious. In this connection, we record the following
result found in Berndt's paper [3, Proposition 2.25]. If a, /? > 0 with a)? = ж2,
then
a ? csch2 (аи) + j8 ? csch2 ^^
Entry 15. Let q> be defined by A4.8). Then
(l) 1 + 2 X —r^: = V1 ~ x sm
„ti coshfay) v
n + 1H x r
sinBn
sinBn
i • -i
cosBn + 1H l t /x/l — x sin2 (jo — y/x cos
Log
Proof. Parts (i)-(iii) are merely reiterations of A4.8) and A4.9).
To prove (iv), we integrate (i) and use A4.10). Accordingly,

18. The Jacobian Elliptic Functions 173
. s sinBnfl) f * Л _ ^2, cosBn0)\ ,.
0 + У, = I I 1 + 2 V, — I ^"
„=! И COSh(MH Jo \ n=l COSh(l>l)>)/
re r*
= I ZyJ 1 — x sin2 (/> d0 = dcp — <p.
Jo Jo
Similarly, to prove (v), we integrate (ii) and use A4.10). Thus,
» sinBn + 1H f* » cosBh + 1H
Г» oo
Jo »=O
„ti Bn + 1) cosh{|Bn + l)y} Jo „tb cosh{|Bn
cos <P ^^
Jo
Cv cos (/) dq>
Jo yj\ — x sin2 <p
yj\
= i sin (^/x sin q>).
Lastly, we integrate (Hi) and use A4.10) to establish (vi). Observe from
A4.11) that в = тс/2 if and only if <p = n/2. Hence,
2, cosBn + 1N
Гя/2 « sinBn -
=1
»—о sinn(i^n + l)y}
[
'x sin
«/2
sin q>
^/l — x sin2 <p
— x sin2 <p — yfx cos
Formula (iv) is due to Jacobi [1], [2, p. 158]. Ramanujan (p. 222) omits the
minus sign on the right side of (vi).
Entry 16 (First Part). Suppose that в and q> are related as in A4.8). // we replace
в by \% — 0 in any formula involving 9, then we have the following table for
converting certain functions of <p:
Old Formula
cot <p
sin <p
cos <p
^/l — x sin2 cp
New Formula
,/1 — x tan cp
COS (p
y/l — x sin2 <p
y/l — x sin <p
yj\ — x sin2 <p
yf\ — x sin2 <p

174 18. The Jacobian Elliptic Functions
Proof. From A4.1) and A4.8),
Kk BK0\ , r
— sn I I = \zJx sin q>.
% \ % J
Replacing 0 by %n — 0, using the identities (Whittaker and Watson [1, p. 500])
sn(« + K) = cd w = -—,
dn«
and employing A4.8) and A4.9), we find that
Kk (IK (n _ \\ _ Kk cn(- 2Щп) Kk cn(z0)
n \n\2 )) ж dn(-2Ke/n) % dn(z0)
COS i
— x sin2 q>
Hence, the second entry in the table follows.
By A4.1) and A4.8),
Kk BКв\ , r
СП I I = 2zy/x cos <P-
n \ n )
Replacing в by jtc - в, using the identities (Whittaker and Watson [1, p. 500])
, sn u
сп(м + K)= -k' sd м = -k'-
dnu'
and employing A4.8) and A4.9) once again, we see that
Kk / 2Кв\ Kkk' sn(z0) . r- sin q>
— en К = = jzJx{l -x) .
л V тс У тс dn(z0) V yi-xsin2^
Hence, the third line of our table has been verified.
The first line of the table is now an immediate consequence of the second
and third lines, and the fourth line follows readily from the second line.
Entry 16 (Second Part). With в and q> related by A4.8),
?, (-l)"cosB« + 1H _i r- cos q>
W „h sinh{iB» 4X
k cosh{iB» + 1),}
v sinB" +
(in) esc в + 4
n=O
v (-l)"cosBn
2
,. v л , . v (-l)cosBn + 1H /- r^-
(iv) sec 0 + 4 2, щ+гГу—\ = z sec <P V1 ~ x sin <?'

18. The Jacobian Elliptic Functions 175
and
( П\ да
7 + л +4 I
4 -"У n=0
Proof of (i). Replace 0 by |тс — в in Entry 15(iii) and employ the first part of
Entry 16 to deduce the desired result.
Proof of (ii). Replace в by jn — 0 in Entry 15 (ii) and use the table in the first
part of Entry 16.
Proofs of (iii), (iv). We employ the Fourier expansion (Whittaker and Watson
[1, P. 512])
dn(zfl) ..m sec0 4 » (-l)"cosBn+lH
___ = йфв) - — + - ? eBn+l)y_l ,
Utilizing A4.1), A4.8), and A4.9), we find that
(-l)"cosBn+ 1H zJ\ - x sin2 <p
= T^
which is (iv). Replacing 0 by ^тс — 0 and using the table from above, we
complete the proof of (iii).
Proof of (v). Integrating Entry 16(iv) over [0, 0] and noting that
d I и
— ogtan^- + ^
we find that
. S (-l)"sinBn+lH
= z sec (jo^/l — x sin2 <p ^0 = sec <p dq> = Log tan I - + — I,
Jo Jo \4 2/
by A4.10).
Entries 16(iii)-(v) are somewhat mysterious in that Ramanujan had not
recorded the Fourier series of dc u, which we used in our proofs, or the Fourier
series of ns и, which could have similarly been employed.
Entry 17. Let 0 and q> be related by A4.8). Define L, as in Section 9 of Chapter
15, by
L(e->)= 1-24 1-^.
Then

176 18. The Jacobian Elliptic Functions
cos в ?, n2 sinBn0) з cos
sin3 0 „=i e2ny - 1 sin3 (/>v
(ii) ^Tq
(iii) cot в
J=i smh(ny)
= z cot (jo^/l — x sin2 <p + 2 ^/l — x sin2 <p <
Jo
20z Гя/2 /- ^r~
Jl-x sin (jo dq>,
71 Jo
2JOV л: Jo
- x sw.2 q> dq>.
We are not sure how Ramanujan deduced these formulas, of which (ii) and
(iv), according to Whittaker and Watson [1, p. 520], are due to Jacobi. In our
proofs below, we rely on formulas from the theory of elliptic functions not
recorded by Ramanujan but which are all found in Whittaker and Watson's
book [1].
Proof of (ii). From Whittaker and Watson's treatise [1, p. 535, Exercise 57],
,.-.,*,-«>. + !(=-*)-,?=???, A7.1)
where
E= у/1-х sin2 q> dq>, A7.2)
Jo
the complete elliptic integral of the second kind. Using A4.1), A4.8), and the
definition of ns u, we may rewrite A7.1) in the equivalent form
42 +
„tl e2ny - 1 sm2 <p
It therefore remains to show that
Це~2') = — -2z2 + xz2. A7.4)
7C
Now (Whittaker and Watson [1, p. 521]),
dK E К

18. The Jacobian Elliptic Functions 177
If we convert this equality into Ramanujan's notation and solve for E, we find
that
E = roc(l - x)-^- + %nz(l - x).
Putting this expression for E into A7.4), we find that it suffices to prove that
L(e~2y) = 6x(l - x)z^- + z2(l - 2x).
But this equality is precisely Entry 9(iv) in Chapter 17, and so Entry 17(ii) is
established.
Proof of (i). Differentiating (ii) with respect to 0, we obtain the equality
cos в » n2 sinBn0) 2 cos <p dq>
sin30 + „ti e2ny - 1 ~ ~ Z sin3» ~dl'
But by A4.10), d<p/d9 = z^/l — x sin2 <p, and so (i) is immediate.
Proof of (iii). We first show that the derivatives of the left and right sides of
(Hi) are equal. Differentiating (iii) with respect to 0 and using A4.10), we deduce,
after some simplification, that
2 а о ? n COSBH0) 222 2z r
-CSC2 0 + 8 ? 2 = Z2 -Z2 CSC2 (p E,
п=1 в — 1 7C
where E is defined by A7.2). But this equality is precisely A7.3), which we have
seen is another form of (ii). Thus, it remains to show only that (iii) is valid for
just one particular value of 9. Recalling that в = тс/2 if and only if q> = я/2,
we readily see that (iii) holds when в = тс/2.
Proof of (iv). We first write (iv) in a more traditional form
A sinB»0) _ KE(q>) _ WKE
nksinh(ny) ж n2 ' K '
where
Л
Е(ф) = \/l ~ x sm2 ^ d<p,
the incomplete elliptic integral of the second kind. We now see from Whittaker
and Watson's text [1, pp. 518, 520] that A7.5) may easily be translated into a
result of Jacobi, and so appealing to Jacobi, we complete the proof.
In Section 18, Ramanujan considers equations of the form
П{х,е->,г,0,<р) = 0. A8.1)
He transforms certain parameters and determines the effect of these changes
on the remaining variables. The procedures he establishes are therefore analo-

178 18. The Jacobian Elliptic Functions
gous to the processes of duplication, dimidiation, and change of sign described
in Section 13 of Chapter 17.
Entry 18(i). // в is replaced by \Q and у by %y in A8.1), then
sin {yfx sin q>))) = 0.
Proof. Changing у to %y yields the process of dimidiation described in Section
13 of Chapter 17. Thus, we only need to examine the effect on q>.
Consider Entry 15(iv). Replacing у by \y and в by \6, we find that the left
side of Entry 15(iv) becomes
i^v A8.2)
„ti n
On the other hand, by Entries 15(iv) and (v),
\(<p + sin (yfx sin <p))
_ i » sinBn0) 2, sinBn + 1H
~2 + h 2n cosh(i27^) 4
2n cosh(i27^) „4 B7Г+1) cosh{iBn + \)y)
It follows from A8.2), A8.3), and Entry (iv) that q> is transformed into
\{q> + sin (y/x sin <p)), as desired.
Entry 18(ii). // в is replaced by\n — Q and e~y by —e~y, then
Proof. Changing e~y to — e~y means that we are "obtaining a formula by a
change of sign," which is discussed in Section 13 of Chapter 17. Thus, we need
only examine the effect on q>.
If we replace в by %n — в and e~y by — e~y in Entry 15(iv), we find that the
left side is transformed into
1
„tlcosh(ny)
by Entry 15(iv) once again. Thus, q> is converted into |тс — q>, and this com-
completes the proof.

18. The Jacobian Elliptic Functions 179
Entry 18(iii). // e~y is replaced by —e~", then
Q( * , -e~>, Z^TX, в, cot- (-p?\) = 0.
Proof. As in the proof of Entry 18 (ii), we need only determine the transforma-
transformation on q>.
From Whittaker and Watson's text [1, p. 512], A4.1), and A4.8),
= cs(z0) = -3-J = cot <p. A8.4)
sn(zW)
Replacing e~y by —e~y, we find that the left side of A8.4) is transformed into
tff iye'2
by A8.4) again. Hence, cot q> is converted into (cot q>)/^/l — x, and this
completes the proof.
Ramanujan (p. 223) incorrectly asserted that cot q> is changed into
у/1 — x cot q> above.
Entry 18(iv). Let
z
zr
(Thus, y' and z' arise when x is replaced by 1 — x or k2 is replaced by k'2.) If
в is replaced by idz/z' and у by y', then
il(l - x, e-y', z', Wz/z', i Log tan (^ + |J J = 0.
Furthermore, sin <p is converted to i tan <p and cos <p to sec <p.
Proof. It is clear that it suffices to examine the effect on q>.
By A4.1) and A4.8), sn(z#) = sin q>. Applying the indicated transforma-
transformations, we find that sn(z#) is converted into (Whittaker and Watson [1, pp. 505,
494])
sn(i0z) = i sc@z') = i—77r~- = i tan q>;
cn(az)
that is, sin <p is transformed into i tan (p.
By A4.1) and A4.8), cn(z0) = cos q>. Applying the given transformations,
we see that cn(z0) is transformed into (Whittaker and Watson [1, pp. 505,
494])

180 18. The Jacobian Elliptic Functions
1 1
cn(i0z) =
that is, cos q> is sent into sec q>.
Lastly, q> is transformed into
cn@z') cos q>'
• -1 / ч т ( COS <P \ (П <P\
Sm (I tan *> = ' L°g (rTsi^j = ' L°gtan D + 2 j'
after an elementary calculation. This completes the proof.
Entry 19(i). Let an ellipse of eccentricity e be given byx = a cos <p,y = b sin <p,
0 < q> < In. Let P = {a cos q>, b sin q>) and A = (a, 0). Then L(AP), the length
of the arc AP, is given by
-г
Jo
L{AP) — a\ J1 — e2 cos2 q> dq>.
Jo
Of course, this formula for L(AP) follows from elementary calculus just as
in Section 3. This formula for arc length is apparently due to Legendre [1,
p. 617].
Entry 19(ii). Let a hyperbola of eccentricity e be given by x = a sec q>, у =
b tan <p, 0 < <p <, 2n. Let P = {a sec <p, b tan <p) and A = (a, 0). Then the arc
length L(AP) is given by
L(AP) = a tan (p^Je2 — cos2 q> — a I ^/e2 — cos2 (p dcp
Jo
dq>
Ie2 — cos2 <p
Proof. Using the standard formula for arc length, we find, after some simplifi-
simplification, that
L(AP) = a sec2 (p-Je2 — cos2 q> dq>,
Jo
where e = (l/a)y/a2 + b2. Integrating by parts, we arrive at
L{AP) = a tan <p*Je2 — cos2 (p — a = dq>
Jo ^/e2 — cos2 (p
С 9
= a tan q>sje2 — cos2 <p — a y/e2 - cos2 <p dq>
Jo
Г*1 ( n 5— 1 - c°s2 9 \ 1
+ a \ I Je2 - cos2 q> , 1 dq>.
Jo \ y/e2 — cos2 q>J
Upon simplifying the last integral above, we complete the proof.

18. The Jacobian Elliptic Functions 181
Entry 19(ii) is due to Legendre [1, p. 652], although it is closely connected
with Landen's [2] earlier work in 1775 on the expression for a hyperbolic arc
in terms of the difference between two elliptic arcs.
Entry 19(iii). Let the perimeter L of the ellipse x = a cos t, у = b sin t,
0 < t < 2%, be given by
L = n(a + b)(l + 4 sin2 Щ, 0 < в < л, A9.1)
where
sin в = y/x sin <p and x = I I . A9.2)
\a + b/
Then when e — 1, <p — 30°18'6", and as e tends to 0, <p tends monotonically to 30°.
Our statement of Entry 19(iii) is somewhat stronger than Ramanujan's,
who says that q> "very rapidly diminishes to 30° when e becomes 0."
Proof. We first show that q> > тс/6. From C.2), C.6), A9.1), and A9.2),
3 - 2,/l - x sin2 ф = 1 + 4 sin2 \6 = ? aBx", |x| < 1. A9.3)
n=O
From A9.3), C.7), and Theorem 1 in Section 3, it follows that
3 - У4-Х < 3 - 2,/l - x sin2 q>.
Solving this inequality, we find that sin2 q> > 1/4, or q> > тс/6.
Second, we calculate <p when e = 1. Thus, x = 1 and в = q>. Therefore, from
A9.1) and C.2),
1 + 4 sin2 \ц> = 2F1(-h -\\ 1; 1) = -, A9.4)
by Gauss' theorem (Bailey [4, p. 2]). Thus,
sin2 \q> = - - X- = 0.0683098861.
It follows that <p = 30° 18'6".
Third, we calculate q> when e = 0. From A9.2) and A9.3),
,. . , .. sin20 ,. 4sin2i0
hm sm q> = hm = hm
x->0 jc->0 X jc->0 X
= lim 1=1 = a = i.
*-o x 4
Thus, <p tends to тс/6 as e tends to 0.
Fourth, we show that q> is a monotonically increasing function of x. From
A9.3),

182 18. The Jacobian Elliptic Functions
4A - x sin2 q>) = B-? eye"?,
or
x sin2 <p =
In order to show that <p is increasing, it suffices to show that if we write the
right side above as a power series in x, then all of the coefficients are nonnega-
tive. Putting
t *nxn) = t К*', A9-5)
we observe that it suffices to show that
К < 4aB) n > 2. A9.6)
We first prove that
- + — > — + ——, 1 < r < \(n - 1). A9.7)
<xr an_r ar+1 aB_r_!
Using the definition of an in C.7), we find that
1 1
<xr+1 <xr (и - rJBr + ^)а„_г
J 1 (r - iJBn - 2r - |)ar
2J---(n-r-lJ
Thus, A9.7) has been established.
Upon successive applications of A9.7),
" ^ <3 " 3 <¦¦¦ <con,
al + an-l a2 + an-2 a3 + an-3
where
Hence, by A9.5),

18. The Jacobian Elliptic Functions 183
n-l n-l
К = ? aJa»-J ^ 2co» Z aj
7=1 J=i
A9.9)
by A9.4).
Next, we determine those values of n for which
First, let и = 2т. Thus, by A9.8), we investigate when
am+l < a2m+2
This inequality is equivalent to
(m - jJ Bm + jJBm - jJ
(m + lJ-Bm + 2JBrn + lJ'
It is easily verified that this inequality holds for each positive integer m, and
hence A9.10) is valid for every even integer n.
Second, let n — 2m + 1. Thus, from A9.8), we wish to determine when
a2m+3
a
2m+l
After some simplification, this inequality is found to be equivalent to the
inequality
(w + jJBw2 + m + |) Bm + |JBw + |J
2m2 + 5m + ^ ~ 4Bm + 3J '
After some additional manipulation and computation, we find that this in-
inequality is valid for m > 3 but not for m = 0,1, 2.
In conclusion, A9.10) is true for n = 2,4,6,7,8,9,.... It follows from A9.9)
that, if n is even,
and if n is odd,
Now, A2 = af = ^ = 4a2, Я3 = 2ata2 = т^ = 2a3, and 15 = 2(aja4 + a2a3)
= 29/215 = ffa5. Hence, A9.6) has indeed been established, and this com-
completes the proof.

184 18. The Jacobian Elliptic Functions
Entry 19(iv). Consider the same ellipse as in Entry 19(iii), but now set
where
fa-b^2
sin в = y/x sin <p and x =
Then when e=l,q> = 60°4'55", and when e = 0, q> = 60°.
Proof. First, we prove that if e = 1, then q> = 60°4'55". When e = 1, it follows
that x = 1 and 0 = <р. Thus, from C.2), A9.11), and A9.4),
• 2 'In л
sin (p sin У j x 4
It can be verified numerically that (/> = 60°4'55" is the solution of this equation.
Second, from A9.11), C.2), and C.6),
. , .. sin2 0 „ ,. sin2 0
hm sin q> = hm = 3 hm
= 3 lira —
~~ —» 11111
X x->o X{2 +
з
Hence, q> = n/3.
Although we have satisfactorily proved Ramanujan's assertions in Entry
19(iv), our result is weaker than Entry 19(iii). As we showed in Theorem 2 of
Section 3,
A9-12)
10 + y/4 - 3x »=o 5 + y/l - x sin2
It easily follows from A9.12) that
—i_< 4;in2(p . (i9.i3)
5 + y/1 — |x 5 + ^/1 + x sin2 (/>
Now the right side is an increasing function of sin2 <p. When sin2 <p = |, we
have equality in A9.13). Thus, A9.13) implies that sin2 (/> > |; that is, (/> > я/3.
It seems quite likely that as e decreases from 1 to 0, q> decreases monotoni-
cally from 60°4'55" to 60°. However, the calculations seemingly needed to
prove this conjecture appear to be rather laborious.
Villarino [1] has strengthened Entries 19(iii) and 19(iv) by developing
power series expansions for <p in terms of the eccentricity e.
Villarino has offered a very credible explanation for Ramanujan's rep-

18. The Jacobian Elliptic Functions 185
resentation A9.11). (A similar argument can be made for A9.1).) Setting
t = x = (a- bJ/(a + bJ in C.5) and C.6) and noting that sin2 (тг/3) = |, we
see that
ix sin2 (я/3)
= 1 +
1 J-ixsin2^/3) -ix sin2 (я/3) -Jx sin2 (я/3) -^x sin2 (я/3)
+ з{ i + i + i + i +•
Now replace the feth numerator of the continued fraction by — ?x sin2 (тс/3 +
<xk). Thus, tx1 = <x2 = a3 = 0, but a4 # 0. Next, replace тс/3 + <хк by (/> for each
k, 1 < fc < oo, and also replace jx sin2 (тс/3) by ^x sin2 q>. Lastly, set
sin в = y/x sin (p. Then, formally,
lj"-jsin20 -jsin20 I
3J \ + I +•••]
= 1 +
= 1 +
i(-l +cos0)
sin2 в
+ cos2 (в/2)'
Hence, by C.2), we have established a heuristic derivation of A9.11).
Corollary (i). Let the perimeter L of an ellipse be given by
L = n(a + b)——, 0<в<п/2,
v
where
tan в = y/x cos <jo and x = I r I . A9.14)
\a + bj
Then as e increases from 0 to 1, q> decreases from n/6 to 0. Furthermore, <p is
approximately given by
a + b{ a + b
Proof. We first examine the case e = 0. Then x = 0 and 0 = 0. From C.2) and
C.6),
tan 0 °°
for |0| < n/2 and |x| < 1. Thus, from A9.14) and A9.15),

186 18. The Jacobian Elliptic Functions
, ,. tan20 ,. 02tan20
lim cos <p = lim = hm -^—
x->0 x->0 X x-+0 X в
= lim -J1Z1 ^ '— = 3ax = j.
x->o x 4
Hence, <p = я/6.
We next determine q> when e = 1. Thus, x = 1 and tan в = cos <p. From
A9.15) and A9.4),
tan0
в
!
п=0
Hence, в = тс/4 and q> = 0.
It appears to be extremely difficult to show that as x goes from 0 to 1, q>
monotonically decreases from тс/6 to 0. However, we can show that 0 < q> < n/6.
As a first step toward this end, we show that g(x) := (tan x)/x is a monotoni-
monotonically decreasing function of x for x > 0. Now,
1 tan x
1 _ 1 Cx du
~xA+x2)~FJ0 Пм?
+ X2 1 + H
<a
It follows that g( is monotonically decreasing for x > 0, as claimed.
Suppose that we can show that, for 0 < x < 1,
\яаГ\\у/гх) -=о " в tan-
Since x/tan-1x is increasing, it follows that
\y/bx < tan в < y/x,
or
/ < cos (jo < 1;
A9.17)
that is, 0 < q> < я/6. Thus, it remains to establish A9.17).
In Chapter 12, Section 18, Corollary 1 (Part II [9, p. 133]), Ramanujan
records the continued fraction
_. x x2 BxJ CxJ
tan 1 x = - —-
1 + 3 + 5 + 7 +•••
where here we take x > 0. (For a proof, see Perron's book [1, p. 155].) It
follows that

18. The Jacobian Elliptic Functions 187
tan x 1
or
7-4r- < i + hx2.
tan i x
Hence, for 0 ^ x ^ 1,
i
which establishes the first inequality of A9.17).
To prove the second inequality in A9.17), first define
G(x) := tan Jx -
F(x)'
where F(x) = 2РЛ~ъ ~h U x). We want to show that G(x) <, 0,0 ^ x < 1.
An elementary calculation gives
ljx(l + x)F2(x)G\x) = F2(x) - A + x)F(x) + 2x(l + x)F'(x)
= -ix + ? Цях", A9.18)
я=2
where цп > 0, 2 rg и < oo. Equality A9.18) shows that G'(x) < Ofor x > 0 and
x sufficiently small. It is also clear from A9.18) that G'(x) = 0 at most once on
[0,1]. But since G@) = 0 and G(l) = 0, by A9.4) (or A9.16)), it follows that
G'(x) = 0 exactly once on @,1). Hence, G(x) < OforO ^ x ^ 1. Thus, the proof
of A9.17) is complete.
Lastly, we establish Ramanujan's unusual approximation for <p. We ob-
observe that
a+b
and
Thus, Ramanujan is attempting to find an approximation to (p of the form
y/l - x(A + B{1 - у/1-х) + Cx), A9.19)
which will be a good approximation both when x is close to 0 and when x is
near 1. Our task is then to determine A, B, and C.
First, we find an approximation for q> in a neighborhood of x = 0. From
A9.15), we seek an approximation for в2 in the form
в2 = 3ajx + px2 + •¦•,

188 18. The Jacobian Elliptic Functions
where p may be determined by the equation
<xtx + <x2x2 + -" = ?C<x1x + px2 + •••) +
Equating coefficients of x2 on both sides, we find that
«2 = IP + fa?.
Solving for p, we find that p = — ^. Hence,
X
COS2 9 = ^ = A + \x
Now write
<p = ^ + px +
Then
! + ш>х + ¦¦¦ = со$2(я/6 + /Jx + ¦ ¦ ¦)
i cosB0x + •••)- (,/3/2) sinB/?x
Equating coefficients of x on both sides, we deduce that ft = —21^/3/160.
Hence, in a neighborhood of x = 0, we have the following approximation
for <p:
<P = |-^x + O(x2). A9.20)
Next, we want to approximate q> in a neighborhood of x = 1. As seen earlier
in the proof, в is then in a neighborhood of я/4. A straightforward calculation
yields
_я
я2 Г 4
0--
A9.21)
Now 2^1 (—i, — i; 1; х) is not analytic at x = 1. However, for x in a neigh-
neighborhood of 1, we can deduce that (Erdelyi [1, p. 110, Eq. A2)])
Comparing A9.21) and A9.22), we deduce the approximation
as x tends to 1.

18. The Jacobian Elliptic Functions
189
Lastly, we need the elementary expansion
1
= 1-J(x-1) + -, |х-1|<1.
Thus, from A9.22)-A9.24), we conclude that, as x tends to 1,
tanfl в
fx
1 \/я я
я
A9.24)
COS (p =
в
8(я — 2)
(x - 1) + o(x - 1)
x A - i(x - 1) + o(x - 1))
4-я ,
= 1 +
4(я-2Г
Since cos <p = 1 — %<p2 + • • •, we find that
4-я
or
A - x) + o(l - x),
1 - X + 0(^/1 - X),
A9.25)
as x tends to 1 —.
Our last task is then to use A9.20) and A9.25) in A9.19) to calculate A, B,
and C. When x tends to 0, A9.19) tends to A. Thus, A = тс/6 by A9.20). Next,
examine (<p - я/6)/х as x tends to 0. From A9.19) and A9.20), we find that
-\A + ifi + С = - :
160 '
A9.26)
Now check (p/y/l — x as x tends to 1 — .FromA9.19)andA9.25),weseethat
Ы-п
A+B+C=
A9.27)
Using the value A = я/6 and solving A9.26) and A9.27) simultaneously, we
conclude that
B =
and
80 -2-aI1Ol93S
= -0.0206291.
80
Converting A, B, and С to the sexagesimal system and substituting in A9.19),
we finish the proof.

190
18. The Jacobian Elliptic Functions
Before stating Corollary (ii), we describe a geometrical diagram given by
Ramanujan. Consider an ellipse x = a cos и, у = b sin u, 0 < и < 2%, with С
as the center, A = (a, 0), and В = @, b). Let AN be perpendicular to AC. With
P and Q on the negative and positive x-axes, respectively, let CP = CB = CQ.
Choose a point M on AN such that LMPN = \LAPM. Furthermore, put
cp = L MQA. Consider now a circle centered at P with radius PA. Suppose
that this circle intersects PN at K. Let J denote that point on the circle
obtained from the radius through B.
We abuse notation below in that, for example, PN may denote either the
line segment with end points P and N or the length of this line segment.
However, no confusion should arise.
(PC
в
Q \
k^$4
Corollary (ii). Let L(UV) denote the length of an arc UV of an ellipse or circle.
Suppose that PKN is rotated until
L(AJ) _ L(AB)
L{AK)~ AN '
Then (p is approximately equal to
30° - x(l - x)(ll'22" + 32'42"x),
where, as before, x = ((a — b)/(a + b)J.
A9.28)
A9.29)
Ramanujan states this corollary in a somewhat retrorse manner, because
A9.28) is given as part of his conclusion. He also made a calculational error
and so incorrectly asserted that (p is approximately equal to
30°
- x)E°19.4' - 6°3.5'x).
A9.30)
Ramanujan [7] also posed Corollary (ii) as a problem in the Journal of the
Indian Mathematical Society. Villarino [1] has established a stronger version
of Corollary (ii) in the spirit of Entries 19(iii) and 19(iv).
Proof. We first show that (p = я/6 when x = 0 or 1. Observe that

18. The Jacobian Elliptic Functions 191
AM
tan (p =
a-b'
and so
AM
tan <p = ,.
a + b
Hence,
f tan'1 (^/x tan 9) = | tan i^A = \L АРМ = LAPN, A9.31)
by Ramanujan's construction. Thus, if L denotes the perimeter of the ellipse,
by A9.31) and A9.28),
,an,i tan- ф «an „) .
a + о (a
L ЦАК)
Ца + b) L(AJ)
л ЦАК) _
4 Ь(Л7) 2 l(~2' ~2' '
by C.2).
We now show that
Since СРВ is an isosceles right triangle, L СРВ = я/4. Thus,
(LAPJ)(a + b) = ^{a + b) = L(AJ).
Also,
(LAPN)(a + b) = L(AK).
Combining the last two equalities, we deduce A9.33).
Putting A9.33) in A9.32) and employing A9.31), we arrive at
f tan (y/x tan <p)
When x = 1, this equality reduces to
tan(ftp) _ 4
by A9.4). Hence, (p = я/6.
A9.32)

192 18. The Jacobian Elliptic Functions
For x sufficiently small,
tan(| tan (yfx tan <p))
| tan (y/x tan (p)
= 1 + i(| tan (y/i tan q>)J + Д(§ tan (yfi tan <p))* + ¦¦¦
= 1 + Кл/x tan q> - &fi tan (pK + ¦ ¦ -J
+ Hijx tan <p + ¦ ¦ -L + • • •
= 1 + f x tan2 (p +. JqX2 tan4 <p + ¦ ¦ ¦. A9.35)
In order to obtain a first approximation for <p, we can ignore the fact that (p
is afunction of x. Thus, combining A9.34) and A9.35) and equating coefficients
of x, we find that
| tan2 (p = i
Hence, we can conclude that (p tends to я/6 as x tends to 0.
Our procedure now is similar to that in the proof of Corollary (i). We find
expansions for <p in neighborhoods of x = 0 and x = 1 and combine them
together.
Thus, first we write
(p = \ + px + ¦ ¦ ¦.
о
From A9.34) and A9.35),
1 1 , 3 .(% \ 1 ,
X + X +- = Xtan4 + pX + -j + X
Equating coefficients of x2 on both sides, we deduce that p = — 11^/3/5760.
Thus, in a neighborhood of the origin,
Next, write
Ф = ^ + 0(х-l) + -",
in a neighborhood of x = 1. From A9.22), A9.34), and a very laborious
calculation, we find that
4 1, n, , n _ tan(| tan-1 (Jx tan(^/6 + p(x - 1) + • • •)))
1 ^X — I) ¦+¦ O(X —I) j=
71 n I tan [y/x tan(^/6 + p(x - 1) + • • ¦))
4 9A/1

18. The Jacobian Elliptic Functions 193
Equating coefficients of (x — 1) on both sides and solving for /?, we deduce that
P 12я-24 8 '
Thus, in neighborhood of x = 1,
We now want to combine the two approximations A9.36) and A9.37) into
an estimate of the sort
(f> x % - x(l - x)(D + Ex). A9.38)
6
Examining (<p — я/6)/х as x tends to 0, we deduce from A9.36) and A9.38) that
5760
Examining (<p — я/6)/(х — 1) as x tends to 1 —, we see from A9.37) and A9.38)
that
Thus,
Putting these values of D and E in A9.38) and converting the numbers into
the sexagesimal system, we complete the proof.
In a note following Corollary (ii), Ramanujan indicates a third value
(besides x = 0 and x = 1) at which <p = я/б. Не also records the values for
which his version A9.30) of A9.29) achieves either a local maximum or mini-
minimum. In fact, from A9.29), we see that on 0 < x < 1, the value я/6 is achieved
only at the end points. An elementary calculation shows that A9.29) has a
local minimum 29°52'33". This value is obtained at x = 0.6213949 or e =
0.9929672, since e = 2x1/4/(l + -Jx).
In Entry 20(i), Ramanujan attempts to "square the circle." His approxima-
approximation in doing so is highly accurate and was given in his paper [1], [10, p. 22].
Since Ramanujan gave few details in [1], and since his argument is not very
long, we give it below.
Entry 20(i). Let О be the center and PR a diameter of a circle. Bisect OP at H
and trisect OR at T; more precisely, ОТ = 2TR. Let TQ be perpendicular to
OR, where Q is a point on the circle. Let RS = TQ, where S lies on the arc QR.
Let OM and TN be parallel to RS, where M and N lie on PS. Let PK = PM,
where К lies on the circle but on the opposite side of PR from Q and S. Draw

194
18. The Jacobian Elliptic Functions
PL perpendicular to OP and of length equal to MN, with L on the same side of
PR as K. Let RC = RH, where С lies on RK. Draw CD parallel to KL, where
D lies on RL. Then
nn2 _ 1
RD ~*
Из'
(We have again abused notation in that, for example, RC denotes the line
segment with end points R and С as well as the length of this line segment.)
In fact, Ramanujan says that the area of the circle is (approximately) equal
to RD2. Thus, he is approximating n by
щ = 3.14159292,
which differs from n by about 0.00000027. According to notes left by
G. N. Watson, this approximation to n was discovered by the father of
Adrian Metius. It might be recalled here that Hardy related that Ramanujan
[3, p. xxxi] had "quite a small library of books by circle-squarers and other
cranks."
Proof. For brevity, set d = PR. Thus, PT = У and TR = %d. Therefore,
(frfJ + QT2 = PQ2 and (idJ + QT2 = QR2.
Adding these equations and solving for QT2, we find that
QT2 = Ы2-
Also,
PS2 = d2- RS2 = d2- QT2 = %d2 B0.1)
and
PK2 = PM2 = (^J - OM2 =

18. The Jacobian Elliptic Functions 195
Thus,
PL2 = MN2 = (%PSJ =
and
RK2 = PR2 - PK2 = \$
So,
RL2 = PR2 + PL2 = f§f d2.
Finally,
RD _ Ш) _ 3KL _ 3 /355 144 _ 1 /355
P~R ~ ARC ~ ~4RK ~ 4V324 113 ~ 2>/Т1з'
The desired result now follows.
In a note following Entry 20(i), Ramanujan remarks that "RD is i^th of
an inch greater than the true length if the given square is 14 square miles in
area." Indeed, for a circle of area j7td2 = 14 x 633602 (in square inches),
= 0.0100653026,
which justifies Ramanujan's claim.
The two mean proportionals between a and b are the two values x and у
defined by
a x у
x у b
Corollary (i). Inscribe an equilateral triangle of side t in the circle of Entry
20(i). Let m denote the first of two mean proportionals between <f and PS. Then
mli^dyfn) differs from unity by approximately 1/30,000.
Proof. First, from elementary geometry, t = \sfbd. From B0.1), PS =
Thus, solving
^d _m_ у
m у
for m, we find that m = ^dC1I/6. Now,
m
~~ 29630>
which justifies the stated approximation.

196 18. The Jacobian Elliptic Functions
We have rephrased and clarified Ramanujan's version of Corollary (i): "One
of the two mean proportionals between a side of an equilateral triangle
inscribed in the circle and the length PS is only less by 3OOOOth part of it than
the true length." In summary, Ramanujan has taken C1I/3 as an approxima-
approximation to n. In fact,
C1I'3 = 3.1413807,
which differs from n by about 0.00021.
Corollary (ii). // we approximate л by (97^ — j^I'4 in the expression %f
then if a circle of one million square miles is taken, the error made is approx
imately l/100th of an inch.
Proof. Let \nd2 = 106 x 633602 square inches. Then
( IQHL Ц1/8Ч
= 103 x 63360 1 - v 2 "; = 0.0101561291,
V / J
which justifies the given claim.
Recall that the approximation (97^ - xi)m to % is given in Section 3. Our
version of Corollary (ii) clarifies Ramanujan's original statement (p. 225).
The appearance below of Entries 20(ii) and 20(iii) is enigmatic indeed; there
does not seem to be any connection between these entries and any other result
in Chapter 18.
Entry 20(ii) is due to Euler [2], [4, pp. 428-458] and was rediscovered a
century later by Hoppe [1]. Because the result is not widely known and a
short proof can be given, we provide a proof of Entry 20(ii).
Entry 20(ii). Parametric solutions of the equation
A3 + B3 = C2
are given by
A = Зи3 + 6и2 - n,
B= -3n3 + 6n2 + n,
and
С = 6n2Cn2 + 1),
where n is arbitrary.
Proof. Assume that
A + B= I2n2. B0.2)
Thus, factoring A3 + B3, we find that

18. The Jacobian Elliptic Functions 197
А2-АВ+в2=ш>
where we may assume that n Ф 0. Thus,
(A - Bf = f(A2 -AB + B2) - $(A2 + 1AB + B2)
= ~- 48n4. B0.3)
9n2
We next assume that С can be written in the form С = n2(om2 + /?) for some
pair of integers a, /?. Furthermore, we would like to write
(A - Bf = $п2(т2 - PJ-
From B0.3), we see that if we choose a = 18 and /? = 6, both of these require-
requirements can be met. Thus, we obtain the proposed formula for C, and we find
that
A - В = 6и3 - In. B0.4)
Solving B0.2) and B0.4) simultaneously, we derive the proffered parametric
equations for A and B.
Entry 20(iii). Parametric solutions of the equation
A3 + B3 + C3 = D3
are given by
A = m1 - 3(p + l)m4 + Cp2 + 6p + 2)m,
B = 2m6- 3Bp + l)m3 + Cp2 + 3p + 1),
С = m6 - Cp2 + 3p + 1),
and
D = m7 - 3pm4 + Cp2 - l)m,
where m and p denote arbitrary numbers.
This classical diophantine equation was perhaps first seriously discussed
by Viete [1] in 1591. Euler [2], [4, pp. 428-458] completely solved the
problem by finding the most general rational solution; Ramanujan's solution
is less general. A general solution may be found in Hardy and Wright's text
[1, pp. 199-201]. Now, in fact, sometime later, Ramanujan did find the most
general solution and recorded it in his third notebook. See Berndt's book [11,
Chap. 23, Entry 50] for details. A general solution in rational integers is not
known. Many papers have been written on this venerable diophantine equa-
equation, and one should consult Dickson's book [1, pp. 550-561] for further
references. In particular, there exist many ways in which to formulate solu-
solutions. Thus, we briefly indicate how Ramanujan's solution can be gotten.

198 18. The Jacobian Elliptic Functions
Proof. Some algebraic manipulation shows that the given equation may be
put in the form
(B + C){3(B - CJ + (B + CJ} = (D- A){3{D + AJ + (D - AJ}. B0.5)
Assume that
B + C
D-A
= m2. B0.6)
(The most general solution, in fact, is obtained by putting (B + C)/(D — A) =
m2 + 3k2.) After some additional manipulation, B0.5) can then be written
3{m2(B - CJ -{D + AJ} = A - m6)(D - AJ.
Now define n by
m{B -Q-(D + A) = nm{\ - m6). B0.7)
Thus,
3n{m(B - C) + (D + A)} = *P~ . B0.8)
m
From these last two equations, we see that multiplying А, В, С, and D by the
same constant has the effect of multiplying n by the same constant, and
conversely. Thus, without loss of generality, we can set n = 1. Solving B0.7)
and B0.8) simultaneously for В — С and D + A, we deduce that
В - С = i(l - m6) + J^(D - AJ B0.9)
от
and
D + A = -\m(\ -me) + —(D- AJ. B0.10)
6m
Lastly, we solve for A, B, C, and D in terms of m and D — A. To avoid
fractions, we introduce another parameter p defined by
D - A = 3m(m3 - 1 - 2p). B0.11)
The remainder of the proof is quite straightforward. Equalities B0.6) and B0.9)
yield the proposed values of В and C; equalities B0.10) and B0.11) give the
stated formulas for A and D.
Entry 20(iii) was discussed by Watson in his survey paper on the notebooks
[5].
Ramanujan offers several examples to illustrate Entries 20(ii) and 20(iii).
For each example, we append the values of the parameters needed to produce
the example.

18. The Jacobian Elliptic Functions 199
3+ AK = 392 (B = 2)
H3 + 373 = 2282 (n = i)
713-233 = 5882 (и = 4)
33 + 43 + 53 = 63 (m = p = 2)
I3 + 123 = 93 + 103 (m = 2,p = 3)
13 + 753 = G0iK + D1iK (m = 3jP=n)
33 + 509З + 346 = 1188з (m = 4) p = M)
183 + 193 + 213 = 283 (m = 2,p=l)
73 + 143 + 173 = 203 (m = 2, p = -1)
193 + 603 + 693 = 823 (m = 2, p = -2)
153 + 823 + 893 = Ю83 (m = 2, p = -3)
33 + 363 + 373 = 463 (w = 2,p=-4)
I3 + 1353 + 1383 = 1723 (m = 2,p=-5)
233 + 1343 = 953 + 1163 (m = 2,p=-10)
1333 + 1743 = 453 + 1963 (m = 2, p = -13)
I3 + 63 + 83 = 93.
Observe that the example I3 + 63 + 83 = 93 does not fall under the pur-
purview of Entry 20(iii). Thus, evidently, Ramanujan was aware that he had not
found the most general solution of A3 + B3 + C3 = D3. The example I3 +
123 = 93 + 103 was immortalized by Hardy, who, when writing about his
recently deceased friend, recalled "I remember once going to see him when he
was lying ill at Putney. I had ridden in taxi-cab no. 1729, and remarked that
the number G • 13 • 19) seemed to me rather a dull one, and that I hoped it was
not an unfavourable omen. 'No,' he replied, 'it is a very interesting number; it
is the smallest number expressible as a sum of two cubes in two different ways'"
(Ramanujan [10, p. xxxv]).
Hardy then asked Ramanujan if he knew the corresponding result for
fourth powers. After thinking a moment, he replied that he did not know the
answer and supposed that the first number is very large. In fact, the smallest
solution
1334 + 134* = 1584 + 594 = 635,318,657
had been found by Euler [2], [4, pp. 428-458].

200 18. The Jacobian Elliptic Functions
Actually, the example I3 + 123 = 93 + 103 was found much earlier by
Frenicle in 1657. Frenicle and J. Wallis each found several additional examples
for two equal sums of two cubes. A bitter argument between Frenicle and
Wallis ensued with each accusing the other of using trivial methods. A descrip-
description of this feud may be found in Dickson's book [1, p. 552]. For more
complete details, see Fermat's Oeuvres [1, Lettre X. Vicomte Brouncker a John
Wallis, pp. 419-420; Lettre XVI. John Wallis a Kenelm Digby, pp. 427-457].
In 1898, С Moreau [1] found the ten solutions of A3 + B3 = C3 + D3,
where the sum is less than 100,000. After 1729, the next largest sum is
4104 = 23 + 163 = 93 + 153.
The example I3 + 123 = 93 + 103 can also be found on a fragment in the
publication of Ramanujan's "lost notebook" [11, p. 341].
Entry 21. Let x be any complex number, except that x cannot be a pole of the
functions given on the left sides below. Then
л cosh(w-4/3x) + 2 cosGtx) 1
3y/3x2 cosh(w«v/3x) — cosGtx)
n
% x4 + n2x2 + n4 M. (x4 + n2x2 + n4)(em^ - (-1)")'
л cosh(w4/3x) + 2 cos(jtx) + 6 cosh(^x/4/3)
З/З2 cosh(w4/3x) — cos(jtx)
I 2 У - v ;
4i 11. A. ' Z—i л *, *, t PZ
+ n x + n n=i <x -j- n*x + и ){e v ( 1) )
M x4 + n2x2 + n4'
where
'-, if n is even,
, if n is odd,
..... In 1
("О —
e2*^* - 2e*^*x cos(toc) + 1
1 1 2тг
= 2x3 +
,2f
n=i (x4
(x4 + n2
1

18. The Jacobian Elliptic Functions
201
and
(iv)
In
•-1 (9x4 - 3n2x2 + п4)(епя^3 - (-1)")
1 S 1
х Jk Зх2 + Зпх + и2'
Ramanujan's formulation of Entry 21(ii) in the notebooks (p. 226) does not
appear to be correct.
Proof of (i). Let
._
2 cos(roc)
3^/Зх2
- cos(jtx)
B1.1)
We shall expand / into partial fractions.
We note that / has a quadruple pole at x = 0 and simple poles at x =
n exp(±?ti/3), 1 <, \n\ < oo. After a moderate amount of calculation and con-
considerable simplification, we find that
R
иехр(-я(/3)
—7=— coth(%nn^/3), if n is even,
V3n2
r
tanh(^7tn4/3), if n is odd,
je
"'73
and
exp(Bi/3)
ie~*il3 г
coth(j7tn4/3), if nis even,
if n is odd,
ie
where 1 < |n| < oo.
Consider now the contributions of the four poles +« ехр(±тп/3) to the
partial fraction decomposition of/. For n even, these contributions total
B1.2)
x4 + n2x2 + n4
while for n odd, they sum to
x4 + n2x2 + n4
1 -
B1.3)

202 18. The Jacobian Elliptic Functions
where 1 < n < oo. We also find, after a straightforward calculation, that
f(x) =
+ 0A)
B1.4)
гузях4
in a neighborhood of x = 0. Summing B1.2) and B1.3) over all even n and
odd n, respectively, 1 < и < oo, and using B1.4), we deduce the partial fraction
expansion claimed in Entry 21(i).
Proof of (ii). Let
h(x) := f(x) + g(x),
where / is defined by B1.1) and
In со&Ъ(лх/^/3)
Зу/Зх2 cosh(K4//3x) — cos(roc)
We shall determine the partial fraction decomposition of g and combine it
with that of / from Entry 21(i) to obtain the partial fraction expansion of h
claimed in Entry 21 (ii).
Observe that the poles of g are the same as those for/, with the same orders.
By routine calculations, for the poles of g,
ie
*
and
le *
ifniseven,
, if n is odd,
, ifniseven,
if и is odd,
where 1 < \n\ < oo.
The four poles ± n exp( + ni/3) contribute
2и совЬ^
and
n2x2
, if и is even,
2n si
-, if «is odd,
B1.5)
B1.6)
«4 + n2x2 + и4)
to the partial fraction expansion of g. Finally, in a neighborhood of x = 0,
g(x) =
+ 0A).
B1.7)

18. The Jacobian Elliptic Functions 203
From B1.5)—B1.7), we may determine the partial fraction decomposition of
g. Combining this with the expansion of / from Entry 21 (i), we deduce the
desired expansion for h.
Proof of (iii). Let
2% 1
/(*):=
cos(roc)
which we wish to expand in partial fractions. Observe that / has a quadruple
pole at the origin and simple poles at x = + иехр(±га/3), 1 < и < oo. Straight-
Straightforward, but not quick, calculations show that
—-j= j=r, if n is even,
^nexp( — ni/3) ч . я(-(з — nn [з~п
-= ¦=, ifnisodd,
and
-= -=-, ifniseven,
_ 2,/Зп2 sinh(|wn,y3)
Knexp(%ii3) — "j . _nij3 _яиу^/2
—-= j=^, ifnisodd,
where 1 < \n\ < oo.
Adding the principal parts for the four poles x = + n exp( + ni/3), we obtain
the expressions
3 sinh(i7tn/3) + n3
, if n is even,
n2 sinh(^7tnx/3)(x4 + n2x2 + n4)
and
x3 cosh(iwn4/3) + n3 sinh(\nnJl) .„ .
^Ц= —-—, if n is odd,
n2 cosh(^n4/3)(x4 + n2x2 + n4)
where 1 < и < oo. Lastly, the principal part about x = 0 is found to be
1 I 2n n2
2^/Зпх4 2х3 З^/Зх2 6х'
Hence,
2% 1
3x — 2e"^3x cos(wx)
1 1 ^ 2%
^*~2?^

204 18. The Jacobian Elliptic Functions
_ *L_ , хз у i . у n
6x + к n2(x4 + n2x2 + n4) „4i x4 + и2х2 + n4
00 / 1 \nrt
»-i (x4 + n2x2 + nA){em^ - (-1)")'
Replacing я2/6 by ^"=i и, we find, after a short calculation, that
n2 . 2, 1 й n
1
X n=l X +
+ nzx" +
1
^ v-4 , „2,.2 , „4
n=l X + П X + П
„2-
nx + n
Substituting this into the penultimate equality, we complete the proof.
Proof of (iv). Let
/(x):=
1
: cosC?tx) + 1
which we now expand in partial fractions. We see that / has a quadruple pole
at x = 0 and simple poles at x = ±in exp( ± ти/ЗУ^/З, 1 < и < оо. Routine
calculations yield
, if n is even,
In2 sm
In2
and
In2 sm
In2
if n is odd,
ifniseven,
if n is odd,
where 1 < |и| < oo.
The contributions of the four poles + in exp( ± лi/ЗУ^/З to the partial
fraction expansion of / total
x3 sinhG
if n is even, and
x cosh(^
n2 sii
ЩП^З)
- §n2x sinh
ih(W3)(;
- f n2x cosr
{%ппу/3) ¦+
с4 - i«2x2
^и3 cosh(^rcnN/3)
+ hn*)
-Vsinh(^nV3)

18. The Jacobian Elliptic Functions 205
if и is odd. Lastly, the principal part of/ about x = 0 is equal to
1 inn2
6x3 4. Ay2 6x
Summing all of the principal parts, we arrive at
In 1
3x2 e2*^3x — 2en^3x cosC7tx)
1 _ J_ л
6x3 З^/Зх2
00 v^ 2
-^ + 9
Я
и2(9х4 - Зи2х2 + и4) + „ll 9х4 - Зп2х2 + и4
(-1)"»
+ 6^
n=i (9х4 - Зи2х2 + п4)(е"я^ - (-1)")'
Replacing я2/6 by ^™=1 n~2, we find, after a simple calculation, that
- — + 9 У 3 13 У n
6x h n2(9x4 - 3n2x2 + n4) kx 9x4 - 3n2x2 + и4
if '
Putting this in the penultimate equality, we deduce the desired partial fraction
decomposition.
Example.
« (-I)"?! 1 11 Я П
V v ' 1 1 i
n=i 81 + 9n + n 324^/Зп 756 21^/3 18.^/3A + со8ЬC«уЗя))
Our version of this example differs from that of Ramanujan (p. 226) in two
respects. He has 25/756 instead of 11/756 and - n/E4y/3~) rather than n/B7y/3).
If in B1.8) below, there appeared 1/108 instead of -1/108, then we would
obtain 25/756. We have no explanation for Ramanujan's other numerical
error.
Proof. Putting x = 3 in Entry 21(iii) and rearranging terms, we find that
A (-I)"»
"=i (81 + 9n2 + и4)(евя^ - (-1)")
1 1 я 1
324^/371 Ю8 27^3 6^9 + Зп + п2
B1.8)

206 18. The Jacobian Elliptic Functions
To evaluate the series on the right side of B1.8), we employ the formula
(Hansen [1, p. 105])
where ф(г) = r{z)/T(z). Letting x = 1, у = f, and z = §C^/3I, we find that
CO 1
2 / r\2 2
1 J 2 L7lL^
i U + 3^/3/ r V2 2
2
1 - 3V3i 42 2
Since (Gradshteyn and Ryzhik [1, p. 945])
ф(Ъ + z) - ф{\ - z) = n tan(rtz), B1.10)
we conclude that
1 я s
Substituting this evaluation in B1.8) and simplifying, we complete the proof.
Ramanujan concludes Section 21 with a note claiming that
CO 1
у l
n=l П2 + ИХ + у
can be evaluated exactly if x is an integer and у is arbitrary. This assertion is,
indeed, correct, for B1.9) can first be used to write the sum in terms of
l/f-functions. Using B1.10), the recursion formula
1
and (Gradshteyn and Ryzhik [1, p. 945])
^A — z) = \p(z) + n cot(?tz),
we can reduce this evaluation to elementary functions.
Entry 22. // 0 < x < 1 and n > 0, then
r\ Г" ( Г" dB \; l x 4 9x 16
ш exp I — n —;—— —=)d(p = - - — — ,
Jo \ Jo J\ - X Sin2 в) И + И + И+ И + И +•••'

18. The Jacobian Elliptic Functions 207
f00 ft" dd \ coscp
(ii) exp -n —, . . ,= <fo
Jo \ Jo ^/1 — x sin в/ у/1 — x sin <p
and
(iii)
Proof of (i).
Г
Jo
Let
1
в
exp
H
(
1
и
1
V n
+
4x
+ и
«Г-
JO ,
1-х
и
9
16x
+ и + и
/l —
4x
— n
i0
xsin'
9A
+
+ •
s
le,
—
n
)r
X)
cos <p
— x sin2 q>
16x
- и +••
- B2.1)
sin20
f* Л0
u:= =
Jo ,/1 — x si
Then sin q> = sn м and dipjdu = dn м. Hence,
expl-n = I dy = e~mdnudu.
Jo V Jo^l —xsin 0/ Jo
However, Stieltjes [1], [2] and Rogers [3] have shown that
f ^ -« , , 1/и х/и2 22/и2 32х/и2 42/п2
Jo 1+1+1+1+1 +
which is easily seen to be equivalent to Ramanujan's formula.
Proof of (ii). Let и be given by B2.1). Then
dcp/du = dn и = у/1 — x sn2 и = у/1 — x sin2 <p and cos q> = en м. Hence,
f00 / Г* d0 \ cos<? , Г" »
exp I—и == I —-p= ^y = e "cnudu.
Jo \ Jo y/l — x sin 0/ yjl — x sin2 <p Jo
But by a result of Stieltjes [1], [2] and Rogers [3] (see also A3.3)),
f°° II2 22x I2 42y
e~nu en и du = - — —- —¦ — m?\
Jo П+П+И+П+П+---
and the proof is complete.
Proof of (iii). Using the same substitutions as in the two proofs above, we
find that
f °° ( [* d0 \ cos <p , f« nu en« ,
exp -и = ^2— d<^= e'm-—du.
Jo V Jo y/l - x sin2 0/ 1 - x sin2 <p Jo dnu
In the notation of Jacobi, sin coam u = en м/dn u, and he [1], [2, p. 147] has

208
18. The Jacobian Elliptic Functions
shown that sin coam и = сп(к'и, ik/k'). Hence, the integral on the right side
above may be written
We may now employ B2.2) with к replaced by ik/k' to find that
1 / 1
22{ik/kf
f
42(tfc/k'J
n/k'
k'\n/k' + n/k' + n/k' +n/k'
_ 1 12A - x) 22x 32A - x)
n + n — n + n — n + ¦¦¦'
In order to ensure that this continued fraction converges, we appeal to a
theorem in Perron's text [1, p. 53, Satz 2.16]. The proof is now finished.
Entry 23. Let x and у be complex with Re(x + iy) > 0. Then
(i) '"'
COS
n2ny
x2 + y2
+ У2 + xI12 U + E
[2 i
cos(n2ny)
and
+ y2 - xI'2 f; e~n2*x sm(n2ny)
У2
exp -
n nx
. n2ny
^ SHI I -s ,
+ y2/ Vх + У
У2 ~
1 °°
Proof. As we shall see, both (i) and (ii) follow from the inversion formula
where a/? = n and Re a, Re /? > 0. This formula has been given three times in
the second notebook: as a corollary to Entry 7 in Chapter 14, as Entry 27(i)
in Chapter 16, and as a corollary in Section 6 of Chapter 17.

18. The Jacobian Elliptic Functions 209
Let a2 = ф + iy), and so 02 = n(x - iy)/{x2 + yz). Then B3.1) yields
(x + 1у)щ\1 + У e-n2*x(cos(n2ny) - i sin{n2ny))
B n=i
n2nx
n2ny \ . ( n2ny
x I cos I ——— ) + i sin '
x2 + y2
x'+y2'"'
B3.2)
where principal values are taken. Letting a2 = n{x — iy), so that
$2 - 7t(x + iy)/{x2 + y2), in B3.1), we find that
fl °°
(x — iyI14<- + У e~"x(cos(n2ny) + i sin(n2ny))
12 „tl
exp -
COS
x +yj\ \x2+y2
— i sin
n2ny
x2 + y2
B3.3)
Elementary calculations give
(
x-iy
and
У + У _
iy )
x — iy
Thus, adding B3.2) and B3.3), we obtain part (i), and subtracting B3.3) from
B3.2), we deduce (ii).
Corollary. // Re(x) > 0, then
100 /O X /l -i- V °° t
- + 2_, e cos(n2nyjl — x ) = x — 2j e sm(nzny/l — x ).
2 „=i ^/l — x "=i
Proof. Putting у = ./I - x2 in Entry 23 (ii) and simplifying, we deduce the
proposed formula.
Examples. Recall from Entry 22 of Chapter 16 that the classical theta-function
q> is defined by
9(q)= Ё <i>

210 18. The Jacobian Elliptic Functions
Then
<p(e-*) = V\/5 - 10<p(e-5*) B3.4)
and
Ф fi-*^513) = C + ^Ъ)ф-Ъ«^). B3.5)
Proof. If we set a2 = Sn, and hence ft2 = я/5, in B3.1), we find that
J5<p(e-5«) = <p(e-«i5). B3.6)
Next, let x = 1 and у - 2 in Entry 23(i) to deduce that
2ф. (i + 2 е~<5иJ"/5 + cos ^ f г"
(.2 b=i 5 „=i
n^0(mod 5)
{1 + f C-5»2" - cos ^ g e~^ + cos ^ ?
(i И=1 J П = 1 J Я=1
- cos y)^(C-5") + cos Ц-
by B3.6). Upon simplification, we may readily deduce B3.4).
To prove B3.5), we set x = ,/5/3 in the corollary to deduce that
Since
У5/3 _
y/l -
the penultimate equality reduces to
l - cos y) <p(e-3*^) + cos
2J3 + J5 + 1 . :
sin
sin («р(с
Multiplying both sides by ^[b — 1 and simplifying, we obtain B3.5).

18. The Jacobian Elliptic Functions
211
Ramanujan commences the last section of Chapter 18 with a geometrical
construction (Entry 24(i)). Let у = L TMM' be any angle, such that 0 < у < n,
where MM' denotes the diameter of a semicircle cutting the bisector of у at
R. Let RP be perpendicular to MM' with P e MM'. Suppose that MP is the
diameter of another semicircle. Let Q be a point on this semicircle such that
PQ = PM'. Let 8 denote the angle QMP. Lastly, let S denote the midpoint of
MM'. (As in Sections 19 and 20, we abuse notation by using X Y to denote
the line segment from X to У as well as its length.)
Ramanujan now makes three claims.
Proposition 1. "If RP divides MM' in medial section, then MQ coincides with
MR."
The words "medial section" indicate the golden mean. Thus, Ramanujan
asserts that if
MP J5+1
B4.1)
PM' 2 '
then MQ and MR are coincident.
Proof. Transcribing Ramanujan's conclusion, we are required to show that
cos 8 = cos jy, or that
From similar triangles,
MQ MP
~MP~~MR'
MR MP
MM' MR
Hence, B4.2) is equivalent to
MP2 = MQMR
B4.2)
B4.3)
= JMP2 - QP2 JMP • MM'
- PM'2 jmp ¦ мм',

212 18. The Jacobian Elliptic Functions
or
MP4 = (MP2 - PM'2)MP{MP + PM'),
or
MP2 - MP ¦ PM' - PM'1 = 0. B4.4)
In summary, we have shown that the conclusion of Proposition 1 is equivalent
to B4.4). But solving B4.4) for MP, we immediately obtain B4.1).
Proposition 2.lftx and t2 denote the times it takes for a pendulum to oscillate
through angles 4y and 48, respectively, then
MM'
z=:wt2- B45)
Proof. From Hancock's book [1, p. 91],
IMM' f1 d(p
4
9 Jo y/l — a sin2 cp
where g is the acceleration due to gravity, the length of the pendulum is MM',
and a = sin2 y. Likewise,
where 0 = sin2 8. Hence,
tt,aFi(U;i;gO B46)
Since QP = PM', a brief calculation shows that
p = sin* 8 = (m - IJ, B4.7)
where m is defined in B4.5). Second, by B4.3),
• 2 л - 2i 2i JRP\2(MP\2
= sin y = 4sin2l7cosiy = 4^ДJ
У1(
ма#7 V мм1
Thus, using B4.7) and B4.8) in B4.6), we find that
T= i7/i i. i. л~ ГГ2\ • B4.9)

18. The Jacobian Elliptic Functions 213
We now apply Landen's transformation (Erdelyi [1, p. Ill, formula E)])
, . 4z
t-zJ^i(ii;l;z2) B4.10)
with z = m — 1. We see immediately that B4.9) reduces to t1/t2 = m, and the
proof is complete.
From the definition B4.5) of m, we observe that m > 1. Moreover, this fact
and B4.9) imply that m < 2.
Proposition 3. If m is defined by B4.5), then
2PS
cos у =
mMP'
The factor MP was inadvertently omitted by Ramanujan (p. 228).
Proof. Using B4.3) and B4.5), we observe that
MPV , „ MP t
) l 2l
IPS IPS
MM' mMP'
In a note ending subsection (i) of Section 24, Ramanujan asserts that P is
of the second degree in a. Indeed, from B4.8),
2 - 2^/1 - a
m = , B4.11)
a
and so, by B4.7),
f2 - 2Jl-<x - a\2 A - J\ - aL a2
B4-12)
However, more appropriately, as we shall see below, p is of degree 2 because
the relation B4.12) is a modular equation of degree 2.
Before proceeding further, we precisely define a modular equation of degree
(order) n. Let K, K', L, and L' denote the complete elliptic integrals of the first
kind associated with the moduli к, к', (, and (', respectively. Suppose that the
relation
L' K'
holds for some positive integer n. Then a modular equation of degree n is a
relation between the moduli к and ( which is induced by the equality above.

214 18. The Jacobian Elliptic Functions
Transcribing this definition into our notation and the terminology of hyper-
geometric functions, we conclude that a modular equation of degree и is an
equation relating a and /? that is induced by
"^гШ, 11;«)"° = ^'(ii; 1;/?Л B4ЛЗ)
Entry 24(ii). Let
m = F(i Li-flV B4> *
2fU2i 2> Ь PJ
as ш Section 24(i). We ca// m the multiplier. TTien modular equations of the
second degree are given by
иц/l -a + УД = 1 B4.15)
and
mVl - a + p1 = 1. B4.16)
Furthermore,
W- 1+^-=, ';/ , B4.17)
1yr^ l+(la)
Proof. First, we show that the equalities B4.15)-B4.17) are valid. Then we
demonstrate that these equalities, indeed, are modular equations of the second
degree.
Both B4.15) and B4.16) are easily verified by substituting B4.7) and B4.8)
into the left sides of B4.15) and B4.16). Likewise, the equalities of B4.17) are
similarly verified.
In order to show that B4.15)-B4.17) are modular equations of the second
degree, by B4.13) and B4.14), we need to show that
2-^1 (hh U 1 — «) tn
By B4.7), B4.8), and B4.10) with z = B - m)/m,
m
2Fi(ii;l;l-/0 2Fi(ii;l;mB-m)) 1 + B - m)/m 2'
which completes the proof.
Entry 24(iii). Modular equations of degree 4 are given by
y/m(l - aI/4 + p11* = 1 B4.18)
and

18. The Jacobian Elliptic Functions
m(l _ а)У* +
Furthermore, the multiplier m is given by
1 + A - a)
- aI'4
+ Л/Г^
215
B4.19)
B4.20)
Proof. It is clear from B4.13) that a modular equation of degree 2", r>2, can
be obtained by iterating a modular equation of degree 2'.
The equality B4.12) may be written in the form
1-V1 -«
1 + jT^
Iterating to obtain a modular equation of degree 4, we find that
B4.21)
I + Vl - a
\
After a considerable amount of simplification, we deduce that
- A - a)
B4.22)
We thus obtain the following curious algorithm to derive a modular equation
of the fourth degree from a modular equation of the second degree: replace ft
by y/p and 1 — a by ^/l — a.
We now find an expression for the multiplier m in modular equations of
degree 4. Using B4.14) and B4.11) and then iterating with the aid of B4.21),
we find that
2F,ft. i; i;«) -
2-2J\-oi
Thus, the new multiplier m is equal to

216 18. The Jacobian Elliptic Functions
¦ih-
{1 + A - <xI/4}2'
B4.23)
after considerable simplification.
If we now substitute B4.22) and B4.23) into the left sides of B4.18) and
B4.19) and all expressions of B4.20), we readily verify each identity.
Observe that B4.18)-B4.20) may be obtained formally from B4.15)-B4.17),
respectively, by replacing m by y/m, 1 — a by ^1 — a, and /? by ^/p.
There appear to be some errors in Ramanujan's modular equations in
Sections 24(iv) and 24(v). In Entry 24(iv), Ramanujan claims that
- <xI/8 + p11* = 1 B4.24)
is a modular equation of degree 8 and that
is a modular equation of degree 16. Equations of the 8th and 16th degrees can
be obtained by one and two further iterations, respectively, of B4.22) and
B4.23). Our calculations indicate that modular equations of degrees 8 and 16
are not nearly as elegant as those claimed by Ramanujan. Because the equa-
equations are not attractive and no new ideas are involved, it does not seem
worthwhile to pursue these details here.
Entry 24(v). // we replace a by 1 — /?, /? by 1 — a, and m by njm, where n is the
degree of the modular equation, we obtain a modular equation of the same
degree.
We call this process of obtaining a modular equation the method of
reciprocation. Alternatively, we say that the latter equation is the reciprocal
of the former. In the theory of modular forms, this modus operandi is called
Fricke involution.
Proof. Making the proffered substitutions in the definition of m given by
B4.14), we find that
m 2F!(ii;l;l-«)"

18. The Jacobian Elliptic Functions 217
But by B4.14), this may be rewritten
2*"i(i,i;i;«) 2*"i(i i; i;/0 '
that is, we obtain the defining relation B4.13) for a modular equation. This
completes the proof.
Ramanujan now offers several examples to illustrate his algorithm. Thus,
making the prescribed substitutions in B4.15), we obtain
^ УП^ = 1, B4.25)
a modular equation of the second degree. Next, solve B4.15) for m and
substitute its value in B4.25) to obtain the second degree modular equation
Using Entry 24(v) in conjunction with B4.18), we derive the modular
equation of degree 4,
4=/?1/4 + A - «I/4 = 1.
Im
This equation and B4.23) lead at once to another fourth degree equation
= 2@A - <x)I/4.
If B4.24) were correct, Entry 24(v) would immediately yield the modular
equation of degree 8,
>m
The purported equalities B4.24) and B4.26) taken together would then give
A - A - «)V*)A - pi*) = 2^2(^A - a)I'8.
Entry 24(vi). Consider again B4.13) and B4.14). Then
Proof. Using Entry 9(i) of Chapter 17, we differentiate both sides of B4.13)
with respect to ft to find that
n da 1
«A - olJF}(\, i; 1; a) dp p(l - pJF?& fc 1; /»)"

218
18. The Jacobian Elliptic Functions
Noting the definition B4.14) of m, we see that the proposed formula follows
immediately.
Entry 24(vi) is due originally to Jacobi [1], [2, p. 130]. See also Cayley's
book [1, pp. 201, 216-217].
Ramanujan appears to remark that if we can find da/df} by differentiating
a modular equation (presumably a modular equation independent of m), then
we can determine m from B4.27).
Section 24(vii) consists of the following statement: "Equations in terms of
ф functions can be transformed to those of cp functions and vice versa while
those of/ and x functions remain unchanged. E.g. the identity
Ф^1'3) , , , r w/ ,1 B428)
becomes
<p(q113)
<p(q3)
= 1 +
= 1 +
B4.29)
We are uncertain about Ramanujan's intention in this claim. The functions
ф and cp are related, and we shall show that B4.28) and B4.29) are readily
equivalent, so perhaps this explains part of the statement.
We first show that B4.28) implies B4.29). From the definition of \j/, we find
that 52@, т/2) = 2qm\j/(q), where q = e™. Thus, B4.28) is equivalent to
S2@, т/6)
«
1/3
S2@, Зт/2) \9$@, 3t/2)
Replacing т by 2т/Cт + 1), we transcribe the formula above into
т/3 \ /_./. т \ \1/3
1 = 1 +
0,
Зт
0,
Зт
3t
'Зт
\
-l
B4.30)
By the transformation formulas for theta-functions (Rademacher [1, p. 182]),
o,
Зт+1
Зт+153@,т),
3t + 1S3(O, Зт),
and
$2 0,
т/3
"Зт+

18. The Jacobian Elliptic Functions 219
Using these equalities in B4.30) and putting the resulting equality in a slightly
different form, we deduce that
Since 53@, t) = <p(q), B4.31) reduces to B4.29).
Ramanujan restates B4.28) and B4.29) in Section 1 of Chapter 20, and so
we defer proofs until then.

CHAPTER 19
Modular Equations of Degrees 3, 5, and 7 and
Associated Theta-Function Identities
In several ways, this is a remarkable chapter. Not only are the results enor-
enormously interesting and often difficult to prove, but many questions arise in
regard to Ramanujan's methods of proof. Undoubtedly, many of the proofs
given here are quite unlike those found by Ramanujan. He evidently possessed
methods that we have been unable to discern. No hints whatsoever of his
methods are provided by Ramanujan.
As the chapter's title indicates, Ramanujan herein studies modular equa-
equations, primarily of degrees 3, 5, and 7. For each particular degree, Ramanujan
appears to first derive a series of interesting identities involving theta-functions
of appropriate arguments. These are then used to establish an astonishing
battery of modular equations of that degree (order). We have not always been
able to follow this process, and so, at times, we have had to reverse this
procedure and employ modular equations to prove theta-function identities.
We emphasize, however, that no circular reasoning is involved in our pre-
presentation. Frequently, we prove modular equations in an order different from
that given by Ramanujan. It could be that, in arranging his numerous modular
equations, Ramanujan gave priority to those he felt were more important
and/or more elegant.
The theory of modular equations began with the work of Legendre and
Jacobi. Informative source about modular equations are the books by Cayley
[1] and Enneper [1]. The latter book also provides much of the history of the
subject. Ramanujan's development of modular equations is vastly more sub-
substantial, however, than that of his predecessors. Most of the modular equa-
equations given in this chapter are not found elsewhere in the literature. Not only
are the results new, but Ramanujan's methods are apparently original as well.
Ramanujan published but one paper [2] in which modular equations are

19. Modular Equations and Associated Theta-Function Identities 221
discussed, but because modular equations, per se, were not the raison d'etre
for this paper, Ramanujan's methods in this theory are not disclosed.
Chapters 16 and 17 are crucial for the development of this chapter. Many
basic properties of theta-functions found in Chapter 16 are repeatedly used
here in proving theta-function identities. Many of the formulas in Chapter 17
are employed to establish modular equations and also to produce theta-
function relations from modular equations.
It is always assumed that \q\ < 1. Following Ramanujan, we frequently do
not use a compact summation notation because the laws of formation of the
signs and exponents are more easily ascertained by explicitly displaying the
first several terms.
Entry 1.
(i) Let q> and ф be defined as in Entry 22 of Chapter 16. Then
l1!» П П2 Л3
q
1/8
ql>* q a2 q3
(p(q) 1 +l+q+l + q2 +l+q3 +
(ii) Recall that Да, b) is defined by A8.1) in Chapter 16. Let
Then
t
and
v + v = ^'WV (U)
Proof of (i). We employ Entry 12 of Chapter 16. Replace a2, b2, and q in Entry
12 by 0, — q112, and q1'2, respectively. This gives
(-qz',q2)ao 1 q qz q3
Part (i) now follows immediately from Entry 22, Chapter 16.
K. G. Ramanathan [6] has also proved Entry l(i).
Proof of (ii). Applying Entry 30(ii) in Chapter 16 with a = iq1'2 and b =
— iq312, we find that
,///;/11/2ч I ./,/¦ i/il/2\ /Утл1/2 1л^/2\ _i /Y .*/, 1/2 .',,3/2^ о ft -.3 _5\
у/\щ ) -r y/\ — щ ) — J[}h 9 —Щ ) ~i J\ — Щ > Щ /~7\ — 4 > —4 )•

222 19. Modular Equations and Associated Theta-Function Identities
Similarly, by Entry 30(iii) in Chapter 16,
Hq^ft-q, -q1).
Thus,
it; =
by Entry 22(ii) in Chapter 16. Now apply Entry 11 of Chapter 16 with a, b,
and q replaced by iqllz, 0, and —q, respectively. Thus, A.1) follows at once.
Next, by Entry 30(i) in Chapter 16, with a = —q and b = —q3,
Thus, by Example (iv) in Section 31 of Chapter 16,
mf4-q, -q1)
and
I_f2(-q\-q5) _<p(q) + <p(q2)
V
The truth of A.2) and A.3) are now evident.
Ramanathan [4] has independently given the same proof of A.1).
Entry 2. Recall that f(-q) is defined in Entry 22 of Chapter 16. Then
(i) f(-q, -q*)f*{-q")
= f(-qs)f(-q6, -q9)f(-q, -<?14)/(-<Л -q11),
f(-q2, ~q3)f4-q15)
= f(~q5)f(-q\ ~q12)f(-q2, -ql3)f(-q\ -<A
(ii) f(-q, _g«)/3(-921)
= f(~q1)f(-q6, ~q15)f(-q, -q20)f(-q8, ~q12),
f(-q\ -q5)P(-q21)
= n-q'm-q9, -q12)f(~q2, -ql9)f(-q\ ~q16),
and
f(-q\ -q*)P(~q21)
Proof. Each of these five equalities is proved in precisely the same fashion by
expanding each side into an infinite product via Entry 19 (the Jacobi triple

19. Modular Equations and Associated Theta-Function Identities 223
product identity) and Entry 22(iii), both in Chapter 16. We give the details for
only the proof of the first part of (i).
By the two aforementioned theorems,
Я-Ч, -4*)f4-q15) = (q; q5Uq*l q5Uq5l <?5U<?15; q15)l
and
f(-95)f(-96, ~q9m-q, ~q14)f(-q\ -q11)
and the proof of the first part of (i) is complete.
In fact, Ramanujan appends the words "and so on" at the end of Entry 2.
Thus, the next formula in this series would be for /(—q, —q8)f3(—q21)-
Entry 3. We have
/:\ _././-2\././ _6\ g g . g "
(ii)
41 — q 1 + qz 1 + q4 1 — q
a1
s
(iii)
and
where in the last sum on the right side, the summation is over all values of n
which are neither a multiple of 3 nor an odd multiple of 2.
Proof of (i). First employ (8.5) in Chapter 17 with a = q and b = qs. Then use
Entries 19, 22(iii), (iv), 25(iv), and 24(iii) in Chapter 16. Accordingly, we find
that
fl6n+l oo fl6n+5
Я-fl4 — fl8)
I j-^ISSa I , V+1° = qf V
n=o i — q и=о i — q j\—q,—

224 19. Modular Equations and Associated Theta-Function Identities
/(-g4)(g6;g12b,*V)
q (*WU<?12; ?12)
q
А-д*М-д6)ФЧд6)
x(-q2)f(-glz)
Proof of (ii). Entry 3(ii) is the same as Entry 8(iv) in Chapter 17, and a proof
was given there.
Proof of (Hi). In part (i), replace qz by q, expand the summands into geometric
series, and sum by columns, combining terms in alternate rows. Hence,
gw g512 g1'2 g1112
со „n+1/2 _ 5n+5/2
n% 1 - g6tt+3
- -T V «3"+3/2 sinB«
lnU 1
where we have put q = ew. Hence,
- q6n+3
Employing A3.10) and A4.12) in Chapter 18 and letting
T
cschz(
n= — со
cschz((n
where now e~y = q3, we deduce that
/ q3n+3'2 sinBn + Щ2
~Xh )
= ^ n cosBn0)
ii q~3n - q3n
V щЪ" S'n2 (n0) ? V n
ii 1-g6" nkq-3n-q3n

19. Modular Equations and Associated Theta-Function Identities 225
by A4.15) in Chapter 18. Using C.2) in C.1), we find that
Зч л * nq3n sm2(n6)
3n + ^5nj oo щЗп
\-q6n 3 к Г^
nqn _ ? nq3n
& \~q2n „ti 1 - q6"'
which immediately yields equality (iii).
Proof of (iv). Writing (ii) as a double series and transforming it by columns,
while collecting terms in alternate rows, we first arrive at
Next, we employ the fundamental identity A4.3) of Chapter 18 along with
C.2) above. Thus,
е-У) - ( f
Y
J
+ f ? = -ф(еУ) - ( f L±MY
2 ii cosh(n);)/ ^ У ' Vn=o sinh{iBn + l)y}J
since c"* = g3. Using C.3), letting e~y = iq3/2 and c'fl = iq1/z, and employing
C.4), we deduce that
<Г1/2 - ЧЩ q'1 + q q~3'2 ~ q3'2
Л Г Ч q + q q
_ iqt/2-iq-112 -q-q'1 -iq3'2 + iq'3'2
~\2+ + +
2+ -iq~3'2 + iq311 + -<Г3-<г3 + iq'9'2 - iq
q6
+ f
2 nti

226 19. Modular Equations and Associated Theta-Function Identities
By Entry 8(ii) in Chapter 17,
Thus, using this equality above, we see that
n{(-l)V-2iV/2
= 1 + 4
12
where we have used the factorization 1 — x3 = A — x){\ + x + x2). After
canceling those terms involving powers ofq3, we transform the remaining odd
powers via the formula
q q 2q2
,2"
1+q \-q l-q
Noting that we have now also canceled those terms involving odd multiples
of 2, we observe that the proof is complete.
Entry 4. We have
(i) q*43253 ^^ *V
l-q2 l-q* l-q8
5V
1 — г10 "* '
(ii)
l+q l-q2 l-q*
{)

19. Modular Equations and Associated Theta-Function Identities 227
and
+ +
и M = i +
AV) cp(q>) 1+
Proof of (i). In C.2), replace q3 by q and differentiate both sides with respect
to в. Upon putting в = я/3, we find that
2 sin{Bn + 1)я/3}\ / S Bя + \)qn+m cos{Bn + 1)я/3}\
2t?o l-q2n+1 J\?o l-q2-+r
= » n2q" sinBym/3)
n=i 1 — q
By Entry 3(i),
S, qn+1/2 sin{Bn
2-i i „2п+1 ~^
B=o 1 — q I
By Example (iii), Section 17, Chapter 17,
S, Bn + l)qn+112 cos{Bn + 1)я/3}
n=0 1 — 9
1 » Bn + l)gn+1/2 3 » Fn
2и+1
„й 1-а
Using these last two results in D.1), we obtain at once the equality
^з »=i x -
which reduces at once to (i).
Proof of (ii). Recall the Fourier series (Whittaker and Watson [1, pp. 512,
535])
and
я 7 я2 7Й 1 - 9И
Here и = 2Кв/л, q = e~2)I, 0 is real, and К and ? are the complete elliptic
integrals of the first and second kinds, respectively. Since ns2 и — cs2 и = 1
(Whittaker and Watson [1, p. 493]),
n « qn sinBn0)\2 . , . o s, nqn cosBn0)
cot в - 4 У ^-—Ц—- and esc2 в - 8 ? -1-—Ц—-
в=1 1 + в / и=1 1 — 4
2K \2 , 4K(K -E) o » ид" cosBn0)
— nsu =csc20H ^5 --8^

228 19. Modular Equations and Associated Theta-Function Identities
differ by a constant. Differentiating with respect to в and then letting в = тс/6,
we arrive at
¦/ /Л Л ^<Zsin(n7c/3)\/ 2, 1€Л , 0 ft
cot W6) - 4 ? -TT7-J (esc (,/6) + 8 ?
E7; D.2)
By Entry 3(ii),
cow) - 4 i q1
while, by Entry 8(ii) of Chapter 17,
Using these two results in D.2), we deduce that
Replacing 9 by — q, we finish the proof.
Proof of (iii). By Entry 8(x) of Chapter 17,
•M92)/2(-9)= ? Cn + 1)93л2+2я
n=— 00
= /(9, 95)^-(Log{z(-9V; 96L(-9/z3; 96)oc(96; 96)oc})L=i
az
- 3
where we have employed the Jacobi triple product identity. Now use Entries
19, 22(ii), (iv), 25(iv), and 24(i)-(iii) in Chapter 16 to deduce that

19. Modular Equations and Associated Theta-Function Identities 229
f(-q,-q5) (q;q6Uq5;q6Uq6iq% (q; q2Uq6; q6)*
Ф2(д)Р(д) =Ф2(дМд)Ф(-д)
ФМ-дЖд3) Ф(д3)х(-д)Яд)
_Ф2(д)х(дЖ-д) = Ф3(д)
Ф(д3М-д) Ф(д3)'
Replacing q by — q in D.3) and then using the result above, we finish the proof.
Proof of (iv). By Entry 8(ix), Chapter 17,
<P2(-q)f(-q)= t F« + l)g{3n2+n)l2
= -?¦ t q{3n2+n)l2z6n+%=x
UZ
f(q, q2)^-(Log{zf(q/z6, q2z6)})\z=1
q3n+1 q3n+2
Replacing q by — q, we are led to examine (p2(q)f(q)/f(—q, —q2)- By Entries
19 and 22(ffi) and B2.4) in Chapter 16,
(?; -e)u (e; -«3)oc(-g2; -
<р3(д)(д3; -д3)* y3(g)
from which the truth of (iv) is evident.
Recall that a modular equation of degree n is defined in Section 24 of
Chapter 18 and in the Introduction. In Section 5, Ramanujan offers several
modular equations of degree 3, and so we now summarize some of the notation
that is used in this and succeeding sections. Let
zi = 2*"i(i,i;l;«) and zn = 2F1(i±;l;p),
where n is the degree of the modular equation. The expressions у/л and y/fi
are the two moduli, and we say /J is of the nth order (degree) in a. Recall, from

230 19. Modular Equations and Associated Theta-Function Identities
F.4) in Chapter 17 and the definition of a modular equation B4.13) in Chapter
18, that
<p(q) = <p(e-') = zf
and
tp{ff) = <p(e->) = z]J\
The multiplier m is defined by the equation
Zi = mzn.
It seems that Ramanujan derived his modular equations of degree 3 by
taking formulas relating (p(q), ^i{q),..., (p{q3), &(q3), ¦ ¦ ¦ and transcribing them
via formulas in Sections 10-12 in Chapter 17.
Entry 5. The following are modular equations, formulas for multipliers, and
identities for degree 3:
(ii) (a/?I'4 + {A - a)(l - p)}1* = 1;
/O3\ 1/8 з /A _ K\ 1/8
(iii) m = l + 2 ?- ; -=1 + 2 ЦL)
-!_ ^_ _ 1
m= 1 - 2(afi)v*
(vii)
9 [«У2 , /1 - а\1/2 /a(l - a)\1/2.

19. Modular Equations and Associated Theta-Function Identities 231
/«3A _ „43\ 1/8
(viii) (a|85I/8 + {A - a)(l - mm (j
= (a^I* + {A - aM(l - p)}1'*
= (id + W2 + {(i - «)(i - P)}1'2)}112;
(ix) {a(l - 0)}1/2 1'2
(x) m(l - aI/2 + A - p)x'2 = -A - pI'2 - A - aI'2
m(X - В112 = -^1/2 + а112 = 2(a0I/8;
m
(xi) m - 1 = 2((a^I'4 - {A - a)(l - 0)}1*);
m + 1 = 4AA + (a^I'2 + {A - a)(l - f})}1'2}I'2;
(xii) ifP = {16a/»(l - a)(l -^)}1/8 and Q
then B + ^ +
(xiii) i/ P = (aj3I/8 and Q = (^/aI/4,
then S_
(xiv) i/ a = sin2(^ + v) and В = sin2(^ — v),
then sinB^) = 2 sin v;
(xv) if a is an appropriately chosen root of the quadratic equation
then
We have written Entry 5(xv) in a more complete form than that given by
Ramanujan (p. 231). The appropriate root a is given in E.13) below.

232 19. Modular Equations and Associated Theta-Function Identities
Proof of (i). It is evident from Entries 4(iii) and (iv) that
2ФЧд)_<Р3(д) <P3(-q2)
Ш3) <p(q3) <P3(-q6Y
We now transcribe this equality by means of Entries 10(i), (iii) and ll(i) in
Chapter 17 and find that
o 2-3'2zP(ae'K/8 _ z3/2 zf(l - aK/8
2-112гУ2(ре3УI18 ~z^ + zf(\ - P) '
Upon simplification, the first part of (i) is obtained.
The second equality of (i) is obtained from the first by employing Entry
24(v) of Chapter 18.
Proof of (ii). It is easily verified from Entries 3(i), (ii) that
= cp(q)cp(q3) - q>(-q)q>(-q3).
(This formula can also be readily deduced from C6.2) of Chapter 16.) Convert-
Converting this formula with the aid of Entries 10(i), (ii) and 11 (iii) in Chapter 17, we
find that
from which (ii) is obvious.
This form of the modular equation is due to Legendre [2, p. 229] and can
be found in Cayley's book [1, p. 196] as well as Jacobi's Fundamenta Nova
[1, p. 68], [2, p. 124].
Proof of (iii). From Entries 4(iv) and 3 (ii), it is readily shown that
Using Entries 10(i), (iii) in Chapter 17, we transcribe this equality and find that
3/2 З/2/i _ 43/8
~Щ + ^~Ш\ Ш/8 ~ ^z]' z\j/z.
Canceling z3/2/zj/2, we derive the second equality of (iii).
The first equality of (iii) is simply the reciprocal of the second (in the sense
of Entry 24(v) in Chapter 18).
Proof of (iv). From parts (i) and (iii),
Л1/8 m-1 /A-0KY/8 m + 1
A - aK^1* 3 - m f<x3\lls 3 + re
aDd )
Taking the product of the cube of the first and the fourth, and then the product

19. Modular Equations and Associated Tbeta-Function Identities 233
of the cube of the fourth and the first equalities of E.1), we find that, respectively,
Hence, by E.1) and E.2),
(m-lKC + m) (m - 1)C + mK
and a=—w—• E-2)
from which (iv) is obtainable immediately.
Proof of (v). In E.1), multiply the first and the second, the third and fourth,
and the first and fourth equalities to deduce that, respectively,
- pK\118 _ m2 - 1 /a3(l - аK\1/8 _ 9 - m2
'{l-')!' *'. к">-»>
and
Solving the first equality for m and the second for 3/m, we obtain two of the
desired equalities.
The remaining two equalities of (v) are readily verified by substituting from
E.3).
Proof of (vi). Our procedure is logically somewhat different from that of
Ramanujan. Define p by the equation
m = 1 + 2p. E.4)
Then the required formulas for a and fi follow immediately from E.2.).
Next, in E.1), multiply the cube of the second equation by the third. Then
take the cube of the third equality times the second. We then deduce, respec-
respectively, that
, (
16m 16m3
Using E.4), we deduce the desired formulas for 1 — ^ and 1 — a.
It might be noted that the first and third equalities of E.1) immediately
imply that p > 0 and p < 1, respectively.
For 0 < p < 1, observe that
da 2A - pJB + pJ
dp A + 2pL
and
dp 6p2(l+pJ
>0
,2 -
dp A + 2p)
There is consequently a one-to-one correspondence between a and p and also
between P and p when p lies between 0 and 1.

234 19. Modular Equations and Associated Theta-Function Identities
The parametric equations for a and ft in (vi) were actually first discovered
by Legendre [2, p. 223] and rediscovered by Jacobi [1, p. 25], [2, p. 76].
Proof of (vii). The proofs of (vii)-(xi) depend on E.2) and E.5). Thus, we first
deduce that, respectively,
(p\>2 m{m - 1) /1 - pV/2 m{m
3 + m \1-<V Ъ-т
The first formula of (vii) now follows from a straightforward calculation, while
the second follows from reciprocation.
Proof of (viii). From E.2) and E.5), respectively,
{cup ) = z: and {A — oc)CX — p) \ — .
8m 8m
Hence,
= 1 -
4m
C - m)(m - 1)
Am
by E.2) and E.5). This proves the first equality in (viii).
Taking the reciprocal of this equality, we find that
(a5/?I* + {A - aM(l - /?)Г = 1 ( (\_flj
by E.6). On the other hand, from E.2) and E.5),
1 + (a/?I'2 + {A - a)(l - p)V12
, , (w - 1JC + mf (m + 1JC - mJ (m2 + 3J
= 1++= E8)
' + Ш? = —&m2~- El8)
The truth of the second equality in (viii) is now manifest from E.7) and E.8).
Proof of (ix). The equality of the first and third expressions in (ix) is an
immediate consequence of (vii), and so is the equality of the first and fourth
expressions.
From E.2) and E.5),
(re + l)(m + 3){(m2 - 1)(9 - m2)}1/2
16m2

19. Modular Equations and Associated Theta-Function Identities 235
(m - 1)C - m){(m2 - 1)(9 - m2)}112
+
-2W-.)(!-ЯГ. E.9)
and so the proof of (ix) is complete.
Proof of (x). From E.2) and E.5), it is easily found that each of the first three
expressions in (x) is equal to
Likewise, from E.2) and E.5), we readily see that the latter three expressions
of (x) are each equal to
m
It is possible that this set of formulas was suggested by the simplicity of the
expression for m2a — /J, given in the proof of (iv); for this indicates the likely
existence of a simple expression for the factor m yfa. — y/fi.
Proof of (xi). From E.2) and E.5),
... (m — 1)C + m) (m + 1)C — m)
4m 4m
m2-3
2m '
from which the first equality of (xi) is apparent.
The second part of (xi) follows immediately from E.8).
Proof of (xii). From E.2) and E.5),
P2 = _
8m2 """ * 9-m2 "
Thus,
„_ m2-l . P 9-m2
and — = -
Q n
The elimination of m from the latter pair of equalities yields
P Jl-PQ
Q y/iPQ + 1
Rearranging this equality, we easily deduce the result claimed in (xii).
Proof of (xiii). From E.2) and E.5),

236 19. Modular Equations and Associated Theta-Function Identities
2 _ (re - 1)C + re) an(J ^ m{m - 1)
p2 _ (re 1)C + re) an(J ^
4m 3 + m '
It follows that
m-1 , P 3 + m
PS = ^ and --_.
Eliminating m from this pair of equalities, we find that
P 2 + PQ
Q~2PQ + V
which, upon rearrangement, yields the desired result.
Proof of (xiv). We assume that \i + v and ц — v are positive acute angles, and
so 2v is also an acute angle. Since it is clear that a > fi, it also follows that v
is positive.
Using the given values of a and /J, we find that the first equality of (ix) can
be written in the form
sinB^) = {4 sinB^ + 2v) sinB^ - 2v)}1/4.
Hence,
sin4B^) = 4 sin2B^) - 4 sin2Bv);
that is,
B - sin2B^)J = 4 cos2Bv).
Thus,
sin2 Bm) = 2A - cosBv)),
and (xiv) follows at once, since v is a positive, acute angle.
We observe that
{A - a)(l - /?)}1/2
On the other hand, by E.2) and E.5),
Thus, we deduce that
3-^- E.10)
We also shall later need an expression for cos v. By (xiv) and E.9), we find
that
cos2 v = 1 — sin2 v = 1 — J sin2B^)

19. Modular Equations and Associated Theta-Function Identities 237
_ 1 (m2 _ 1)(9 _ W2)
16m2
_ (rn2 + 3J
16m2 '
that is,
cos v = —- . E.11)
4m
Proof of (xv). Recall that p is defined by E.4) and that, after the proof of (vi),
we showed that 0 < p < 1. Define q by
q = P + p2, E.12)
so that 0 < q < 2. We are then given that
Solving this quadratic equation, we find that either
U3 +
П 2D
2D,
or
2H 2D, + ХГ
Suppose that a is given by E.13). Then from (vi) it follows that
Hence,
as desired.
Suppose that a is given by E.14). If p = p3B + p)/(l + 2p), then by
the one-to-one correspondence established after the proof of (vi),
a = pB + pK/(l + 2pK, which is a contradiction. Suppose that
p = A + pK(i _ p)/(i + 2p). Then 1 - a = pB + pK/(l + 2pK and
1 _ p = p3B + p)/(l + 2p). It follows that
./»; \i-pj i + 2p i + 2p
However, this contradicts (i). Since the two values specified for P are the only

238 19. Modular Equations and Associated Theta-Function Identities
two values that satisfy the equation /J(l — /?) = q3{2 — q)/(l + 4q), we must
conclude that when a is given by E.14), the value of /J(l — ft) is not the one
required in (xv). Hence, the appropriate root a in the statement of Entry 5(xv)
is that specified by E.13).
Suppose that the root given by E.13) is the smaller root, that is,
\3
This reduces to
2q2 + lOq - 1 < 0.
Hence, pB + pK/(l + 2pK is the smaller root when q < iC -Jb - 5) and the
greater root when q > jC yJ3 — 5).
Entry 6.
(i) Let p be defined by E.4). Then
(ii) Let q be defined by E.12), where pis defined by E.4). Then ifq<\$Jb- 5),
(iii) // tan \{A + B) = A + p)tan A, then
(iv) J/ tan \(A - B) = (A - p)/(l + 2p))tan B, then
,Г d(p
J(
¦2p)
2 +p\ . 2 "I1'2
(v)//
. ч 2 tan В + 2A - x)tan3 В
tan |(C + B) = v '
then

19. Modular Equations and Associated Theta-Function Identities 239
dq> , CB d<p
It is tacitly assumed that 0 <> А, В, С < тс/2.
Proof of (i). Part (i) is simply a version of the formula E.1), that is,
2Fi(U; i; <*) = »»2Fi(i,i;i; A
when a and ft are given by the parametric equations of Entry 5(vi).
Proof of (ii). By Entry 33(ii) in Chapter 11 (Part II [9, p. 94]),
2F1(l i; 1; 4a(l - a)) = 2F&, \; 1; ^A - {1 - 4a(l - a)}1'2))
where, to obtain the last equality, we merely repeat the prior steps, but in
reverse order. The equality in (ii) now formally follows from Entry 5(xv).
Observe that in taking the square root above, it was assumed that a < \,
that is, pB + pK/(l + 2pK < \. It is easily seen that the latter statement is
equivalent to the inequality q < ^C ^/3 — 5), and so the proof of (ii) is com-
complete. (With respect to the restriction on q, recall the remarks made at the
conclusion of the proof of Entry 5(xv).)
Proof of (iii). This formula is the general transformation of the third order.
There is no evidence as to how Ramanujan obtained it. Thus, rather than
derive a proof ab initio, we merely content ourselves with demonstrating how
it can be derived from the form of the general third-order transformation
discovered by Jacobi [1], [2, p. 76]; namely,
r <* л
Jo f, ,/2 + p\.. 2 I1'2 J«
1/2>
when
. _ A + 2p)sin A + p2 sin3 A
~ 1 + pB + p)sin2 A
Solving this quadratic equation for p, we find that
1 — sin A sin В ± cos A cos В
P ~ sin ,4(sin В - sin A)
Since A and В vanish simultaneously, it is clear that the ambiguous sign above

240 19. Modular Equations and Associated Theta-Function Identities
must be replaced by a minus sign. So,
1 - cos(A - B) sin |(B - A)
sin Л (sin В — sin A) sin A cos \{B + A)
cos A sin i(B + A)
sin A cos j(B + A)
In other words, tan %(A + B) = A + p)tan A, and this establishes (iii).
Proof of (iv). In Jacobi's result F.1), replace p by — B + p)/(l + 2p), A by
—B, and В by A. We then deduce that
— Г
1 + 2р Jo
iv) S1" m 1 ~"
when
. 3A + 2p)sin В - B + рJ sin3 В
S1D " A + 2рJ - ЗрB + p)sin2 В '
Solving for B + р)/A + 2р), or employing F.2) with the designated substitu-
substitutions for p, A, and B, we find that
2 + p 1 - cos(,4 + B)
1 + 2p sin B(sin A + sin B)
_ sin \{A + B) _ cos В sin \{A - B)
sin В cos \{A — B) sin В cos \{A — B)
that is
tan $(A - B) = (fj? - 1) tan В = rj^^tan B.
\1 + 2p у 1 + 2p
Therefore, the proof of (iv) is completed.
Proof of (v). By replacing В with С in (iii) and comparing (iii) and (iv), we
deduce that
ГC *<P _ з fB dep
Jo {1 - x Sin2 q>Y12 Jo {1-х Sir
:sin2(p}1/2'
where x = pB + pK/(l + 2pK, and В and С are connected by the relation that
is obtained by eliminating A from the equations
tan ftA + C) = A + p)tan A and tan \(A - B) = -?—? tan B. F.3)
1 + 2p
From the addition formula for tan и and the latter equality of F.3), it follows
that

19. Modular Equations and Associated Theta-Function Identities 241
.an ЦА + щ .
Hence,
tan A = tan {^4 - B) + j{A + B)}
F.4)
_ 1 + 2p A + 2p) - A - p)tan2 В
~~~ A - p)B + p)tan2B
~ A + 2pJ - A - p)(l + 2p)tan2 В
_ 3A + 2p)tan В - A - pJ tan3 В
A+2pJ-3(l-p2)tan2B '
Using both equalities of F.3) and then F.4), we find that
tan i(C + B) = tan{i(,4 + C) - %(A - B)}
_ A + p)(l + 2p)tan A - A - p)tan В
1 + 2p + A - p2) tan A tan В
_ 2 tan В + 2A - x)tan3 В
1 - A - x)tan4 В '
after a somewhat lengthy computation. This finishes the proof of (v).
Although a triplication formula of this type is due to Jacobi [1, p. 29],
[2, p. 80], this form of the triplication formula with the relatively simple
expression connecting В and С is due to Ramanujan. The simpler relations
between A and В in (iii) and (iv) evidently made it possible for Ramanujan to
discover his elegant rendition of the triplication formula, whereas the analysis
needed from Jacobi's relations would, indeed, be more formidable and less
discernible. Jacobi's work is recapitulated in Cayley's treatise [1, pp. 201-202].
Entries 7(i), (ii). Recall the definition of z in F.2) of Chapter 17. Let x =
pB + pK/(l + 2pK. Let В denote an acute angle. If cos В = A - p)/B + p),
then
«) r* *
Г if
Jo {1-х sin2
J/ sin В = A + 2p)/B + p), then
(ii) Г '?
{ ' Jo {1-х sin2 -11/2 ?
л
Proof. We show that these two formulas are consequences of Entry 6(v) and
the associated conditions F.3). From F.3), we see that as В increases from 0

242 19. Modular Equations and Associated Theta-Function Identities
to я/2 to n, %{A — B) does the same. Thus, A increases from 0 to Зя/2 to Зтс.
Also, if A increases from 0 to к/2 to тс, \{A + С) does likewise. So, С increases
from 0 to я/2 to я. Thus, when С is equal to n, so is A, and, furthermore, В is
a positive acute angle.
Using Entry 6(v) and recalling F.9) in Chapter 17, we thus deduce that
Г—
Jo {!-•
dtp л
: sin2 q>}1/2 3
when В is the positive acute angle that satisfies the equation
1 -P
tan \{л - B) = i~ tan B.
Using the identity
tan \{л - В) = cot \B =
sin Б
we eventually find that cos В satisfies the quadratic equation
B + p)cos2 В + A + 2p)cos В + (p - 1) = 0.
Solving this equation for cos В and taking the proper root, we find that
cos В = A — p)/B + p), as desired.
Also, when С is equal to тс/2, so is A. Thus, by the same type of argument
as that above,
Г
Jo
dcp я
Jo {1-х sin2 p}1'2 6"
where В is the positive acute angle that satisfies the equation
tan^it — \B) = -—r-tan B.
u 2 1 + 2p
From elementary trigonometry, this equation may be put in the form
sin В — 1 + cos В _ 1 — p
sin В + 1 - cos В ~ 1 + 2p
It is easily checked that sin В = A + 2p)/B + p) is the solution to this equation.
In the classical notation of elliptic functions, formulas (i) and (ii) assume
the respective forms
2K\ 1-p (K\ l + 2p
1 - and sn — = ¦
2 + p V3/ 2 + p'
where k2 = x = pB + pK/(l + 2pK. Other formulas of this nature are due to
Forsyth [1], Glaisher [3], and Burnside [1].

19. Modular Equations and Associated Theta-Function Identities
243
Entry 7(iii). Let AP denote the diameter of a circle Ш with center O. Let ТВ be
perpendicular to AP, with Be4> and T e AP. Draw chords PR and PRt equal
in length to ТВ with R nearer to B. Form AB, AR, and ARX. Then a pendulum
oscillating through LABAR^ takes (AR - ОТ)/АО or 3A0/(AR + ОТ) times
the time required to oscillate through L.4BAR.
Proof. As we shall see, Entry 7(iii) can be derived from Entry 5(xiv).
Let OP = a and LPAB = ц. Then AB = 2a cos ц and
Thus,
ВТ = AB sin \i — 2a sin /x cos /x.
плп RP ВТ .
sin LPAR = -— = —- = sin ^ cos ^ = sin v,
2a la
when v is defined by the equation sinB^) = 2 sin v. By the converse of Entry
5(xiv),
a = sin2(^ + v) = sin2 LBARX
and
P = sin2(ju - v) = sin2 LBAR.
Let tx and t2 denote the respective periodic times that it takes for a
pendulum of length {to oscillate through the angles 4BAR1 and 4BAR. Then
(Hancock [1, p. 91])
*2lt dq>
7- . , ,T,j and t2 =
{1 - a sin2 ер}112 V0Jo
By Entry 6(i), tjt2 = m.
Now observe that LBOT = 2ц. Thus,
ОТ = a cosBfi) = a—-
2m
dcp
G.1)

244 19. Modular Equations and Associated Theta-Function Identities
by E.10). Also,
AR = la cos v = la
Am
by E.11). Substituting these values for ОТ and AR into (AR - ОТ)/АО and
3A0/(AR + ОТ), we find quite easily that each reduces to m, and so the proof
is complete.
Corollary. Suppose that T coincides with 0. Then LBAR = тс/12, LBARX =
5тс/12, and
AR-OT _ ЗАО r-
АО AR + OT~^ '
Furthermore, a pendulum oscillating through 300° takes yfb times the time
required to oscillate through 60°.
Proof. The hypotheses immediately imply that ц = тс/4, v = n/6, and m =
^/3, by G.1). So, LBAR = ц - v = тс/12 and LBARX = ц + v = 5тс/12. The
assertion about the pendulum also follows at once from Entry 7(iii).
According to notes left by G. N. Watson, this special case concerning the
angles 60° and 300° is due to Legendre several years before the discovery of
the general cubic transformation.
The following geometrical description is an extensive elaboration of that
in Entry 7(iii). For convenience, we have divided Entry 7(iv) into three parts.
Entry 7(iv) (First Part). Let AP denote a diameter and PQ a chord of a circle
<?. Let В denote the midpoint of the arc PQ. Draw AB and PB. Let Bt be the
mirror image of В in AP and construct ABt and PB±. Let R be a point on %>
such that PR = \PQ and so that R is on the same side of AP as В and Q. Let
К t be the image of R in AP. Form PR, PR1,QR,andQR1. Draw ARandARu
cutting PB and PB± at С and Cu respectively. Construct a line perpendicular
to AP at P. Let the extensions of AB and ABX meet this line at M and Mt,
respectively. Extend BP and AR1 to their point of intersection C2, and extend
BtP and AR to their point of intersection C3.
Then a circle V will pass through M, C, Cx, MltC2, and C3, and this circle
will be orthogonal to the circle (€. Furthermore, <ё" will be tangent to the straight
lines AB and AB^ at M and Mj, respectively. Let О denote the center of <ё".
Form ОМ and OM^.
The circle <g" also passes through the intersections of the circles with centers
A and P and radii AB and PR, respectively. The distances of any point on the
circumference of V from A and P bear a constant ratio. Lastly,
QRQRi = 3RP2.

19. Modular Equations and Associated Theta-Function Identities
245
Proof. Our procedure is logically somewhat different from that of Ramanujan.
Let the radius of ^ be denoted by a. Let L PAQ = 2ц, where Q is any point
on « Since the arcs PB and BQ are equal, it follows that LPAB = ц. Also,
AB = 2a cos ц and
Letv= LPAR. Then
sinB/x) = |^.
RP QP
sin v = —— = -—.
2a 4a
Hence, G.2) and G.3) imply that
sinB/x) = 2 sin v.
G.2)
G.3)
G.4)
We now reorder the steps in Ramanujan's line of reasoning. Draw a second
circle <g" that is tangent to AB and ABX at M and Mu respectively. We show
that <€' passes through C, C1,C2, and C3.
Since, by construction, A is the pole of MMt with respect to (€', A and P
must be inverse points with respect to <ё" (Coxeter [1, p. 78]). Consequently,
the circle <6 with AP as its diameter is orthogonal to %>' (Coxeter [1, p. 80]),
and <ё" is the locus of points whose distances from A and P are in the constant
ratio (Court [1, p, 173])
AM
1
PM sin ц
To prove first that С lies on <?', observe that, by the law of sines,
AC sin LAPC sin LAPB cos ц _ 2 cos ^ _ 1
PC ~ sin Z./MC ~ sin LPAR ~ sin v '
by G.4). Thus, by G.5), С lies on «".
G.5)

246 19. Modular Equations and Associated Theta-Function Identities
Examining next C3, we have, by the law of sines,
АСЪ sin LAPСъ sin LAPBt cos ц 1
PC3 sin LPAC3 sin L PAR sin v sin ц
as above, and so, by G.5), C3 lies on V.
Since Ct and C2 are the images of С and C3, respectively, in AP, Cx and
C2 must also lie on <«?'. Thus, since C, Cj, C2, and C3 lie on <g", our definition
of <«?' is reconciled with that of Ramanujan.
Next, draw the circles with centers A and P and radii rx = AB and r2 = PR,
respectively. If X is either of their points of intersection, then
AX _rt
Thus, by G.5), X lies on <ё" provided that r2 = rt sin /x. Since rt = AB = 2a cos ^
and r2 = РЯ = 2a sin v, we see that this condition is indeed met.
By the law of sines, elementary geometry, G.4), E.10), and E.11),
QR sin LQPR sinB/x — v) sinB^)cos v — cosB^)sin v
KP~sin LRQP sinv ~ sin v
= 2 cos v - cosB/x) = m. G.6)
Similar considerations show that
QRt sin LQPRX sinB^ +v) „ /Л1 ч З
^ww = -¦ ^-7^ = —V- = 2 cos v + cosBu) = -. G.7)
RP sin LR&P sinv m K '
Thus, G.6) and G.7) imply that QR ¦ QRt = 3RP2, as desired.
Most of the content of Entry 7(iv) (first part) was submitted by Ramanujan
as a problem to the Journal of the Indian Mathematical Society [5], [10, p. 331].
A solution to this problem was never published. However, more recently, the
first part of Entry 7(iv) was the basis of a Ramanujan Centenary Prize
Competition [1] held in Mathematics Today, an Indian journal aimed at
students of mathematics in high schools and colleges. A total of 24 solutions
were received, and three were published.
Entry 7(iv) (Second Part). A pendulum oscillating through the angle 4BAR1
takes QR/RP or ZR^/R^Q times the time required to oscillate through the
angle ABAR.
Proof. By the last statement of Entry 7(iv) (first part) and G.6), the two ratios
QR/RP and 3R x P/R x Q are each equal to m. The given result now follows from
the proof of Entry 7(iii), wherein it was shown that the ratio of the two
respective designated times is equal to m.
Entry 7(iv) (Third Part). With the notation of the first part of Entry 7(iv) and
Entry 5(xiv),

19. Modular Equations and Associated Theta-Function Identities 247
«1/2 *• /?l/2 (\ ^Al/2 /i ff\l/2 /T o\
a ~ лг- ' P ~ ~7^' " ~ а> —^7Г> \l — P) --ГЯ' \.'-°)
AC,
= (bcbcY = (bmV'2 = bp
{ac.acJ \am) ap' ( '
{A-«)A-ЯГ« (~ -^, G.10)
m = R~P' m = R~i' G11)
G.12)
ЛМ AM
A - а)зу8 а1/2 A - аI'2
— р J (ар; (Ц — ад! — р))
_ ВС, АР АР _ PC, АР
~ ~ВР~АС, ~ АС, ~ ВР АС,
PC, AM
AC, PM
and
!/8 Л .
= 1, G.13)
[-a) \aJ {A — а)A — уЗ)}1/8
АР^ВСАР АРСР СРАМ
~А~С А~СВ~Р~А~СВ~Р~А~СР~М~ ' (' '
Proof. Since LBAC, = ц + v and LBAC = ft - v, it follows that
ВС, . ВС
—^- = sin(// + v) = a1/2, -— = sm{fi - v) = )?1/2,
—- = cos(/i + v) = A - aI/2, and — = cos(// - v) = A - PI12.
Thus, G.8) is established.
From G.8),
AC3ACj
By G.4),
v)sin(// - v)}1/4
= {sin2 ft cos2 v — sin2 v cos2
G.15)

248 19. Modular Equations and Associated Theta-Function Identities
From
We thus
Thus, all
Next,
From
and so
similar triangles,
BP
AP =
find at once that
MP
"Ш and
(BPV _
[ap) -
BP BM
А~Р~Ш>'
BM
AM'
the equalities of G.9) are established,
by G.4),
{A - a)(l - /?)}
similar triangles,
1/8 = {cos(//
= {cos2 ц
= COS jl =
+ v)cos(fi- v)}1/4
! cos2 v — sin2 ц sin2 v}1/4
AB
"ap'
AB AP
A~P~A~M'
(ABV _
\AP ~
AB
A~M'
G.16)
G.17)
Thus, G.10) has been proved.
The equalities of G.11) have already been established in G.6) and G.7).
Equality G.12) is a trivial consequence of G.9) and G.10).
Next, by G.15), G.17), and Entry 5(xiv) or G.4),
аЛ1/8
Р)
ll
- аK\1/8
-t)
а1/2
(аДI/8
sin fi
sin v
(
{A-
v) cc
1 - aI'2
а)A-Д)}1/8
>s(/i + V)
COS fl
sinB/i)
sin /г cos /г 2 sin /г cos /г
On the other hand,
sin(fi + v) cos(fi + v) ВС, АР AB AP
sin ft
COS ft
AC,
ВС,
BP
PC7
BP
AP
AC,
AP
AC,AB
АР ВС, - BP AP
AC, BP AC,
PC, AM
BP AC, AC,MP'
by G.16). Thus, all equalities of G.13) have been established.

19. Modular Equations and Associated Theta-Function Identities 249
Lastly, by G.15), G.17), and G.4),
I-a) \ol) {A - a)(l - jS)}1'8 (a/?I/8
cos(fi — v) sin(fi — v) sin v
= 1.
cos ц sin ц sin ц cos
On the other hand,
cos(// - v) sin(// - v) _ AB АР ВС АР
cos ц sin/г ~ ACAB AC BP
AP BCAP APBP-BC APCP
AC ACBP AC BP ACBP
_ СРАМ
~ac~pm'
by G.16). This completes the proof of G.14) and all of Entry 7 as well.
This concludes, for this chapter, Ramanujan's study of modular equations
of degree 3, with the concomitant theory of theta-functions and associated
geometry. In Section 8, we begin the corresponding theory for degree 5.
Entry 8. We have
(i) #3(«Ж«5) - 5q2t(q№3(q5)
q 2q2 3q* 4q* 6q6
T '
1 - q2 \-q* 1 - q6 T 1 - q8 T 1 - q12
(ii) 5<p(q)q>3{q5) -
l+q l-q2 l+q3 l-q4 l-q6
+ q l+q3 l+q1 l+q9
and
(iv)
q 2q2 3q3 4q*
l+q l + q2 1 + q3 1 + q* 1 + q6
The proofs of Entries 8(i), (ii) are rather difficult. Some results from Section
13 are employed in our proofs. However, no circular reasoning is involved,

250 19. Modular Equations and Associated Theta-Function Identities
because the results from Section 8 are not subsequently utilized, except in
Entries 9(i), (ii), (v), and (vi), which are not used in Section 13.
Proof of (i). From C.4),
(f n+i/2sin(w + 2Hy _ у Щ" sin2(jnfl)
\,ko 1 - q2n+1 ) ~' »=i 1 - q2n '
Let в = Зл/5 and в = я/5 in turn and then subtract the two equalities to
deduce that
¦ 2
. BИ + 1O1
Ot . B« + lOt{\
sin—To—j> (8Л)
The cycles of values for the three expressions
4 ( . J3nn\ . Jnn\\ 2
io7 sin
and
./. Fn + 3)n . Bп + 1)я
2 sin — sin
10 10
are, respectively,
1,1,-1,-1,0,-1,-1,1,1,0;
1,1,0,1,1,-1,-1,0,-1,-1; (8.2)
1,-1,-4,-1,1,-1,1,4,1,-1.
Thus, (8.1) is equal to
q 2q2 3q3 4q4 6q6 Iq1
J I I -* * i
'^ А. л fi * Я. л 1 О л 1^1
1 - q2 1 - q* l-q6 1 - q8 1 - q12 l-,»Tl-4"
9q9 / gl/2 q3/2 qll2 q9/2 ^11/2
^17/2 ^19/2
x-«"
^13/2 4 15/2
+ . 13+i IS
1 — q 3 1 — q
q11'2 q1912

19. Modular Equations and Associated Theta-Function Identities 251
Let Sx denote the first series on the right side of (8.3). Expanding the
summands into geometric series and then summing by columns, we find that
оо „я+1/2 i -Зп+3/2 , -7В+7/2 ¦ _9п+9/2
^Юп+5
We now apply (8.5) in Chapter 17 with a = q1'2 and b = — q9'2 and then with
a = q3'2 and b - —q1'2. It follows at once that
_ -Lzrrv (8'4)
Applying Entries 29(i), (ii) of Chapter 16 with a = q,b = —q*,c= —q2, and
d = q3, we find that
(8.5)
and
(8.6)
Adding (8.5) and (8.6), we see that
f(q\ q')f(-q3, -q1) + qftf, q%)f{~q, ~q9) = /(«, -q*)f(~q2, q3)-
Substituting this into (8.4), we find that
с qil2f(q> -<z4)/(-<z2><z3)<p(-(
/(-«, -q9)f(-q\ -q1)
(8.7)
/(-«, -«9)/(-«3, -<z7)'
by the corollary to Entry 28 in Chapter 16. By the Jacobi triple product
identity (Entry 19, Chapter 16),
by Entry 22 in Chapter 16. Thus, (8.7) can be written
о g1/2f(q)f(q5M-q5M-q5mq10)
Letting S2 denote the latter series on the right side of (8.3), we rewrite it in
the form

252 19. Modular Equations and Associated Theta-Function Identities
ql/2 qW qbjl qV2 q9l2 qll/2 q13/2
^ + + +
q15/2 qn/2 q19/2 / q5l2 ql5l2
" !_gi9 \l-q5 l-q1
°), (8-9)
by (8.6) in Chapter 17.
Now put (8.8) and (8.9) in (8.3) and change the sign of q. Employing Entries
10(i), ll(iii), 12(ii), (iii), (v) in Chapter 17, we find that
q Iq1 3q3 4q4 6q6 Iq1 8qs
l-q2 l-q* l-q6 I-qs l-q12 l-q1* I - qi6
1/2 3/2 /5/1 _ 454 1/24/ /R\V2
m \ V 5-m
4 5 — m
1/2 3/2
ZlZ5
4 {р?Р)*\т-5\^ j, (8.10)
by Entry 13(iv) and A3.13), where p = (m3 - 2m2 + 5mI/2. On the other hand,
by Entry ll(i) in Chapter 17, we easily deduce that
/fl\l/4\
(8.11)
Combining (8.10) and (8.11), we at last complete the proof of Entry 8(i).
Proof of (ii). The proof of (ii) is not unlike that of (i). Examining Entries 16
(second part) (iii) and 17(ii) in Chapter 18, we observe that the difference of
/ . , S q2n+1 sinBn + 1N>\2
esc в + 4 У -— ~r——
V »=o i - q J
and
is independent of в. To that end, letting в = л/10 and в = Зя/10 in turn and

19. Modular Equations and Associated Theta-Function Identities 253
subtracting the two formulas, we deduce that
,(n\ 2f3n\ » nq2n f (m\ Cnn\
csc [w) ~csc {то J -8„5 n^ |cos Ы -cos Ы
7^ +CSC — +4 > чг+тШП
я\ (Ъп
W - csc(io
10
„2и+1
. BИ + 1O1 . Fл + Ъ)П
х < sin — sin
10 10
Now,
2 / /ия\ /Зия\\ 2 / . Bи + 1)я . (би + 3)я
and
. . Bп+1)я . (бв-
-2 sin — sin-
10 10
repeat in cycles of length 10, respectively, according to (8.2). Consequently,
q2 bf 3q6 4qs 6q12 Iq1* 9q16
+ + + +
llq
18
1-я l-aJ 1-е 1-е 1-е 1-е1
1-е13 1-е15 1-е17 1-е
Letting St denote the former series on the right side of (8.12), we transform
it just as in the previous proof. We then apply, from Chapter 16, the corollary
to Entry 33(iii) twice, Entry 29(i), Entry 19, and Entry 22. Accordingly,
» qn + qin + qln + q9n
1 ~ +«=i l + e10
/(e, q) , f(q, e )

254 19. Modular Equations and Associated Theta-Function Identities
_ (g4; g20Ug8; <?20U<?12; <?20U<?16; g20Ug20; <?20?
(«; g10Ug3; <?10U<z7; gi0Ug9; g10Ug10; glo)l
_(дА;дАид20;д20ид5;д1Х 2(_ 10)
(д;д2Ш0;д10I 9 к q '
x(-g)f4-g10)
<p\-q10). (8.13)
Letting S2 denote the latter series on the right side of (8.12), we rewrite it
in the form
s =1 g_ + ^_ <^ + A_ g9 , g11 g13
°2 l+l3 l5'l-l7 l9+l" l13
g15 g17 g19 ^ q5 q15
(8.14)
by Entry 8(i) in Chapter 17.
Denoting the left side of (8.12) by ?S and putting (8.13) and (8.14) in (8.12),
we deduce that
Invoking Entries 12(iii), (iv), and (vi) in Chapter 17 and Entry 13(iv) below,
we find that
E-mJcp(q)(p(q5)cp2(-qi0)
4m(l - aI/4
p-3m + 5 5p-m2-5m\q>(q)q>(q5)q>*(-q10)
m )
4m m 4m2 ) A - aI'4
\~g
where we have used A3.10) and A3.5) below, Entries 10(i) and (ii) in Chapter
17, and Entry 25(iii) in Chapter 16. Replacing q2 by — q, we complete the proof.

19. Modular Equations and Associated Theta-Function Identities 255
Proof of (iii). For brevity, in the proofs of (iii) and (iv), we put
+ ¦ "
l-q2 l-q4 1 - q6 " 1 - q8 1 - q12
q Iq2 3q3 4q4 6q6
q 3q* Iq1 9q9 Uq11
SM = TT-Z ~ TT-3 - ТТ-Г7 + 77-5 +
l+q l+q3 l+q7 l+q9 l+q11
and
q 2q2 3q3 4q4 6q6
~ 1 + q l+q2 l+q3 l+q4 l + q6
First,
ite) - Si(-e) =
1-q2 1-q6 1-е1* ¦ 1-е1
l+q l+q3 I+q^ l+q9" l+q11
(8.15)
Second,
q3 V
+ 1 Z +
1 - g2 1 - g3 1 - 4* 1 -
'л 1'.. н ж а л ill
l-q l-q3 " l-q7 l-q9 l-q11
4q2 12q6 28q14 36q18
q 3q3 7q7 9q9
Iq1 9q9 llq
11
+ q l+q3 l+q7 l+q9 l+q11
Combining (8.15) and (8.16), we see that
5S3(q) = S2(-q) + 2S2(-q2) + 3{Si(q) - S^-q)}.
Hence, by parts (i) and (ii), Entries 10(ii), (iii) and ll(i), (ii) in Chapter 17, and

256 19. Modular Equations and Associated Theta-Function Identities
A3.10) and A3.11) below,
24 + 40S3(q)
= lO(p(-q)(p3(-q5) ~ 2cp3(-q)cp(-q5) + 20cp(-q2)<p3(-q10)
<p(q)cp4q5)(l0{(l - <x)(l - р?}Щ ~ 2m{(l - aK(l - fi)}*
+ 20{(l - a)(l - pK}11* - 4m{(l - aK(l - 0)}1/8 + 6m(a3j8I/e
+ 6m{a3(l - aK)?(l - )?)}1/8 - 3O(a03I/8
/ч -к, чч( -чк "• т -"«у (р - Зт + 5J 5(р — т2 + Зт)
= Ф<«>Ф V)( ^ ^ + ~
+
5)
2m 8m
V)B5 - т2)
which is formula (iii).
Proof of (iv). Expanding the summands of S^q) into partial fractions, we see
that
в»- « V V + V + V
+
q l-q2 I - q3 T 1 - q* T 1 - q6
q 2q2 3q3 4q* 6ql
1 + q 1+q2 1 + q3 1 + q4 T 1 + q6
11
1 — q 1 — q3 1 — q1 1 — q9 1 — q
q i 2^2 3^Э У 6g6
Hence,
Applying parts (i)—(iii), Entry ll(ii) in Chapter 17, and A3.12) and A3.11)
below, we arrive at

19. Modular Equations and Associated Theta-Function Identities 257
4 + 20S4(-q)
5q>Hq)<p(q5)
V)Em - m2)
-^Ц - 200
# (-r)
-200 +
8A - J8).
4Em - m2)
- <x)/93(l - pK}1'8
4m2 — p2 ^л/ч 64Em — ш2)т2
^ - 200 + l '
m2(m - IJ p2 - (m2 - 3mJ
(Am2
(8.17)
But by Entry ll(ii) in Chapter 17 and A3.12) below,
ФЧ-д) _ 2 /a(l - a)\1/2 _ Dm2 - p2J _ (m - 5J
«2«A4(-«5) ~ Ш Ul - W ~ ™> - IL " (m-1J'
Utilizing this in (8.17), we find that
4 + 2OS4(-q) = q2il/(-q)il/3(-q
Replacing q by —q, we finish the proof.
Entry 9. 77ie following identities are valid:
where the powers of q are not multiples of 5 but are otherwise all the odd
multiples of 22k, к > 0, and where the signs of the terms are +, —, —, +,

258 19. Modular Equations and Associated Theta-Function Identities
according as the power of q is congruent to 1,2, 3, 4 (mod 5), respectively;
(iii) (p2(q) - cp2(q5) =
(iv) {cp(q5) + 2qx>5f{q\ q1)}1 + {(p(q5) + 2q«5f(q, q9)}2
= vHq1'5) - 2q>\q) + 3(p2(qs);
(vi)
/(-«, -q*)fi-q2, -q3) = f(-q)f(-q5);
and
/(«. qg)f(q\ q1) = xW(-qs)f(-q20).
The most fundamental result in Entry 9 appears to be (iii), and so we prove
it first.
Proof of (iii). Employing Entry 8(i) of Chapter 17 and summing by columns,
we first see that
1 - q l-q3 1 - q1 1 - q9 \-q
11
q13 q11 q19
,g + <? t« + q ,
l+q20 l+q30
Now apply the corollary to Entry 33(iii) in Chapter 16 with a = q and
b = q9 and then with a = q3 and b = <?7. We next employ Entry 29(ii) of
Chapter 16 with a = q,b = q9,c = —q3, and d = — q1. Accordingly,
= 4qf(q,q9)f(q\q1), (9.1)
by two applications of Entry 30(iv) in Chapter 16.

19. Modular Equations and Associated Theta-Function Identities 259
By Entries 19 (Jacobi triple product identity) and 22 of Chapter 16, and
also B2.2) of the same chapter,
ч1) = (-«; q10U-q3; Л(-<г7; q10U-q9; q10Uqi0; qlo)i
= X(q)n-q2°)f(-q5)- (9.2)
Substituting (9.2) into (9.1), we complete the proof.
Proof of (ii). By Entry 24(iii) in Chapter 16 and Entries 12(i), (ii), and (iv) in
Chapter 17,
, /V) , , /V) 4qx(q)zl/2{P4l-pM\1124
4q=4qx{q) = \)
_zi'2{<p2(q)-<p4q5)}
<p(q)
by part (iii), and so (ii) is now immediate.
Proof of (i). Part (ii) may be rewritten in the form
a(l - a)
by Entry 24(iii) in Chapter 16 and Entry 12(v) in Chapter 17. The reciprocal
of this modular equation, in the sense of Entry 24(v) of Chapter 18, is
m~1==2V0(l-]G
Transcribing this via Entry 12(v) in Chapter 17, we find that
V(«) X\q) >5
by Entry 24(iii) in Chapter 16. It follows that
which is complementary to (ii).

260 19. Modular Equations and Associated Theta-Function Identities
Hence, by Entries 8(ii), (iii),
40| = 25q>{q)q>3(q5) - ^ - 5{5<p(q)<p V) - <p4q)<p(q5)}
q 3q3 Iq1 9q9
+ + ¦
~ 4 +
q 2q2 Ъч3 . 4q*
J+q l-q2 l+q3 l-q* l-q6
Iq1 8q8 9q9
For each value of n which is an odd multiple of 22k, for some к > 0, we employ
the trivial identity
q" _ q" _ 2q2n
1 ±q" ~ 1 + q" + 1 - q2n'
Upon simplification, we find that
fs(q) 1 . e( q V V V , V
/to5) ¦ u-e i-«3 i + «4 1-е7 1-е9
, Ив11 , 12«12
12
1 1 1 ' л
-q11 l+q
where each of the indices is an odd multiple of 22\ к > 0, and the signs of the
terms are +,+,—,—,—,—,+, + according as the power of q is congruent
to 1, 2, 3, 4, 6, 7, 8, 9 (mod 10). Replacing q by -q, we complete the proof.
Proof of (iv). By Corollary (i) of Entry 31, Chapter 16,
<p2(q115) = {<P(q5) + 2q1/5f(q3, q1) + 2q^f(q, q9)}2
+ 2q^f(q,q9)}2
If we now employ (9.1), (iv) follows at once.
Proof of (v). From the Jacobi triple product identity, observe that
f(-q2,~q3) d^Jtfiq'Uqtiq5).
_ »
En + I)q5"+1 » Eи + 4)q5n+*
„=o 1 - q l „=o l-q
(SnA
A En + 2)q _
2L ^ „5«+2 L, \ _5n+3
B=o i — q и=о l ~ 9

19. Modular Equations and Associated Theta-Function Identities 261
We now use the elementary identity
2mq2m mqm mqm
\-q2m \-qm l+qm
on each even indexed term above when m is an odd multiple of 22k, Ic^O.
Observe that all expressions of the type n<f/(l — q") cancel, and so we are left
with only expressions of the form ±nqn/(l + q"). Furthermore, note that we
obtain a plus sign when и = 1, 4 (mod 5) and a minus sign when и = 2, 3
(mod 5). Hence,
d Я-д\-д3)_ q 3<?3 V V . V
where the powers of q are odd multiples of 22k, k>0. The truth of (v) is now
manifest from (i).
Proof of (vi). Proceeding as we did at the start of the previous proof, we find
that
( Eи + 2)д5и+2 ^ Eи
и=о 1 + q n=o i
Eл + l)q5n+i ™ En
by Entry 8(iv), and the proof is complete.
Proof of (vii). By the Jacobi triple product identity,
/(«, qA)f(q2, i3) = (-«; 95)»(-92; «5)со(-«3; «5)c
<p(-q5)f(-q5)
by B2.3) and Entry 24(iii), both in Chapter 16.
The second identity in (vii) is found in the corollary of Entry 28 in Chapter
16.
Lastly, the proof of the third equality in (vii) is given in (9.2.).
There are, in fact, several proofs of Entry 9(i) in the literature. The first is
due to Darling [1], who employs a heavy dosage of the theory of theta-
functions. Mordell [1] shortly thereafter gave a shorter proof based on a

262 19. Modular Equations and Associated Theta-Function Identities
certain Hauptmodul in the theory of modular functions. Bailey's [2] first proof
of Entry 9(i) depends on a certain formula for a well-poised basic bilateral
hypergeometric series, while his [3] second proof rests on the theory of the
Weierstrass ^-function. A more recent proof of Raghavan [1] depends on the
theory of modular forms.
Now, in fact, Entry 9(i) also appears in a manuscript of Ramanujan [11]
on the partition function p(n). The formula is mentioned as a companion to
a formula forfs(q5)/f(q), which leads to a rapid proof of Ramanujan's famous
congruence pEn + 4)sO (mod 5), n > 0. For a further elaboration of this fact,
see the papers mentioned above. Another proof by Ramanujan for this con-
congruence is discussed by Hardy [3, Chap. 6].
An application of Entry 9(iii) to lattice sums has been given by I. J. Zucker
[3].
Ramanathan [1] has utilized Entry 9(v), which has also been proved by
Bailey [3], in providing proofs of some formulas from Ramanujan's "lost
notebook" [11].
Entry 10. We have
(i) Ш15) - q3/4(i5) = f(q2, q3) + <?1/5/(<z, <z4).
(ii) cp(q115) - <p(q5) = 2qll5f{q\ q1) + 2q^f{q, q9),
И n-q){f(-q115) + qll5f(-q5)} = f\~q\ ~q3) - q2/5f2(~q, ~q%
(iv) q>\q) - (P2(q5) = Mf(q, q9)f(q\ q1),
(v) *2{q) - #2(«5) = f{q, q4)f(q2, q3),
(vi) /4q2,q3) + qf4q,q4)
(vii)
= (^ - #z5)Wta) - 4
\(p(q ) /
and
(viii) flo(~q2, ~q3) ~ q2f10(~q, ~q*)='
Proofs of (i)-(iv). To prove (i), merely replace q by q1/5 in the third equality
of Corollary (ii), Section 31, Chapter 16. Likewise, (ii) follows from the second
equality of Corollary (i), Section 31, Chapter 16.
Part (iii) is simply a repetition of Entry 38(iv) of Chapter 16.
Part (iv) follows immediately from Entries 9(iii) and (vii).

19. Modular Equations and Associated Theta-Function Identities 263
Proof of (v). The proof of (v) is more difficult and is similar to that of Entry
9(iii), which is obviously an analogue of Entry 10(v), a fact made even more
transparent by Entry 10(iv). To avoid fractions as much as possible, we shall
work with ф2(ч2) - q W°)-
Employing Example (iv) in Section 17 of Chapter 17, expanding the sum-
mands into geometric series, and summing by columns, we find that
1 + q* q + q13 q2 + q
13 q2 + q22
+
l-q5 l-q15 l-q25
q + q3 q* + q10 q1 + q11
+ l-q5 l-q15 + l-q25 '"'
We now turn to (8.5) in Chapter 17 and apply it twice, with a,b = iq112, — iq912
and a,b = iq312, — iq1'2. Adding the two results, we see that the equality above
is equal to
Aq,q9)f(q\q7)
Next, add the two formulas in Entries 29(i), (ii) of Chapter 16 with a - q,
b = q*, с = q2, and d = q3. Obtaining a formula for the expression in curly
brackets above, we deduce that
_ <Р2(-д10Жд10)
x(~g2)f(-g20)
_<p(-glo)f(-g10)
x(-g2)
= f{q2,q8)f(q\q6),
where we have successively applied Entry 9(vii), Entry 25(iii) in Chapter 16,
Entry 24(iii) in Chapter 16, and Entry 9(vii) once again. Replacing q2 by q, we
obtain the required result.
For another approach to (v) via modular forms, see a paper of Raghavan
and Rangachari [1].
Proof of (vi). Formulas (vi) and (vii) are the first of a type which is rather
numerous.

264 19. Modular Equations and Associated Theta-Function Identities
Let С denote an arbitrary fifth root of unity and replace q1'5 by q1/5? in
Entry 10(i). Hence,
П Ш150} = П {/(«2, I3) + qll5mq, q*) + q3l4Mq5)}, (Ю.1)
where each product is over all fifth roots of unity. From the product rep-
representation of tl/(q) given in Entry 22(ii) of Chapter 16, we see that
\imq ш fe^ Д 1-е2-1 *(«*)•
n#0(mod 5)
Thus, on multiplying out the product on the right side of A0.1) and using part
(v) and the equality above, we find that
J^j = « W) + /V, q3) + qf4q, qA)
/2fe <?4)/V, <?3)
V
V)
which, upon factorization, yields the result we sought.
Proof of (vii). Formula (vii) is obtained in the same manner as (vi). Let С again
denote an arbitrary fifth root of unity. Employing (ii), we find that
П {<P(qV50} = П Ш) + 2«1/5C/(«3, q1) + 2<ГС7D, Я9)}-
Multiplying out and using the same argument as above, we arrive at
^ = cp5(q5) + 32q/V, q1)
Thus, by (iv),
32q/V, q7) ^
-5<p(qs){<p2(q)-q>2(qs)}2,
which, upon simplification and factorization, yields the proposed result.
Proof of (viii). Proceeding in the same fashion as above, we find from (iii) that
f4-q)Y\{f(-qil50}
с
= П {/2(-42. -<73) ~

19. Modular Equations and Associated Theta-Function Identities 265
Upon expanding the products, we see that
Substituting the expression for /(— q, — q*)f{—q2, — q3) given in Entry 9(vii)
and simplifying, we complete the proof.
See Ramanathan's paper [4] for another proof of Entry 10(viii).
Entry 11.
(i) There exist positive functions ц and v such that
<p(q115) = <p(q5) + V115 + v1/5, A1.1)
where
л + v = <pHq)~(qf){q5){<p*(<i) ~ V(«)*V) + и? V)},
+ Wiq5)}112,
and
(fivI'5 = cp2(q) - q>\q5).
(ii) There are positive functions ц and v satisfying the equations
<f1/4Vta1/5) = qS'W) + ^ + v1/5, (П.2)
where
and
(iii) (f

266 19. Modular Equations and Associated Theta-Function Identities
then
f{-q, -
(iv) For certain positive functions ц and v,
/( ql'5) = E + /'5 - v1'5I'3, A1.3)
qll5f(-q5)
where
( )Ц5=;5 , 3 /6(~g)
ant/
<?/ (-<? ) 9 / (~9 ) <7 / (~9 )
The functions ц and v have different identities in different parts. In fact,
explicit identifications will be made in the proofs. In the sequel, when we speak
of a "positive function" ft (or v), we mean that ft (or v) is positive for sufficiently
small positive values of q.
Proof of (i). A consideration of Entry 10(ii) immediately shows that A1.1) is
valid with
fi1'5 = 2qll5f(q\ q1) and v1/5 = 2q^f(q, q9). A1.4)
The formulas for ц + v and (fivI15 then follow at once from Entries 10(vii),
(iv), respectively.
It remains to prove the formula for ц — v. Observe that
(fi — vJ = (ft + vJ — 4//v
W
- 4{cp2(q) - cp2(q5)}
<p(q )
- \00(p2(q)<p6(q5) + 125<p V)}-
Upon factoring the expression in curly brackets on the far right side above
and then taking the square root of both sides, we complete the proof.
Proof of (ii). From Entry 10(i), we can at once deduce A1.2) with
p1l5=ql*0Aq2,q3) and v1'5 = q9'Mf(q, q%

19. Modular Equations and Associated Theta-Function Identities 267
The formulas for ju + v and (juvI/5 now follow from Entries 10(vi), (v),
respectively.
It remains to prove the displayed formula for ц — v. Now,
(ц - vJ = (ц + vJ - 4/xv
+ llq W)}2 - 4q5l*{r(q) ~ # V)}5-
The remainder of the calculations are identical with those of the previous
proof, with (p(q) and q>(q5) being replaced by ij/(q) and qll2\j/(qs), respectively.
Some care must be exercised in the determination of the proper square root.
Proof of (iii). By the celebrated Rogers-Ramanujan continued fraction,
Entry 38(iii) of Chapter 16,
q1^ 4 ? ? = ll5f(-q,-q4)
1 +1+ 1 + 1 +••• q f{-q*,-q*y
Let each of these equal expressions be denoted by J. Thus,
qil5f(~Q, -q4)f(-q\ ~q3)
qll5f(-q)f(-q5)
f(-qv5)
+ 1 = 2v
№7) = 2v' AL5)
by applications of Entries 10(iii) and 9(vii). Also, by Entries 10(viii) and 9(vii),
Qf5(-q, -q4)f4~q2, -q3)
fll(-q)+nqf4-q)f4~q5)
qf4-q)f4-q5)
+П2"
Solving each of A1.5) and A1.6) for J, we deduce that
(v2 + II'2 - v = J = {(ju2 + II'2 - ju}1'5,
and the proof is complete.
Before proceeding with the proof of part (iv), we derive parametric rep-
representations that will subsequently be useful.
From Entry 11 (iii) and the binomial theorem,
(ц2 + 1I/2 ± ц = {(v2 + II'2 + v}5
= A6v4 + 12v2 + l)(v2 + II'2 + A6v5 + 20v3 + 5v).

268 19. Modular Equations and Associated Theta-Function Identities
Hence, upon subtraction of these two equalities,
2/x = 32v5 + 40v3 + lOv = A + wM + 5A + wK + 5A + w),
where w is defined by v = ^A + w); that is, from the definition of v,
f(-q115)
qll5f(-45Y ( }
Furthermore, by the definition of д and the formula for ц above,
{}q\ = w5 + 5w4 + 15w3 + 25w2 + 25w. A1.8)
qf (-r)
Thus, from A1.7) and A1.8),
qxl5f6(-Q) f4-q)
f4~q115) qf4~
_ ws + 5w4 + 15w3 + 25w2 + 25w
Proof of (iv). Let J be as in the previous proof. Now, from A1.5),
f4-q115) П t
q3ISf4-q5)~\J
In light of A1.3), we are motivated to define
We then need to verify the proffered formulas for (//vI/s and v - ц.
First, by A1.6),
= 3(J-S - J») - 8 = 3
Second,
/ 5V Л -
A1.9)
= (J15 - Г15) + 258(J10 + J-10) - 315(J5 - Г5) + 540
= 1056 + 312(J-5 - J5) + 258(J-5 - J5J - (J-5 - J5K
f6t-a) f12(-a) fi8(-a)
= 34375 + 5625 J^\ H' +225- У 4) J K 4}
qf4-q5) q2f12(~q5) q3f18(-q5Y
upon the use of A1.6). The proof of (iv) is now complete.

19. Modular Equations and Associated Theta-Function Identities 269
In Entries 12(i)-(iv), we have used the variable Q instead of q in order to
present the proofs more clearly. As in Entry 11, the functions ц and v change
from formula to formula. J. M. and P. B. Borwein [6] have employed Entry
12(iii) in devising a quintic algorithm for calculating n.
Entry 12.
(i) There are positive functions ц and v such that
where
;uv = 1 A2.2)
and
fH-O5)
(ii) For certain positive functions д and v,
where
and
v M-Ull5*Q/(-6)|5-15^/(-6) 25^
v n ii + 15 G /6@ + G /12@ G
|515^ 25^
/6(_0 + G /12(_0 G /8(_e) ¦
(iii) There exist positive functions ц and v such that
where
ц + v =
(iv) For certain positive functions ц and v,

270 19. Modular Equations and Associated Theta-Function Identities
where
and
С„ФЧ<25)
f(-q115)
_ ,/5 _ «
f(-q2,-q3)
Proof of (i). Given q, define Q by the equation
5 Log(l/g)Log(l/e) = 4n2.
Letting a = \ Log(l/q) and /? = \ Log(l/Q5) in Entry 27(iii) of Chapter 16 and
noting that a/? = л2, from above, we find that
qi/24 Log1/4(l/q)/(-q) = Q5'24 Log1/4(l/G5)/(-G5)- A2.6)
Replacing q and g by g1/5 and Q5, respectively, and then by q5 and Q1'5,
respectively, we deduce that
qi/i2o Log^(i/q)/(-qV») = ^G25'24 Log^^l/G'm-G25) A2.7)
and
q5/24 Lotfmyq'M-q*) = G1'24 Log1/4(l/G)/(-G)- A2.8)
Dividing A2.6) by A2.8) and then A2.7) by A2.8), we deduce that, respectively,
П-Q)
and
Л-*?175) =5o/(-62S)
qll5f(-q5) У /(-Q) '
Now suppose that we can show, for certain positive functions ц and v, that
1 + \1, ч_ ' = ц1'5 - v1/5, A2.11)
where
juv = 1 A2.12)
and
M-v =.{!!"!+11. A2.13)

19. Modular Equations and Associated Theta-Function Identities 271
Then, from A2.10),
1 I n-qV5) - 1 I 5Qf{-Q25)
+ qll5f(-Q5)~ + Q fi-Q)'
and from A2.9),
f4-g) _1256/6(-65)
Thus, we see that A2.11)—A2.13) translate into A2.1)—A2.3), respectively, and
so it remains to prove A2.11)-A2.13).
From Entry 10(iii),
f(-q115) _ f4-q2, -Q3) qil5f4-q, ~qA)
qll5f(-q5) qll5f(-q)f(-q5) f(-q)f(-q5) '
This dictates to us the choices
_ flo(~q2, -q3) . _ qflo(-q, -q4)
. _ qf
qf(-q)f4-q) f\-q)f5{-q5)'
and so A2.11) is established. Second, formula A2.12) is an immediate con-
consequence of the second part of Entry 9(vii). Lastly, divide both sides of Entry
10(viii) by qf5{—q)f5( — q5), and we arrive at A2.13) at once to complete the
proof.
Proof of (ii). From A2.10) and Entry 11 (iv),
ft—o25} f(—a115)
nn ^ ' - n ч ' - iK 4- и*115
е7ёГ"^7?~5E + м "
25
where we have replaced ju and v in Entry 11 (iv) by ц* and v*, respectively.
Thus, ju* and v* are defined in A1.10). Therefore, by A2.4), we are compelled
to define ц and v in Entry 12(ii) by /x1/5 = i/x*1/5 and v1/5 = iv*1/5. This
establishes A2.4).
Now by Entry 11 (iv) and A2.9),
and
/6(-65)
/6(-e>
^ ГЧ-Q) * P4-Q)r
which finishes the proof of (ii).

272 19. Modular Equations and Assodated Theta-Function Identities
Proof of (iii). In Entry 27(i) of Chapter 16, let a2 = Log(l/<j) and /?2 =
Log(l/G5). Thus,
, A2.14)
where
5 Log(l/e)Log(l/e) = n2.
Replacing q and Q by q1'5 and Q5, respectively, and then by q5 and G1/5>
respectively, we derive
Log1/4AAM41/5) = v^ Log1/4A/G5)(P(G25) A2.15)
and
Log^il/q'Mq5) = Log^l/GMG), A2.16)
respectively. Thus, A2.14) and A2.16) yield
V^ A2.17)
while A2.15) and A2.16) imply
By A2.18) and Entry U(i),
v*1/5
MQ) = 9(q)_.
fl»(Q) 4>(95)
where we have replaced ц and v in Entry 11 (i) by ц* and v*, respectively. Thus,
A2.5) is established if we define
*l/5 v*l/5
and
By Entry ll(i) and A2.17),
and
Upon employing A2.17), we complete the proof.
Proof of (iv). We proceed as we did in Section 24(vii) of Chapter 18. Thus,
we transcribe the formulas involving ф into formulas involving 92. We then

19. Modular Equations and Associated Theta-Function Identities 273
use classical transformation formulas for theta-functions, which apparently
are not found in the notebooks. The transformed formulas are now given in
terms of cp. Appealing to (iii), we then complete the proof.
Put Q — ехр(шт), where Im(r) > 0. In the notation of Whittaker and Wat-
Watson [1], 52@, т/2) = 6VVF)- Tnus> transcribing Entry 12(iv), we want to
show that
where
(juvI/5 = 1 — 5 д2 '—-r— A2.20)
and
We further replace т by 2т/B5т + 1). Now, from Rademacher's book [1, p. 182],
we may readily deduce that
^) lI/2S3@, 25t),
0, r),
and
Hence, replacing т by 2т/B5т + 1) in A2.19)-A2.21), employing the three
equalities above, and using the fact that 33@, т) = q>(g), we find that A2.19)—
A2.21) are transformed into the equations
'-1-*1**'-- 22)
and
We now apply Entry 12(iii), but with ц replaced by -ц. Then A2.22)-
A2.24) follow immediately. Examining A1.4), which gives rise to the values of
ju and v in Entry 12(iii), we see that v is always positive for real q but that ц
takes on both positive and negative values for real values of q. However, the

274 19. Modular Equations and Associated Theta-Function Identities
positivity of v and the formulas for (juvI'5 and ц — v in Entry 12(iv) clearly
imply that ju > 0 for Q sufficiently small and positive.
Proof of (v). By Entries 10(iii) and 9(vii),
f{-q\ -q3) 2/5 f(-q, ~g4) _ f(-q){f(-qil5) + qll5f(-q5)}
/(-«,-«*) f(-q\ -q3) f(-q\-q3m-q,-q*)
f(-q)f(-q5)
from which (v) is apparent.
In fact, Entry 12(v) is a special instance of a more general theorem which
has been established independently by Ramanathan [8] and Evans [1]. Since
Entry 17(v) in this chapter and Entries 6(iii), 8(i), and 12(i) in Chapter 20 are
also particular case of this theorem, we state and prove it here.
Theorem 12.1. Let nbea natural number with n=±l (mod 6). If n = 6g + 1,
where g > 1, then
(n-n/2 f( — a2k -ап~Щ
+ Y r_n*+go(*-g>C*-3g-l)/BB)-H 4,4 )
к K ' q f{-q\ -q»-k)
while if n = 6g — 1, where g > 1, then
j( — q ) /
k —nn~2k\
4 V
+ к ( l> q f(-q\ -q"-") ¦
Before proving Theorem I2.l, we note that setting и = 5 in A2.26) immedi-
immediately yields Entry 12(v).
Proof. Let Uk = J*n-»i2tf»-w and yk = a*(*-D/^*(*+i)/2 From C1 2) in
Chapter 16, if и is odd,
<n-l)/2 /jT Л/ \ (n-l)/2
fc=l \ Uk Uk J t=i
Putting U1 = a = -q1'" and Vt = b = -q2ln, we find that
jy ч ) — jy н > —q )
4- У I _ 11*я*C*-1)/Bл)/-/ (Зп+6*-1)/2 „(Зл-бИг+Хугч
+ ^ (_1)*д«3*+1)/B»)у^_-(Зп + 6*+1)/2^ _ Cn-6lfc-l)/2\ A2 27)
lc=l

19. Modular Equations and Associated Theta-Function Identities 275
We now assume that n = ± 1 (mod 6). Suppose first that n = 6g + \,g > 1.
The gih term in the second sum on the right side of A2.27) equals
We partition the remaining terms in the two sums on the right side of A2.27)
into six subintervals. Note that the first term on the right side of A2.27) is
equal to the term when к = 0 in the second sum on the right side of A2.27).
Accordingly, we find that
A-q11") - (-
(в 2д
V + V +
kl k=g+l k=2g+l
у + у + у J/_ ^*д*(Зк+1)/<2и)Г/-(Зи+6к+1)/2^ _ (Зп-6к-1)/2л
k=0 k=sr+l *=2g+l/
A2.28)
We now combine the first and sixth sums above. Replacing к by 2g + к in
the latter sum, we find that the sum of the fcth terms equals
LJ ' q ' 1<к<а
f(_qk+gqn-k-gj ' 1SKS0,
by an application of the quintuple product identity C8.8) in Chapter 16 with
x = qk+e and к = q"-^-^ Replacing к by к — g, we conclude that the sum of
the first and sixth sums on the right side of A2.28) equals
i
y—q» —q )
Next, we combine the fourth and fifth sums together on the right side of
A2.28). Replace к by — к and к by Ig + k, respectively, in these two sums.
Then by identically the same argument as above, these sums equal
W) t (-
Third, we combine the second and third sums on the right side of A2.28).
In the third sum, we replace к by 4g + 1 — k. Upon simplification, we find
that the sum of the /cth terms equals
/ iyUkC*-l)/Bii)f Л_дC11+6*-1)/2 __Cn-6k+l)/2\
_ g4«+l-2*fl_g21,+4-3*f _д-Э,-1 + Э*)}9 0 + 1 < к < 20.
Applying the quintuple product identity, C8.2) of Chapter 16, with В =

276 19. Modular Equations and Associated Theta-Function Identities
^De+i-2*)/2 an(j g replaced by — q2, we deduce that the expression above
equals
Г(_ап-2к-2д _a2k+lg\
(—\)q "' f( — q") -T? k+g n-k-g\—' в ~^~ ^ — ^ — ^9'
Replacing к by к — #, we conclude that the sum of the second and third sums
on the right side of A2.28) equals
b-D/^/w 2k' -q2k) A231)
Finally, substituting A2.29)-A2.31) in A2.28), we readily deduce A2.25).
Now suppose that и = 6g — 1. The 3 th term in the first sum on the right
side of A2.27) equals
The remainder of the proof of A2.26) is parallel to the proof above with the
roles of the two sums on the right side of A2.27) reversed. This concludes our
proof.
Evans' [1] version of Theorem 12.1 is more general because it holds for all
odd n. His proof is different from that of Ramanathan and will be given in
Chapter 20.
The proof of (vi) that we give here is very difficult. Ramanujan must have
had an easier proof. Before proving (vi), however, we establish a "rational"
version of (vi), namely,
) t'(-q')
which will be used later in this chapter. We first prove a needed lemma.
Lemma. f(-q)f(q) =f(-q2M~q2).
Proof. By Entry 22 of Chapter 16,
fi-q)f(q) = (9; q)J-q; e)« = to; q)*(-q; я2)М2; q2)*,
Proof of A2.32). By Entries 10(v) and 9(vii) in this chapter, Entry 25(iii) in
Chapter 16, and the lemma above,
1-q'T", W-q5)

19. Modular Equations and Associated Theta-Function Identities 277
_ (р(—д5Ж—д5)ф(д5Жд5) гФ2(дъ)Ф2(~д5)
х(-д)х(д)Ф'
<рЧ-д1от-д10) 2WV(-g10)
(-q2№4q2)<f>4-q2) Ч ФЧд2)<Р2(-д2)
X(~q2)
(p4~ql°)
(Р2(-д2)Ф2(д2)
_<{>2(-д10)
which completes the proof of A2.32).
Proof of (vi). Unless otherwise stated, all references in this proof are to results
in Chapter 16.
Using the third equality of Corollary (ii) in Section 31, Entry 25 (iv), Entry
30(i), and Entry 25(iii), we deduce that
™ = 215(Ф(дЖд) Ф(дЖ
' q V Ф2(д) ФЧ-ч)
_ #V) + д215Ф(д5)Яд2, дъ) + д315Ф(д5Шд, д4)
ФЧд)
дф2(-д5) - д215Ф(-д5)Яд2, -д3) + д315Ф(-д5)Я-д, д*)
ФЧ-д)
_ дФ(д10Ыд5) + <?2/№, g*)f(g3, д1) + <?3/5/(<?, g9)f(g\ g6)
Ф(д2Ыд)
дф(д10Ы-д5) - g2l5f(g2, д»Ж-д\ -д1) + g3l5f(-g, -д9Жд\ д6)
Ф{д2Ы-д)
1
(дф(д10){<р(д5М-д) - я>(-
Ф(д2)<рЧ-д2У
+ g2l5f(g\g»){f(g3,д'Ж-д, -я) + Я-ч\ -д'Жд,Ф}
+ g3l5f(g4,дв){Яд,д9Ж-д, -я)-/(-«, -g9)f(g>Ф})- A2.33)
То simplify this expression, we first apply C6.2) with A = l,B = — 1,д = 3,

278 19. Modular Equations and Associated Theta-Function Identities
and v = 2. Using also Entries 18 (Hi), (iv), we find that
= q5f(-q40, -q20)f(-q16, -q~*) - <?27(-<?60, -
+ q49f(-q*°, -q-20)f(-qb2, -<T2°)
= -2qf(-q\ -q«m-q20, -q*°).
Second, we apply C6.1) with A = q1, В = — 1, ц = 3, and v = 2. Again using
Entries 18 (Hi), (iv), we arrive at
f(q\ «')/(-«, -q)+ f(-q3, -q7)f(q, q)
= 2{f(-q3\ -q28)f(-q«, -q4) - q8f(-q52, -q*)f(-q16, -<Г4)
+ q2«f(-q12,-q-12m-q24,-q-12)}
= 2f(-q*, -ftW-q", -q") + q*f(-q52, -q*)}.
Third, utilize C6.2) again with A = qA, В = — 1, ц = 3, and v = 2 to realize
that
f(q, q9)f(-q, ~q) - f(-q, ~q9)f(q, q)
= 2q4{«z5/(-<?44, -q")fi-q«>, -q-*)
- q2Sf(-q6\ -q-4)f(-q2*, -q~16)
+ q57f(-q8\-q~2*)f(-q36,-q-24)}
= 2qf{-q\ -q8){-f(-q4\ -q16) + <ff(-q*, -q56)},
by Entries 18(iii), (iv) again. Using the last three calculations in A2.33) above,
we deduce that
+ 2q2'5f(q2, q8){f(-q28, -qi2) + q*f(-q8, -q52)}
- 2q8>5f(q*, q6){f(-q16, -q") - q4f{-q\ -q56)}). A2.34)
We now simplify this expression by making several substitutions. First, by
Entry 24(iii),
f(-q4) = ^(q2)x(-q2)- A2.35)
Second, applying Entry 25 (Hi) and Entry 24(iii) twice, we see that

19. Modular Equations and Associated Theta-Function Identities 279
Third, in C8.6), replace q by — q10 and let В = l/q2. We accordingly deduce
that
f(-q\-q36)f(q8,q42)
= /(-<Л -ОШ-416, ~q44) - q4f(-q\ ~q56)}- A2.37)
Similarly, replacing q by — q10 and putting В = q6, we find that
By Entry 18(iv),
/(-<T12,-<Z52)=-<T12/(-<?28,-<Z12)
and
Л-9-8,-968)=-^8/(-952,-Л
Using these two equalities in the foregoing equality, we find that
f(-q2\-q12)f(q\q16)
= f(-q40, -q«°){f(-q28, ~q32) + 9*/(-«8, -q52)}. A2.38)
Substituting A2.35)—A2.38) into A2.34), we arrive at
(P2(-q2)\
2q*l5f(q\q6m-q\-q36)f(q*,<
f(-q*°)
By the Jacobi triple product identity, Entry 22(i), and B2.4),
f(-q6, -q^fig6,'
<p(-q20)
= fo12;<
Similarly,
Thus, A2.39) above may be written in the form
2q2l5f(q2,q«)f(-q6,-q1*)f(q6,
f(-q*°M-q20)
A2.39)

280 19. Modular Equations and Associated Theta-Function Identities
2q8l5f(q\ q6)f(-q2, -q18)f{q2, q18)f(q8, q12)
f(-q40M-q20)
f4-q20)
where we have employed Entry 24(iii) and Entry 30(i) twice, first with a = q6
and b = q4 and second with a = q2 and Ь = q8.
Invoking next Entry 24(iii) of Chapter 16 and Entries 9(vii), 9(iii), and 10(ii)
of this chapter, we deduce that
-q2, -q")}
q>{-qlo){2qWf(-q*, -q") - 2q8'5f(-q2, -
_ 1 <p(-q10M-q215)
<p2(-q2) '
which, at last, is the proposed formula.
We come now to a panoply of fifth-order modular equations, some of which
we have already utilized in Section 8.
Entry 13. The following are modular equations and formulas for multipliers for
degree 5:
(i) (a/?I'2 + {A - a)(l - 0)}1'2 + 2{16a/?(l - a)(l - p)}1'6 = 1;
«

19. Modular Equations and Associated Theta-Function Identities 281
/1 _ OM\ 1/24 5 /а5ц _ y>\ 1/24
(v)
{A -
m
{A - a)(l -
(viii) i/ a and Ь are arbitrary complex numbers, then
i _
a(l - а) У V «A - a) )
1+41/3
(x) {a(l - p)}1'4 + {fi(l - a)}1'4 = 41'3{a/?(l - a)(l -
= m{a(l - a)}1'* +
= {a(l-a)}1'4 + -
ш
2
a5V'8 /A - aMY/8 _ 5 (\ + (apI'2 + {A -
) +) v
(xii) "*~w ' vi-«; v«d-«); '

282 19. Modular Equations and Associated Theta-Function Identities
5 /ау/1-аУ /аA-а)У.
m
l-pj W-PV '
, .... 5 Ч(ф112 - {A - a)(l - p)}1'2)
m /1 + (a/?I'2 + {A - a)(l -
m + - = 2B + (a/?I'2 + {A - a)(l - p)}1'2);
m
(xiv) ifP = {16a/J(l - a)(l - /?)}1/2 and Q =
(xv) i/P = (a/?I/4 and Q = ()?/aI/8,
The formulas (i)-(iii) do not appear to be translations of any simple combi-
combinations of formulas given in Sections 8-12. It is likely that Ramanujan's
method of attack proceeded along the following lines. He first discovered
(iv)-(vi), and then when he had deduced (i)-(iii) from (iv)-(vi), he decided to
give (i)-(iii) priority in placement because of their elegance and simplicity. This
conjecture is supported by an apparently similar rearrangement in Section 15.
Proof of (iv). Transcribing Entry 9(iii) via Entries 12(ii), (iv), and (v) in
Chapter 17, we obtain the formula
4.91/6 .1/2 1/2
P) P 4^(
Dividing both sides by z5 and simplifying, we deduce the first formula of (iv).
The second formula is the reciprocal of the first, in the language of Entry
24(v) of Chapter 18.
Proof of (v). Replacing q by — q in Entry 9(iii), we may deduce the formula
<P2(-q)-<P2(-q5)_ x(-g)f(g5)
cp2(q) - cp2(q5) x(q)f(-q5)'
Utilizing Entries 10(i), (ii) and 12(i), (ii), (v), (vi) of Chapter 17, we translate the
formula above into
zt(l - aI/2 - z5(l - p)
1'2
z~~^5 21'6{a(l - a)
which reduces to

19. Modular Equations and Associated Theta-Function Identities 283
m(l - aI'2 - A - pI12 (\ -
m - 1 V1 - P.
On solving for m, we deduce the first part of (v).
From Entries 9(iii), (vii) and 10(v),
*V)-«VV°) <f>(-qlo)f(-q10)
<?2{q) ~ Ф2(«5)
By Entries 11 (iii), 10(iii), and 12(ii), (iii), (iv), (v), (vii) in Chapter 17, the formula
above transforms into the equality
m - 1
after simplification. Solving for m, we obtain the second formula of (v).
Proof of (vi). The formulas of (vi) are simply the reciprocals of the respective
formulas of (v).
Before proving the remainder of the formulas in Entry 13, we derive
parametric representations for m and various radical expressions in a and /?.
Put
Then, from (v) and (vi), respectively,
1 - i;3 5 1 + и3
m = - j- and -= ,. A3.1)
1 — и v m 1 4- uv
To eliminate v, we rewrite the first equation as
m — 1 = v(mu2 — v2),
square this, and then substitute the value
, m(l + u3) - 5
v < '
5u
given by the second equation of A3.1). Accordingly, we obtain the cubic
equation in u3:
125u3(m - IJ = (тиъ + m - 5)Dmu3 + 5 - mJ.
It is easily checked that u3 = — 1 is one root of this equation. Upon dividing
out the extraneous factor u3 + 1 and performing some tedious algebra, we
arrive at the quadratic equation in м3
16m3u6 - (8m3 + 40m2)u3 + m3 - 15m2 + 75m - 125 = 0.
The roots of this polynomial are

284 19. Modular Equations and Associated Theta-Function Identities
, m2 + 5m + 5p
4m2 '
where
p = {m3- 2m2 + 5mI12. A3.3)
If q tends to 0, then m tends to 1 and v approaches 0. Thus, from A3.1), u3
tends to 4. Thus, we are forced to take the plus sign above in the determination
of u3; that is,
з m2 + 5m + 5p
ir =
Am2
We next want to determine i;3. This is most easily accomplished by first
realizing that if u, v, and m are replaced by — v, — u, and 5/m, respectively, in
the equations A3.1), they are invariant. Thus, using A3.2), we find, after a brief
calculation, that
3 — m — 1 + p
4
Since v tends to 0 as m tends to 1, we must take the plus sign above, and so
We now summarize, in terms of a and /?, the formulas that we have derived,
namely,
Their reciprocals are, respectively,
We are now in a position to prove (i)-(iii).
Proofs of (ii), (Hi). By A3.4) and A3.5),
= _5_
PJ \1-P ) 2m2 2 2m
Using the formula for 5/m from (iv), we complete the proof.
Formula (Hi) is the reciprocal of (ii). Alternatively, A3.4) and A3.5) can be
employed once again.
It might be noted here, that by Entries 10(i), (iii), 1 l(i), and 12(i) in Chapter
17, Entry 13 (ii) is equivalent to the formula

19. Modular Equations and Associated Theta-Function Identities 285
<P5(g) <P5(-q2) 7Пя[ в , ФЧд)
<p(q5) <p{-q10) f(qS) *(«5)'
for which no direct proof has been constructed.
Proof of (i). Multiplying the equalities of A3.4) and those of A3.5), we deduce
that
_ \Qp2 - 2m3 - 12m2 - 10m _ m2 - 4m + 5
16m1 2m
= 1 - \{»i - 1)A - l) = 1 - 25'3{«0A - a)(l - y?)}1'6, A3.6)
by part (iv). This completes the proof of (i).
It might be observed that, by Entries 10(i), (ii), 11 (iii), and 12(iii) in Chapter
17, (i) is a translation of
<p2(q)(P2(q5) - v4-
of which a direct proof has not been given.
We now derive some parametric equations for further radicals that will be
useful in the proofs of the remainder of the formulas of Entry 13.
From A3.6),
1 + (a/?I'2 + {A - «)A - /?)}1/2Y'2 _ № ~2m + 5Y'2 _ P A37)
4m J Im
by A3.3). Furthermore, from A3.6),
_„(?-Л _
)
4V \m ) m
Now substitute for m — 1 from Entry 13 (iv) and deduce that
n — fiMY/12
m_ ,a(l-a); \«Q-*)J
Next, by combining A3.4) and A3.5) with Entries 13(v), (vi), we may easily
derive the formulas
-, {(l-aK(l-/?)}1/8 =
4m 4m

286 19. Modular Equations and Associated Theta-Function Identities
W = ' + I^-3W, and {(!-«)(!-mi/8 = P~
A3.11)
Hence, by division,
'V
and
m(m — 1) \1 — P/ m{m — 1)'
and by inversion,
S-m \\-л) S-m
Multiplication of A3.10) and A3.11) yields
1/2 _ 4m3 - 16m2 + 20m + p(m2 - 5)
and
From A3.4) and A3.5),
,1/8 J
-4m2+3Om -p{m2+lOm
1 {6p2m + 8m3 - p(p2 + 12m2)}
and (\=?Г.*±Л. |ВД
(a/?)V2 = ""• "Г ' "'" "-I A3.14)
16m
16m3
_ Bm - pf
16m3 '
Therefore,
Proof of (vii). If we properly combine A3.10), A3.11), A3.7), and A3.16), we
deduce all of the equalities of (vii).
Proof of (viii). Multiply the first equality of (iv) by a and multiply the
numerator and denominator of the right side of A3.9) by b. The first formula
for m in (viii) is now easily verified.
To prove the second, first observe that
5 /6m - m2 - 5 3\2 5
4m / V 4m / 4 V 4m 2/4
/6m - m2 - 5\ /6m-m2-5\2
4m / V 4m

19. Modular Equations and Associated Theta-Function Identities 287
Therefore, by A3.8) and the equality above,
= |E - m2) = 1 - > - 1) - i(m - IJ
= I _ 2
by Entry 13 (iv), and so we obtain the second equality of (viii).
Proof of (ix). Again, from Entry 13(iv),
m3 - 2m2 + 5w _ p2
Am Am
The desired result now follows from A3.7).
The second formula follows from the first by reciprocation.
Proof of (x). Rewrite A3.16) in the form
?5A - aMV/24 _ 2m-p
a(l - B) ) 2M3m '
Using the process of reciprocation, we derive the companion formula
:4 2w + p
0A - a) / 2A/3m '
Upon adding the last two equalities, we readily derive the first part of (x).
From (iv) and A3.8),
m {«A-«)}"¦ + { 0A-/»)}V*
1 Г /5 \5/4 /5 V/4
-Mi-1) tir oW^i^
which is the second equality of (x).
The last equality follows from the second equality, reciprocation, and the
invariance of a/?(l — a)(l — 0).
Proof of (xi). The first formula follows immediately from A3.4), A3.5), and
A3.7), while the second is the reciprocal of the first.
Proof of (xii). The first equality follows from A2.32), the rational version of
Entry 12(vi), by the use of Entries 10(iii) and ll(i), (ii) in Chapter 17. An
alternative proof can easily be constructed with the aid of A3.13). The second
part is simply the reciprocal of the first.

288 19. Modular Equations and Associated Theta-Function Identities
Proof of (xiii). The former formula is apparent from A3.14), A3.15), and A3.7).
The latter formula is obtained from squaring A3.7), substituting for p2 via
A3.3), and then rearranging the terms.
Proof of (xiv). Let P and Q be as defined in (xiv). Then from A3.8) and A3.13),
respectively,
and
Hence,
m V * 5__j"
m
IP 5
2PQ = m - 1 and — = 1,
Q m
from whence it follows that
The desired result now follows by rearranging the terms.
Proof of (xv). With P and Q as defined in (xv), we write parts of (v) and (vi)
in the forms
1 - PQ3 5 1+ P/Q3
m = and Ш = ТТЖ'
respectively. It is now obvious that
whence (xv) follows upon rearrangement.
Entry 14.
(i) Let /? be of the fifth degree in cc. If a = sin2(/z + v) and ft = sin2(/z — v),
then
sinB/z) = sin v(l + cos2 v).
(ii) // p is defined by
m = 1 + 2p, A4.1)
then for 0 < p < |E v^ - П),
and

19. Modular Equations and Associated Theta-Function Identities 289
(iii) // p is defined by A4.1) and 0 < p < 2, then
м- A + 2рJ \l + 2p)
and
(iv) // a and ft are given by the equalities immediately above, then
(v)
Observe that (i) is an analogue of Entry 5(xiv). Formulas (ii) and (v) are
more accurate versions than those stated by Ramanujan.
Proof of (i). Substituting in Entry 13(i), we find that
sin(/z + v)sin(/z — v) + cos(/z + v)cos(/z — v)
+ 2{4 sin(/z +
or
cosBv) + 2{sin 2(ц + v)sin
or
sin2B/z) - sin2Bv) = ?A - cosBv)K,
or
sin2B/z) = ?A - cosBv))C + cosBv)J
= sin2 v(l + cos2 vJ,
and so (i) is established.
Proof of (ii). By Entry 13(iv),
m-l). A4.2)
Using A4.1), we find that the foregoing equality takes the shape
YT^p) =:/(p)> (R3)
Using elementary calculus, we can easily show that / increases as p goes from
0 to ^E y/5 — 11), and then decreases back to 0 as p varies from ^E y/5 — 11)

290 19. Modular Equations and Associated Theta-Function Identities
to 2. Hence, for each value of a between 0 and 1, there are two values of p such
that A4.3)is satisfied. Now,as a tends to 0,m approaches 1. By A4.Цр then
tends to 0. Thus, the appropriate value of p is that with 0 < p < ^E v/5 — 11).
By Entry 13(iv) and A4.1), for p as above,
)(??) A4.4)
and this completes the proof.
Proof of (iii). For brevity, set
[2-p
Then solving for a from part (ii), we find that
a = Kl ± y/l - a).
If p = 0, then m = 1 and a = 0. Thus, we must take the minus sign above.
Hence,
{l + 2pJ
Denoting the far right side by g(p) and employing elementary calculus, we
find that g decreases monotonically from 1 to — 1 as p goes from 0 to 2. Thus,
for each value of a, 0 < a < 1, there exists just one value of p, 0 < p < 2, such
that 1 — 2a = g(p). Clearly, this representation for a is valid for 0 < p < 2.
The proposed formula for 1 — 2K follows in the same fashion. Thus,
Proof of (iv). By A4.1) and part (iii), we observe that (iv) is simply a version
of the equality zt = mz5.
Proof of (v). By (ii), the proposed formula reads
2F,(l i; 1; 4a(l - a)) = 2F1(l |; 1; 40A - P)).
The proof of this is exactly the same as that for Entry 6(ii).
In Section 15, we assume that /? is of the fifth degree in a and у is of the
fifth degree in /?, so that у is of the 25th degree in a. Let m denote the multiplier
connecting a and /?, and let m' be the multiplier associated with /? and y. Put
(see A3.3))
p = (m3 - 2m2 + 5mI12 and p' = (w'3 - 2m'2 + 5m')m. A5.1)
Ramanujan's formulation of Entry 15 is in terms of hypergeometric series; for
simplicity, we employ the notations m and m'.

19. Modular Equations and Associated Theta-Function Identities 291
Entry 15. // a, /?, and у are as defined above, then
A1*8 , (lz«\118 (*(\-*)\lia
A - a)(l - у)
т
ay/ \A — <х)A — у)/ \<хуA — а)A — у)
-2(—т. гг. :) ^1 + 1 — 1 +
.ay(l - a)(l - y)/ I V«y/ Vd - «)A - 7).
m
and
- У)
4
/?2A-/JJ
It seems likely that, in arranging these formulas, Ramanujan gave (i) and
(ii) priority over (iii) because they involve a and у only and not the intermediate
modulus y/p. As we shall see, (i) and (ii) arise from (iii).
Proof of (iii). Replacing q by q5 in Entry 12(vi), we find that
<f>(-g2M-g50) , 2{ФШ(д25)
+q {
срЧ-q10) +q { Viq5) фЧ-q5) )~
Employing Entries 10(iii) and 11 (i), (ii) in Chapter 17, we translate the formula
above into
zP{a(lq)}^ig{y(ly)}_
{)S(lW1/4
Multiplying both sides by z5/{z1z25I'2 = (m'/m)V2, we finish the proof.
Proof of (i). We first derive, from (iii), two equalities connecting m, m', p, and
p' by using the trivial equalities

292 19. Modular Equations and Associated Theta-Function Identities
and a corresponding set for {A - a)(l - y)/(l - pJ}1'8. First, by A3.4) and
A3.10),
""' m'(p-m-iy
Second, by A3.5) and A3.10),
{A - ««(I - y)\» = P'~
Upon multiplication, we obtain the equality
a)(l - y)\118 (m'-l)E-mf
m'2(m-lK '
where, of course, A5.1) was utilized. Thus, Entry 15(iii) now assumes the form
p' + 3m'-5 pi - 3w' + 5 (w'- 1)E - w'J fm'^1'2
m'(p — m — 1) m'(p + m+l) m'2{m - if \m
Combining terms together, we obtain our first new form of Entry 15 (Hi),
namely,
2mpp' + 2m'(m + l)Cm' - 5) + (m' - 1)E - m'J _ fm'V12
— I — I . A5.2)
w'2(w-lK \m
Similarly, by A3.11) and A3.4),
mEp' + m'2 + 5m')'
By A3.11) and A3.5),
_pM) m{5p'- m'2 - 5m'\
Upon multiplication of the two equalities above, we deduce that
(<xy(\ - a)(l - y)\1/8 m'3{m - 1JE - m)
m{S-m'f
Hence, Entry 15(iii) may be written in the form
m'\p + m2- 3m) m'\p - m2 + 3m) m'3(m - 1JE - m) (m'\12
m{5p' + m'2 + 5m') m{5p' - m'2 -5m') wE - m'K \m
which simplifies to
lOm'pp' - 2mm'2{m - 3)(w' + 5) + m'3(m - 1JE - m)
wE - w'K
the second desired new form of Entry 15(iii).
_ (mV

19. Modular Equations and Associated Theta-Function Identities 293
We remark that A5.2) and A5.3) make it evident that pp' and (m'/mI12 are
expressible as rational functions of m and m'; this fact will be useful as a guide
in proving (iv). In fact, in the proof of (i), only A5.3) is used, but the foregoing
work should provide some rationale for the seemingly unmotivated proof of
(iv).
By A3.4) and A3.5),
ЛЛ VB Л _ ?y/8 _ ,y y/8 /0541,8 ( X_y y/8 fll-
w +u-«; "w w +v<i-«v w
_ w'2(p - m ~ 1) m'2(p + m + 1)
5p' + m'2 + 5m' 5p' ~ m'2 - 5m'
IQm'pp' + 2m'2(m + l)(w/ + 5)
Hence, by A5.3),
-3 {2w'2(w + l)(m' + 5) + 2mm'2(m - 3)(m' + 5)
E -
- m'3(m - 1JE - m)}
(m - IK + 2 (« - IJ
m' V
^Y A5 4)
by Entry 13 (iv). Hence, formula (i) is established.
Proof of (ii). Formula (ii) is simply the reciprocal of (i).
It might be remarked here that, with the help of Entries 10(iii), ll(i), (ii),
and 12(iii) in Chapter 17, it can easily be shown that Entry 15(ii) is a translation
of the formula
) Ф(-Я25)~ Н Hf(-q5°) Ч <p(~q50)'
for which no direct proof has been constructed.

294 19. Modular Equations and Associated Theta-Function Identities
Formula (iv) appears to be more recondite than the preceding three for-
formulas, and it is not obvious how it can be deduced from them in any simple
manner. Undoubtedly, Ramanujan had some ingenious method of obtaining
it, for it is inconceivable that it could have been discovered by the process of
verification given below.
Proof of (iv). We first express m and m' as simple functions of a parameter t.
By Entry 13 (iv),
(m - lM^ - l) = 28/?(l - P) = (m' - 1)(^7 - 1J. A5.5)
Now put
t = -^—-, A5.6)
so t approaches 0 as m tends to 1. It is also evident from A5.5) that
s m'- 1
1 =~5 ¦ A5.7)
m
Moreover, from Entry 13 (iv) again,
I^V'24 A5-8)
When we substitute in A5.6) the value of m' obtained from A5.7), we obtain
the quadratic equation in w,
m2(t5 - 1) + m{(t - l)(t5 - 1) - 5(t5 - t)} - 5t5(t - 1) = 0.
Solving this quadratic equation, we find that
2m(t5 - 1) = -(t - l)(t5 - 1) + 5{t5 - t)
±({(t - l)(t5 - 1) - 5(t5 - t)}2 + 20?5(r - l)(f5 - 1)I/2.
A5.9)
In order to simplify the radical, we write
{(t - l)(t5 - 1) - 5(t5 - t)}2 + 20t5(t - l)(t5 - 1)
= {(t - l)(t5 - 1) + 5(t5 + t)}2 - lOOf6
= {(t- l)(t5 - 1) + 5(t5 + t)- 10t3}{(t- l)(f5 - 1) + 5(f5 + t)+ 10t3}
= {(* - l)(t5 - 1) + 5t(t2 - IJ} {(t - l)(t5 - 1) + 5t(t2 + IJ}
= (t - lJ(t2 + 3t + \J{t6 + 4t5 + 10t3 + At + 1}.

19. Modular Equations and Associated Theta-Function Identities
295
Substituting this in A5.9) and letting t tend to 0 in order to determine the
proper sign on the radical, we conclude that
m =
A + BR
A5.10)
where
A5.11)
В = (t - l)(t2 + 3t+ 1),
R = (t6 + At5 + 10t3 + At + II'2,
B2R2 -A2 = 2Ots(t - l)(t5 - 1).
We now determine mm' in terms oft. From E.6), m' = 5t/(w + t — 1). Also,
from A5.10) and A5.11),
B2R2 - A2 lOts(t - 1)
m =
2(t5 -
Hence,
5f
- A) BR-A'
5t 50t6
mm =
t - 1 BR-A BR-(A- 10t5)
50t6(BR + A - 10t5)
" B2R2 -A2 + 20At5 - 100t6
5t(BR + A- 10t5)
~ 2(t - l)(t5 - 1) + 2A - 10t5
= iA0t5 - Л - BR),
by A5.11). Thus, by A5.11) again,
4mm' = 2t6 + 8t5 + it + 2 - 2BR
and
2у/тп' = R-(t- l)(t2 + 3t + 1).
From this, A5.10), and A5.11), it follows that
m(t5 - 1) = 5(t5 - t3) + B(mm')il2.
A5.12)
Combining A5.12) and A5.7), we find that (ww'I/2 satisfies the quadratic
equation
mm' + B(mm'I12 - 5t3 = 0. A5.13)
After these preliminary calculations, we are now in a position to establish
(iv). Multiplying (Hi) by 2@2A - 0J/{ay(l - a)(l - y)})m and using the
proposed formula (iv), we see that we are required to prove that

296 19. Modular Equations and Associated Theta-Function Identities
1/4 / П _ й\2 V/4
- «)(i - у).
which can be rewritten in the form
/ \ 1/4 //j _ aW|
L:=l —(-4) —I— „., n 1 +5
m
m'V p2a-PJ
= 2A-1-1 I. A5.14)
m
We now prove A5.14) by using several previously derived formulas to
reduce the left side. Employing the following formulas in turn: A3.12) and
A3.13), A3.10) and A3.11), Entry 13(iv), A5.4) and A5.6), A5.12), A5.13),
A5.11), and A5.11) and A5.12), we find that
Bm + p)Bm' - p') Bm - p)Bm' + p') 5(w' - 1)E - m)
m(m 1)E m') m(m 1)E m')
m(m - 1)E - m') m(m - 1)E - m') (m - 1)E - m')
2pp' + 2m(m - 3)E - 3m')
m(m — 1)E — m')
(p + m2 - 3w)(p' - 3m' + 5) + (p-m2 + 3m)(p' + 3w' - 5)
m(m — 1)E — m')
16m'
(m - 1)E - m')
+ {A - a)(l - PK}
16т'{ф-<х)Р3A-
(^
h
mE - m')
m(t
t
t3 + 2t2 + (m'))
_ {5(t3 - 1) - B(ww'I/2}(t3 + 2t2 + (ww'I/2)
-E(t3 - l)(f3 + 2t2) + {5(t3 - 1) - B(t3 + 2t2)}(mm'I12
- 5t3B + B2(mm'))
t
{10(t4 - t2) - Bt4 - 2t2 - At + 4)(ww'I/2}
m(t5 -
{(У},
and this completes the proof of A5.14) and hence of (iv).

19. Modular Equations and Associated Theta-Function Identities 297
Proof of (v). By Entry 13 (iv), the left side of (v) is equal to
m Am' + (m— 1)E — m') m
~m' Am + (m' - 1)E - m) = w7'
which completes the proof.
Entry 16(i). Let0 be of the fifth degree in a. If
*B dq>
for some pair A, B, with 0 < А, В <, я/2, then
tan{|H - B)} = 1+xA+p + {A + 2p)A + p2)}1/2)tan2B>
where w = 2p + 1, as in A4.1).
Proof. Entry 16(i) gives the general transformation of the fifth order, which
is due to Jacobi [1, pp. 26-28], [2, pp. 77-79]. As in the corresponding
analysis of Entry 6(iii), it seems sufficient to derive Ramanujan's formulation
from that given by Jacobi. Converting Jacobi's result into our notation, we
see that
. a sin В + a' sin3 В + a" sin5 В
where
1 + b' sin2 В + b" sin4B ± (a sin В + a' sin3 В + a" sin5 B)
= A + sin B)(l ± p sin В + q sin2 BJ, A6.2)
where q is the positive root of the equation
2q{q + p + 1) = p3;
that is,
2q + p + 1 = {A + 2p)(l + p2)}1/2. A6.3)
Moreover, a = m, a' = 2q + p2 + 2pq, a" = q2, b' = 2p + 2q + p2, and b" -
2pq + q2.
We first prove that
By the "half-angle" formula, tan(i7t - ^A) = cos A/(l + sin A). Thus, A6.4)
can be written in the form

298 19. Modular Equations and Associated Theta-Function Identities
cos A cos В 1 — p sin В + q sin2 В
1 + sin A ~~ 1 + sin В1 + p sin В + q sin2 B'
Squaring both sides and simplifying, we determine that it now suffices to show
that
1 — sin A 1 - sin В A — p sin В + q sin2 BJ
1 + sin A 1 + sin В A + p sin В + q sin2 BJ'
A6.5)
Now substitute A6.1) for sin A on the left side of A6.5). Simplifying while
employing A6.2), we immediately ascertain the truth of A6.5), and hence of
A6.4).
Hence, by A6.4),
i(A - B)}
1 — p sin В + q sin2 В
1 + p sin В + q sin2 В
sin2 В
- ?B)— . -— .-.
l+psmB + q sin2 В
2p sin
1 + p sin В + q sin2 В + tan2(^n - iB)(l - p sin В + <j sin2 B)
2p sin В sm
"" со?2(?я - ^B)(l 4- p sin В + q sin2 B) + sin2(i7t - |B)A -psinB + q sin2 B)
p sin В cos В
1 + q sin2 В + p sin2 В
ptanB
~ 1 + (q + p + l)tan2 B'
and, because of A6.3), this is Ramanujan's formula.
Ramanujan begins Section 16(ii) with the following geometrical construc-
construction. Let О be the center and AP a diameter of a circle <? of radius a. Let G
denote the point of medial section on AP; that is, AG/GP = ^A + ,/5). Let
T be any point between G and P. Draw a perpendicular line segment RRy to
AP at T, where R, Rt 6 #. Form PR, ЛК, and ЛК^ Letting M denote the
midpoint of TP, draw MN parallel to PR, with N on AR. With В е <ё on the
same side of ЛР as R, draw BD perpendicular to AP such that BD = MN. Let
Q e # be such that the arcs BQ and BP are equal. Form AB, QR, and 6RX.
It appears, at first glance, that the introduction of the point of medial
section G is irrelevant. However, it can be shown that the condition that T be
to the right of G is necessary to ensure that MN < a, which, in turn, is
necessary for the construction of BD. In fact, if T = G, MN = a.

19. Modular Equations and Associated Theta-Function Identities
-BR
299
Entry 16(ii). Let tl and t2 denote the times required for a pendulum of length (
to oscillate through the angles 4BARt and 4BAR, respectively. Then tl = mt2,
where m is the multiplier of degree 5. Furthermore,
1 + m =
, 5
2QR
and
5 OD
m = 8—.
m AR
A6.6)
A6.7)
A6.8)
Proof. Let v = L.PAR. Then AR = 2a cos v. Since also cos v = AT/AR, we
see that AT = 2a cos2 v. Now,
AT = АО + ОМ - TM = a + ОМ - (a - ОМ) = 2OM.
Thus,
and
AM = a + ОМ = a
= a{\ + cos2 v)
MN = AM sin v = a{\ + cos2 v)sin v. A6.9)
Now define ц to be the angle between 0 and л/4 determined by the equation
sinB/z) = A + cos2 v)sin v. A6.10)
Then, by A6.9), A6.10), and the fact BD = MN,
Df\ Dp ПП
sinB/z) = — = 2—— = 2 sin LPAB cos LPAB = sin 2LPAB.
a 2a BP

300 19. Modular Equations and Associated Theta-Function Identities
Hence, fi = LPAB. By the converse of Entry 14(i), we conclude that
a = sin2(/z + v) = sin2 LBARt
and
P = sin2(/z - v) = sin2 LBAR.
Furthermore, /? is of the fifth degree in a.
The equality tt = mt2 now follows just as in the proof of Entry 7(iii), but
now, of course, Entry 16(i) is utilized in place of Entry 6(i).
By Entry 13(xiii),
m = 7 : \Tn
m /1 + sm(/z + v)sin(/z — v) + cos(^ + v)cos(/j — v)\ '
2
4 cosB/z) 4 cosB/z)
1 + cosBv)V/2 cos v
A6.11)
2
OD/a o OD
which is A6.8).
From Entry 13(xiii) again and A6.10),
- + w + 2 = 6 + 2 cosBv) = 4A + cos2 v) = 4smB^. A6.12)
m sinv
Subtracting A6.11) from A6.12), we find that
^_cos^\ = 4JinB^
sin v cos v / sinBv)
Since LBOP = 2ц, LROP = 2v, and arc QB = arc BP, it is not difficult to
see that LOQR = \n + v - 2ц. Thus, by the law of cosines,
sinB/* - v) = cos(i?r + v - 2ц) = " " = Щ-.
2aQR 2a
Hence, from A6.13),
which establishes A6.6).
Adding A6.11) and A6.12), we find that
5 /cosBAt) sinB/z)\ 4 sinB^ + v)
г 1 = Zl 1 : I = . . A0.15)
m \ cos v sin v / sinBv)

19. Modular Equations and Associated Theta-Function Identities 301
Since LQORi = Ац + 2v, we find that LOQR^ = \n - 2ц - v. Thus, by the
law of cosines,
QD^ i „2 _2 OR
sinB^ + v) = cos(i* - 2^ - v) = ^ = i
Hence, by A6.15),
5 QR1m
m+ 4 КТ/а ^'
which is A6.7).
Example (i). // ЛР = 1, then
TP = {ШРA ~ *)A - P)
DT = №112,
OD + ОТ = {A - а)A - p)}1/2,
and
+ {A - а)A - Р)}112
The last equality above is a trivial consequence of the first three, since
AP = 1. Thus, Ramanujan has found an interesting geometric derivation of
the fifth-order modular equation Entry 13(i).
In our proof below, we proceed without the assumption AP = 1, which
yields only a trivial simplification.
Proof. First, by A6.10),
TP = RP sin v = 2a sin2 v
= 2a {sin2 v(l + cos2 vJ — 4 sin2 v cos2 v}1/3
= 2a{sin2B/4 - sin2Bv)}1/3
= 2a{sinB/z + 2v)sinB/z - 2v)}1/3
= 2a{4a1/2(l - aI/201/2(l - pI'2}113
Second,
DT = ОТ - OD = a cosBv) - a cosB/z)
= 2a sin(/z + v)sin(/z - v) = 2a(a/?I/2.
Third,
OD + OT = a cosB/z) + a cosBv)
= 2a cos(/i + v)cos(/z - v) = 2a{(l - a)(l - P)}
which completes the proof of Example (i).

302 19. Modular Equations and Associated Theta-Function Identities
Example (ii). Let Q = A. Then D = 0, m = ^5, and T divides AP in medial
section, that is, T = G.
Proof. Since the arcs BP and BQ are equal, it is clear that D = 0. Thus, from
A6.11), 5/m-m = 0, that is, m = ^/5.
Lastly, from similar triangles,
AT _ AT _AR2-RT2
TT ~ RT2/AT RT2
From A6.14), with m = ^/5 and Q = A, we conclude that
AT
TP \ 2
that is, T divides AP in medial section.
Our formulation of Example (ii) is somewhat different from that of Rama-
nujan (p. 238).
The last three sections of Chapter 19 are devoted to modular equations of
degree 7 and associated theta-series identities.
Entry 17.
(i)
q q3 q5 q9 q11 q13
where the cycle of coefficients is of length 14.
(ii)
q11 g12 q13 q15
q16 q11 q18 q19
q20 q22 q23 q24
l-q20 l-q22' l-q23 1 - q2*
where the cycle of coefficients is of length 28.

19. Modular Equations and Associated Theta-Function Identities 303
(iii) ф*) - <ptf) = VW, q9) + 2q^f(q3, q11) + 2q9nf(q, q13).
(iv) <A(<?1/7) - qt'Mq1) = f(q3, <?4) + qwf{q2, q5) + ЧЗПАч, Ч6)-
(v)
f(-q,~q6) Ч f(-q2,~q5)
The first two formulas are of extreme interest, since they appear to indicate
that Ramanujan was acquainted with a theorem equivalent to the addition
theorem for elliptic integrals of the second kind. Although it would appear to
be very difficult to prove (i) without this addition theorem, it is apparently not
found in the notebooks.
Proof of (i). In the notation of the notebooks, the addition theorem assumes
the form
qf(-a, -q*/a)f(-b, -q2/b)f(-ab, -q2/ab)
abf(-aq, -q/a)f(-bq, -q/b)f(-abq, -q/ab)
A direct proof of this theorem has been constructed by Glaisher [1] who wrote
the left side of A7.1) as a product of three Laurent series and multiplied them
together. It does not seem worthwhile to give the details here.
To deduce (i) from A7.1), replace q, a, and b by q1, q2, and q*, respectively.
Thus,
qlln _ q13n
1 - q1*"
= f4-q14)f(-q2, -q12)f(-q\ -qlo)f(-q6, ~<Z8)
q vi-q1) Я-q5, -qg)f(-q\ -qlx)f{-q, -<z13)
= q vi-q'm-q^M-q)
In the analysis above, we expanded six of the theta-functions by means of the
Jacobi triple product identity (Entry 19 of Chapter 16), simplified, and then
employed Entry 24(iii) in Chapter 16 to obtain the final form.
Now take the summands in the series on the left side and expand them into
geometric series. After inverting the order of summation in the resulting
double series, we obtain the series displayed on the right side of (i) to complete
the proof.

304 19. Modular Equations and Associated Theta-Function Identities
We remark that, in more classical notation, A7.1) is equivalent to the
identity
E(u) + E(v) — E(u + v) = k2 sn и sn v sn(u + v),
where E(u) denotes the incomplete elliptic integral of the second kind.
Proof of (ii). In order to prove (ii), we appeal to a modular equation
of the seventh order given in Entry 19(i). Multiply both sides of it by
{A - a)(l - 0)}1/8 to put it in the form
{A - a)(l - Р)У* - {A - a)(l - /?)}1/4 = {a/»(l - a)(l - /?)}1/8.
Transforming this equality by Entries 10(ii), (iii) and 11 (ii) in Chapter 17, we
obtain the theta-function identity
cpi-qM-q") - <p(-q2)<p(-q14) = -
Replace q by q2" and sum both sides on n, 0 <, n < со. Using part (i), we arrive
at
<f>(-q)<p{-qn)- 1 = -2
n=O
4
г
1+ q 1 + q2 1+ q3 1+ q4 1+ q5
q6 ,
\+q
I12 <?13 , q15 , ....
,12 1 , „13 + 1 , „15 + I'
l+q1
where the cycle of coefficients is of length 7. Replacing q by — q, we deduce that
¦7)=l + 2[ * ** ^ q* qS
l-q l+q2 l-q3 l+q4 l-q5
q6 q* q9 q10 q11 q12
1+q6 1+^8 l-q9 l+q10 l-q11 l+q
12
«13 . + . «15
BT|_ Л15
l-q13 1-,
where the cycle of coefficients is now of length 14. For each even value of n
above, write
qn q" 2q2"
In-
l+qn l-qn l-q
We now readily see that the right side of (ii) is obtained, with a cycle of
coefficients of length 28. This completes the proof.

19. Modular Equations and Associated Theta-Function Identities 305
Proof of (iii). In Entry 31 of Chapter 16, put a = b = q and n = 7. Then
9(q) = <P(Q*9) + 2qf(q35, q63) + 2q*f{q2\ q11) + 2q9f(q\ q91).
Replacing q by qip, we complete the proof.
Proof of (iv). In Entry 31 of Chapter 16, let a — 1, b = q, and n = 7. Using
Entry 18(ii) in Chapter 16 and the definition of ф, we find that
Ш = /(«",12*) + 4f{q'\ q35) + q3f{q\ <?42) + <Z
Replacing q by qin, we complete the proof.
Proof of (v). Set n = 7 in A2.25).
Entry 18(ii) below serves as the basis for a septic algorithm for calculating
к that has been devised by J. M. and P. B. Borwein [6].
Entry 18.
(i) There are positive functions u, v, and w such that
where
ut) - uw + vw = 289 + 126—-4- T + 19^-^- T + 3 12 yr,
qfi-q) q fi-q) «v (-« )
A8.3)
ww = 1. A8.4)
(ii) With м, г, and w as above,
1 + lqlf{~q*9} = u1/7 - t;1'7 + w1/7,
where
и — v + w =
and
V /
«v - uw + vw = 289 + 18-7V 1_ ! + 19'7V fgr_ /
J \ 4) J У. 4.)
в /12() ¦

306 19. Modular Equations and Associated Theta-Function Identities
(iii) f(q, q6)f(q2, q5)f(q\ q*) = ^Ь^ «,(_,').
(iv) f(-q, -q6)f{-q2, -qs)f(-q3, -<?4) = f{-q)f2{-q1).
(V) f(q, q13)f(q\ qll)
(vi) If
f(-qin)
v
then
2ц = 7(v3 + 5v2 + 7v) + (v2 + 7v + 7){4v3 + 21v2 + 28v}1/2.
Proofs of (i), (vi). Our primary aim here is to establish (i), but, along the way,
we also prove (vi).
In view of Entry 17(v), define a, p\ y, v, u, v, and w by
r " „1/7 fl „2 _5\'
-j-, and v = ^y— j-.
Thus, from Entry 17(v), we deduce immediately that v+l=a + fj + y, which
is A8.1). It is also clear that A8.4) is trivially satisfied. It remains to establish
A8.2) and A8.3).
Let the cubic equation satisfied by a, /?, and у be given by
z3 -pz2 + sz-r = 0.
Thus, p = v + 1, by A8.1). It also follows from A8.4) that r = -1.
For brevity, we let Jo, J1,..., J6 denote power series in q. From Entry 24(ii)
in Chapter 16, it is easy to see that v3 has the form
v3=J0 + <Гб/7Л + q~snh + q~inh- A8.5)
But on the other hand,
v3 = (a + p + у - IK = Jo + <Гб/7Л + q-5pJ2 + q'^Jz
+ q~wh + q'2ph + q'ipJ6, A8.6)
where
Jo= -1 + 6ap>= -7,
= 3(a - p12 + p>2), L A8.7)
= 3@ - y2 + ya2). I

19. Modular Equations and Associated Theta-Function Identities 307
By comparing A8.5) and A8.6), we find, from A8.7), that
y-a2 + aj52 = 0,l
« - /»2 + /»y2 = 0, I A8.8)
That these three formulas are equivalent to a single relation may be seen
by multiplying them by у, а, and /J, respectively, and observing that, since
а/fy = — 1, they merely undergo a cyclical interchange. To obtain this relation
in a symmetrical form, multiply them by a, /J, and y, respectively, and add them
to get the equation
= о, A89)
where each sum is over all cyclical interchanges of the summands. Easy
calculations show that
2>3 = p3-3ps + 3r A8.10)
and
?02y2 = s2 -2pr. A8.11)
Thus, A8.9) becomes
0 = s - (p3 - 3ps + 3r) + s2 - 2pr
= s2 + Cp + l)s - p3 - 3r - 2pr A8.12)
= s2 + Cv + 4)s - v3 - 3v2 - v + 4,
since p — v + 1. Solving A8.12) for s, we find that
2s = - Cv + 4) - Dv3 + 21 v2 + 28vI/2. A8.13)
To see why we chose the minus sign on the radical, observe that, from the
definitions of a, /J, y, v, and s,
v~q-2n and s~-q-311,
as q tends to 0.
Next, replace qip by Cqip, where С is any seventh root of unity. Then, since
Taking the product of each side over all seventh roots of unity and using an
argument analogous to that used in the proof of Entry 10(vi), we deduce that
1?1?(t>'+w+{1'1) A8Л4)
We now face the challenging task of calculating the product on the right side
above. For brevity, in the sequel, we put sn = a." + /Г + у".

308 19. Modular Equations and Associated Theta-Function Identities
First, replacing ? by l/?, we determine the product
(C5a + C6P + С3У ~ 1)(C2* + CP + C47 - 1)
= 1 + s2 - (« - ya)(C2 + C5) - (/» - «/»)(C + С6) - (Г - )Sy)(C3 + С4)-
There will be two additional products of this type. Multiplying these three
equalities and the equality a. + P + у — 1 = p — 1 together, we deduce, from
A8.14), that
*2K + A + s2J?(x - ya)
- 2A + s2)]T(a - УаJ + 3A + s2)Z(i9 - «/»)(У - ft)
A8.15)
where the summations are extended over all cyclical (not symmetrical) inter-
interchanges of a — ya, P — a.p, and у — /fy. We now are presented with the task
of evaluating all terms on the right side of A8.15).
First, since
s2 = p2- 2s, A8.16)
A + s2K = 1 + 3(p2 - 2s) + 3(p4 - 4p2s + 4s2)
+ p6 - 6p*s + 12p2s2 - 8s3
= -8s3 + 12(p2 + l)s2 - 6(p2 + lJs + (p2 + IK. A8.17)
Second, by A8.16) and the definitions of p and s,
(l+s2J]>>-ya)
= A + 2(p2 - 2s) + (p2 - 2sJ)(p - s)
= -4s3 + 4(p2 + p + l)s2 - (p* + 4p3 + 2p2 + 4p + l)s
+ (p5 + 2p3 + p). A8.18)
Next, by A8.8) and A8.16),
;>>2y = ;>>2-;>> = p2-2s-p. (is.i9)
Thus, by A8.19), A8.16), and A8.11),
-2(l+s2)]>>-yaJ
= -2A + p2 - 2s) X (a2 - 2a2y + a2y2)
= -2A + p2 - 2s)(p2 - 2s - 2(p2 - 2s - p) + s2 + 2p)
= 4s3 + F - 2p2)s2 - (8p2 - 16p + 4)s
+ Bp4 - 8p3 + 2p2 - 8p). A8.20)

19. Modular Equations and Associated Theta-Function Identities 309
With the help of A8.19), it can readily be verified that
? fi2y = ps - 3r - p(p - 1) + 2s. A8.21)
Thus, by A8.16) and A8.21),
= 3A + s2)?@y - ф - P2y + a/32y)
= 3A +p2 - 2s)(s -3r-ps + 3r + p2-p-2s + pr)
= 6(p + l)s2 - Cp3 + 9p2 - 9p + 3)s + 3(p2 + l)(p2 - 2p). A8.22)
By A8.19),
X a3y = (pa2 - sa + r)y + (pj82 - s/S + r)a + (pv2 - sy + r)jS
= P(p(p - 1) - 2s) - s2 + rp
= -s2-2ps + p3-p2-p. A8.23)
By A8.11), A8.21), and A8.16),
?a3y2 = {pa2 -stz + r)y2 + (P02 - sfi + r)a2 + (py2 - sy + r)/32
= pZaV - sXav2 + ^Z012
= p(s2 - 2pr) - s{ps -3r-p2 + p + 2s) + r(p2 - 2s)
= -2s2 + (p2 - p - l)s + p2. A8.24)
By A8.10), A8.23), and A8.24),
Za373 = (РУ2 - ^ + r)a3 + (pa2 - sa + r)j?3 + (pjS2 - s/S + r)y3
- s(-s2 - 2ps + p3 -p2-p) + r(p3 - 3ps + Ъг)
= s3 + 3ps + 3. A8.25)
Hence, by A8.10) and A8.23)-A8.25),
= -(p3 - 3ps - 3) + 3(-s2 - 2ps + p3-p2-p)
- 3(-2s2 + (p2-p- l)s + p2) + s3 + 3ps + 3
= s3 + 3s2 - Cp2 - 3p - 3)s + Bp3 - 6p2 - 3p + 6).
A8.26)
Next, by A8.21),
= p{ps ~3r~ p(p ~ 1) + 2s) - s2 + pr. A8.27)

310 19. Modular Equations and Associated Theta-Function Identities
Thus, by A8.21), A8.27), A8.16), and A8.19),
- P3y + 2/J - 2/J2 - «/» + a/J2)
= 4{ps - 3r - p(p - 1) + 2s - p(ps - 3r - p(p - 1) + 2s) + s2
- pr + 2p - 2(p2 -2s)-s + p2-2s-p}
= 4{s2 - (p2 + p - 3)s + (p3 - 3p2 + 3)}. A8.28)
Now, by A8.11), A8.19), and A8.16),
= p(s2 - 2pr) - s(p2 - 2s - p) - (p2 - 2s)
= s2(p + 2) - s(p2 - p - 2) + p2. A8.29)
Hence, by A8.19), A8.11), A8.29), and A8.21),
- арУ - 2j52y2 + 2aj52y2 + j53y2 - aj93y2)
+ у - 2j52y2 - 2j5y + 03y2 + iS2y)
= -3(p2 - 2s - p + p - 2(s2 - 2pr) - 2s + (p + 2)s2
- s(p2 - p - 2) + p2 + ps - 3r - p(p - 1) + 2s)
= - 3(ps2 + Bp - p2)s + p2 - 3p + 3). A8.30)
Lastly, a direct calculation gives
py) = p-s-2. A8.31)
Substituting A8.17), A8.18), A8.20), A8.22), A8.26), A8.28), A8.30), and
A8.31) into A8.15), we at last derive the formula
q2f*(li) = (P - l)(-7s3 + A4p2 + 7p + 35)s2
- Gp4 + 7p3 + 35p2 - 14p)s + P6 + p5
+ Sp* - 6p3 - 13p2 - 6p + 8}
= -7(p - l)s3 + A4p3 - 7p2 + 28p - 35)s2
- Gp5 + 28p3 - 49p2 + 14p)s
+ p7 + 7p5 - 14p4 - 7p3 + 7p2 + 14p - 8. A8.32)
Since p = v + 1, A8.32) may be written in the form

19. Modular Equations and Associated Theta-Function Identities
311
/ ( _g) = -7vs3 + 7vBv2 + 5v + 8)s2
- 7v(v4 + 5v3 + 14v2 + 15v + 5)s
+ v7 + 7V6 + 28v5 + 56v4 + 42v3 - 7v2 - 7v.
Using A8.12) to reduce the right side to a linear function of s and then
employing A8.13), we find that
?f8\z\i(++)
+ v7 + 21v6 + 126v5 + 322v4 + 294v3 - 147v2 - 343v}
= 2v7 + 63v6 + 532v5 + 2009v4 + 3724v3 + 3087v2 + 686v
+ 7v(v4 + 12v3 + 49v2 + 84v + 49)Dv3 + 21v2 + 28vI/2.
Taking the appropriate square root of both sides, we deduce that
2ц = 2
HJ v 4. )
5v2 + 7v) + (v2 + 7v + 7)Dv3 + 21v2 + 28VI'2,
A8.33)
which is formula (vi).
We next calculate s7. To do this, we employ a general formula for sn, which
may be found in Littlewood's book [1, p. 83]. Omitting the rather tedious
algebraic details, we deduce that
p 10 0 0 0 0
2s p 1 0 0 0 0
Ъг s p 1 0 0 0
0 r s p 1 0 0
0 0 r s p 1 0
0 0 0 r s p 1
0 0 0 0 r s p
= -Ips3 + Gr + 14p3)s2 - Gp5 + 21p2r)s + p1 + lp*r + Ipr2.
Hence, from A8.32) and A8.34),
Cr*7q\, - si = 7s3 + (~ V + 28p - 28)s2
A8.34)
7p5 _ 7p4 _ 1рз + lp2 + 7p _ g.

312 19. Modular Equations and Associated Theta-Function Identities
By A8.2), it remains to show that
2 l_\ = -* + (P - 4P + V + DP - V + 2P)S
4J У 4 )
-Ps + P* + P3-P2-P-1. A8.35)
With the use of A8.12) and A8.13), it is a straightforward, but laborious, task
to reduce the right side of A8.35) to the right side of A8.33). This concludes
the proof of A8.2).
Examining A8.3), we are led to calculate ]Г/Гу7. То do this, we can use
A8.34). Thus, in A8.34), suppose that a, /f, and у are replaced by a/3, /Jy, and
ya, respectively. Then r, p, and s are replaced by r2, s, and rp, respectively.
Hence,
= s7 + 7ps5 + 7s4 + 14pV + 21ps2 + 7(p3 + l)s + 7p2. A8.36)
By A8.3), we want to show that the right side of A8.36) is equal to
-289 - 126/* - IV - Ц3. A8.37)
To do this by hand would be a superhuman feat. Therefore, we employ
MACSYMA. In A8.36), we set p = v + 1 and substitute the value of s given
by A8.13). In A8.37), we substitute the value of ц given by A8.33). Both A8.37)
and the right side of A8.36) then reduce to
-i{21v10 + 595v9 + 6468v8 + 37229v7 + 127421 v6
+ 270445v5 + 355103v4 + 275723v3 + 113484v2 + l9208v + 578
+ (v9 + 63v8 + 910v7 + 5929v6 + 21007v5 + 43099v4 + 51107v3
+ 32907v2 + 98OOv + 882)Dv3 + 21v2 + 28vI/2}.
Of course, the proof that we have given is quite unsatisfactory, because it
is a verification which could not have been achieved without knowledge of
the result. Ramanujan obviously possessed a more natural, transparent, and
ingenious proof. A proof of A8.2) via the theory of modular forms can be found
near the end of the introductory material in Chapter 20.
Compared to our proofs of (i) and (vi), the proofs of the remaining four
parts are almost trivial.
Proof of (ii). Applying Entry 27(iii) of Chapter 16 with a = i Log(l/Q7)
and P = \ Log(l/g), where q and Q are chosen so that
7 Log(l/e)Log(l/?) = 4я2,
we find that
gV2* Logv*(i/g)/(_e) = QW Log1/4(l/67)/(-e7). A8.38)
Replacing q by qip and Q by Q'', we find that

19. Modular Equations and Associated Theta-Function Identities 313
Log«*(l/e7)/(- б49)- A8.39)
Lastly, reversing the roles of g and Q ш A8.38), we deduce that
Dividing A8.38) by A8.40) and then A8.39) by A8.40), we deduce that,
respectively,
-у *• ' A8.41)
and
тоО)2ет- A842)
Thus, by A8.42) and part (i),
Furthermore, by A8.41) and part (i),
The formula for uv — uw + uw is proved in the same way.
Proof of (iii). By Entry 19, B2.4), and Entry 22, all in Chapter 16,
f(q, q6)f(q\ qs)f(q3, q*)
= (-«; q'U-q2; q'U-q3; e7U-e*; q7U-q5; q\
which completes the proof.
Proof of (iv). The proof employs Entries 19 and 22 of Chapter 16 and is
even easier than the proof of (iii) above.
Proof of (v). With the aid of the Jacobi triple product identity, it readily
transpires that
f(q, q ^^КУ^
i q ¦> q
by Entry 22 in Chapter 16.
Chapter 19 ends with a battery of modular equations of degree 7.

314 19. Modular Equations and Associated Theta-Function Identities
Entry 19. // P is of the seventh degree in a., and m is the multiplier for degree
7, then
(i)
+ {A - «)A -
7_ V/?(!-/?)
7\ 1/8 /,< _ ч7\ 1/8 ?
1 -a / vr/ (\ 1 -a / \tx
a7\l/8 fAv7\l/8
49 _ /«y2 /1 - aV'2 _ /«A - a)V/2 _ /яA - a)V/3
1 J V i« /
{A -
49/l+(a^I/2 + {(l-
~2
m
+\JJ +
x'24 _ 3 + m2
i/24 з 49

19. Modular Equations and Associated Theta-Function Identities 315
and
(viii)
m - - = 2((a/JI/8 - {A - <z)(l - /J)}1/8)B + (<ф)* + {A - «)A - /J)}1'4).
m
(ix) i/
«а-«)У '
(x)//
P = («/»I/2 and Q = (W2,
t/ien
P + I = Q + i + (pl/8 _ p-l/8)8
(xi) // a = sin2(/x + v) and )S = sin2(/x — v), tften
cosB^) = B cos v - 1)D cos v - 3I/2.
The seventh-order modular equation given in the first equality of (i) is due
to Guetzlaff [1] in 1834. Fiedler [1] in 1835 and Schroter [1], [2], [4] in 1854
also proved this modular equation. More complicated modular equations of
degree 7 have been discovered by Schlafli [1], Klein [1], Sohncke [1], [2],
and Russell [1].
Proof of (i). Let ц = 4 and v = 3 in C6.8) of Chapter 16. This yields the
equality
q*), A9.1)
where we have used the equality
deducible from Entry 18(iv) in Chapter 16. Transforming A9.1) by means of
Entries 10(v) and ll(i), (iii), (v) in Chapter 17, we deduce that
Simplifying, transposing, and taking the square root of each side, we arrive at
the first equality of (i).

316 19. Modular Equations and Associated Theta-Function Identities
Second, square the first formula of (i) and transpose. Then square the result
and transpose again. This gives, in succession,
(ajSI'4 + {A - a)(l - jS)}1'4 = 1 - 2{a/3(l - a)(l - /3)}1/8
and
1 + (a/3I'2 + {A - a)(l - P)}112
= 2 - 4{a/3(l - a)(l - ft}118 + 2{a/?(l - a)(l - jS)}1'4.
Dividing both sides by 2 and then taking the square root of both sides, we
deduce the second equality of (i).
In order to derive further modular equations, we need to obtain expressions
for a, /?, and various radicals in a and /J in terms of a positive parameter t,
which we define by
<x0 = t8. A9.2)
Hence, from part (i),
A - «)A - P) = A - t)8. A9.3)
It now follows that a and ft are roots of the quadratic equation
x2 - {1 + t8 - A - t)8}x + t8 = 0. A9.4)
Now,
A + t8 - A - t)8J - 4t8
= {1 + t8 - A - t)8 + 2r4} {1 + t8 - A - t)8 - 2t4}
= {A + t*J - A - t)8} {A - t*J - A - t)8}
= {A + t4) - A - if} {A + t4) + A - tL} {A - t4)
= I6t2Bt2 - 3t + 2)A - t + t2JB - t + t2)(l - tJ(l - t + 2t2).
Hence, solving A9.4), we obtain the roots
xj = i(l + f8) - i(l - t)8 ± 2t(l - t)(l - t + t2)
x {Bt2 - 3t + 2)B - t + f2)(l - t + 2t2)}1/2. A9.5)
Clearly, from the definition of a modular equation, a is the larger root. For
brevity, we then write
a. = A + BR and fi = A-BR,
where
л = i(l + t8) - i(l - r)8,
В = 2t(l - t)(l - t + t2),

19. Modular Equations and Associated Theta-Function Identities
317
and
R = {B - 3r + 2f2)B - t + f2)(l - t + 2t2)Yt2.
From A9.4),
(a ± ?f = {A + t*J - A - 08}a,
and so by the same factorization process as used above,
a + t* = 2A - t + ?2){afB - 3t + 2t2)}112
and
a - t* = 2A - 0{at(l - t + 2t2){2 - t + t2)}1/2,
for clearly, by A9.2), a. > t*. Consequently, by addition,
у/л = A - t + t2){tB -3t + 2t2)}1*2
+ A - r){t(l - t + 2t2)B - t + t2)}1*2.
A9.6)
A9.7)
A9.8)
If a is replaced by /f, the only change in the preceding argument is that the
sign of the second radical must be changed, since /? < t*. Hence,
(l-t + t2){tB -3t + 2t2)}1!2
- A - t) {?A - t + 2t2)B - t + t2)}1/2. A9.9)
It is quite obvious from A9.2) and A9.3) that expressions for ^1 — a and
yjl — P can be obtained by replacing t by 1 — t and choosing the appropriate
signs of the radicals. Therefore,
3t + 2t2)B - t + t2)}
1/2
A9.10)
and
y/l-P = О - t + t2){(l - 0A - t + 2t2)Y'2
+ t{(l - 0B - 3r + 2t2)B - t + t2)Y12. A9.11)
To calculate ^a(l — a) in its simplest form, it is perhaps wise to let
t = ^A — u). Thus, replacing t by 1 — t has the effect of changing the sign of и.
Under this change of variable, A9.8) and A9.10) take the shapes, respectively,
8^ = C + «2){A - и)B + u + «2)}1/2
+ A + и){A - и)B - и + u2)G + «2)}1/2
and
8N/l-a = C + u2){(l + u)B - u + u2)}1/2
h и)B + « + u2)G
Hence,

318 19. Modular Equations and Associated Theta-Function Identities
64
+ C + u2)G + u2I/2{(l + и)B -u + u2)
- A - и)B + u + u2)}
= B + 12u2 + 2u4){4 + 3u2 + u4I/2
+ 2uC + 4u2 + u4)G + u2I/2
= |{D + 3u2 + и4I'2 + иG + u2I/2}3.
Therefore,
(T = i{D + 3«2 + u4I'2 + иG + u2I'2}. A9.12)
Similarly, from A9.9) and A9.11),
1/2 2 A9.13)
On squaring A9.12), we deduce that
(gjj ~ "M = i{2 + 5u2 + u4 + uB8 + 25u2 + 10u4 + и6I'2}
= i{2 - 7t + 1 It2 - 8t3 + 4t4 + A - 2t)R}, A9.14)
where R is defined by A9.6). It is seen from A9.13) that an analogous analysis
yields
! - л)=^{2+5+~uB8+25+io+мбI/2}
= ^{2 - It + lit2 - 8t3 + 4t4 - A - 2t)«}. A9.15)
Hence, by A9.14) and A9.15),
are the roots of the quadratic equation
x2 - iB + 5u2 + u*)x + ^A - u2J = 0, A9.16)
or
x2 - B - It + lit2 - 8t3 + 4t4)x + t2(l - tJ = 0.
In the proofs of Entries 13 (iv), (v), we derived formulas for the multiplier
m from Entry 9(iii). In the absence of any formula analogous to Entry 9(iii),
we must proceed differently here. Thus, in order to find a parametric rep-
representation for m, it seems necessary to use Entry 24(vi) of Chapter 18 with

19. Modular Equations and Associated Theta-Function Identities 319
n = 7, namely,
a(l — a)
By differentiating A9.4) with respect to t, we find that
Ba - {1 + ts - A - t)8})^ = 8({t7 + A - 0> - t7),
with a similar equation involving /J. Hence, from A9.17) and A9.2),
2_ /?(l-/?){t7 + (l-t)>-t7
a(l - a) {t7 + A - tO}P - t1
1 - Jg t8 + t(l - tO - Jg
1 - a t8 + t(l - tO - a
where in the last step we multiplied out the numerator and denominator and
then substituted for /J2 and a.2 by A9.4). Hence, upon cancellation and the use
of A9.2) and A9.4),
= -7
= -7
t2{\
m
t
t2
t2
— a
—
—
(t
-tf
t(\
(t
(a
{1
-
—
-Pf
+ P)t + a/S
(t - РУ
+18 - A -
PJ
-1 +12J
t-p
0A -1 +
1
1
J
-08}t + t8
t2y
A9.18)
i U -VU -i "I" »""Г
Consequently,
t- R
A9.19)
We have, indeed, chosen the proper square root, because from the first equality
in A9.18) and the fact that a > P, it follows that P < t.
We are now in a position to easily prove (ii)-(iv).
Proof of (ii). By A9.5), A9.6), and A9.19),
_ t - |A + t8) + j(l - Q8 + 2t(l - Q(l - t + t2)R
=-3 + St-6t2+ 4t3+ 2R. A9.20)

320 19. Modular Equations and Associated Theta-Function Identities
It follows that
A - 2t)m = -3 + 14t - 22t2 + Ш3 - St* + 2A - 2t)R.
Hence, from A9.15),
(l_2r)m = um=l-4(i^—^) , A9.21)
and from A9.2) and A9.3),
l-4(
_ , a(l - a)
fit "™~ ^^ f ? л \ / Л Г\\ \ \ /fi
and this is the first formula of (ii).
The second formula of (ii) follows from the first by the process of
reciprocation.
Proof of (iii). Observe that, by A9.2), A9.3), and A9.19),
( = =
1 - a / \ a / 1 - t t t(l-t)
= m{\ - t(l - t)}
Employing part (i), we readily deduce the first formula of (iii).
The second formula of (iii) is simply the reciprocal of the first.
Proof of (iv). The first formula in (iv) is achieved by substituting for t and
1 - t from A9.2) and A9.3) in the obvious identity
l-t \l-t
The second part of (iv) is the reciprocal of the first.
Proof of (v). Let
First, by A9.3), A9.10), and A9.11),
- t){2 -3t + 2t2){2 - t + r2
Using an analogue of A9.7) and the identity above, we deduce that
T = 2A - t){/fe(l - t + 2t2)B - t + t2)}112
x 2t{(l - /J)(l - t)B - 3t + 2t2)B - t + t2)}»2

19. Modular Equations and Associated Theta-Function Identities 321
4t2(l - tJB - t + t2J{(l - t + 2t2)B -3t + г*2)}1'2^!^
^A - u2JG + u2)D + 3u2 + u4I
where, as above, и = 1 — 2t and where we utilized A9.12) and A9.13). Re-
Rearranging terms, we arrive at
(\-u2J[P{\-P)\213 A-й2K
+ (Аи2
by A9.16). Now, from A9.21),
Hence,
I — RW2/3 M —
?mV(l - u2)}
2
= 8f2A" °2 (fr^f) +14A ~tL(m2 ~

322 19. Modular Equations and Associated Theta-Function Identities
Solving for m2, we deduce that
p, l-p
- if t*(l - t)* 8 \t*(l - t)*J
7*0-/0У2
which is precisely the first formula of (v).
The second formula of (v) is the reciprocal of the first.
We remark here that Entries 10(iii), ll(i), (ii), and 12(iii) in Chapter 17 can
be employed to convert the former equality of (v) into the theta-function
identity
<?4(-<Z14) , з/VV) *iq)\ *г/(д)_л
*B) Ч U*() n)) Щ П1) ~
No direct proof of this fascinating identity has ever been constructed.
Proof of (vi). By A9.18),
A - t + t2fm2 = ¦ ~ ~ ~
t2(l-tJ A-tJ t2 t2(l-tJ'
Hence, by A9.2), A9.3), and part (i),
1-я ) +\a) \a(l-a)
= m2{\-ti\-i)}2
= тЩA + (арI* + {A - «)A - iS)}1'2)I'2,
which is the former formula of (vi). The second is the reciprocal of the first.
Proof of (vii). From A9.20),
m = -A - 2t)C -2t + It2) + 2R,
and so
m2 = C - it + 6t2 - 4t3J + AR2 - 4ЯA - It) - 8ЯA - 2t)(l - t + t2).
Using A9.5) and A9.15), we deduce from the foregoing equality that
2 zJU
m -\w^f

19. Modular Equations and Associated Theta-Function Identities 323
= C - 8t + 6t2 - 4t3J + 4Я2 - 4B - It + lit2 - 8t3 + 4t4)
_ 2-4t 8_/i_ 48)
" t(l -1)
1 + 3r
~ 1-t*
after a rather tedious calculation. Hence, from the equality above, A9.2), and
A9.3),
t(\-t)J t(l-t)
0A - j5)\1/3
t(l - t) / t 1-t
which yields the first formula of (vii). The second is the reciprocal of the first.
Proof of (viii). From the analysis leading to A9.20), it is clear that similar
reasoning yields the companion formula
- - = - 3 + it - 6t2 + 4t3 - 2R. A9.22)
m
Adding this formula to A9.20), we find that
m--= -6 + Ш- \2t2 + 8t3
m
= 2{t - A - t)} {2 + t2 + A - tJ}. A9.23)
Formula (viii) now follows at once upon the use of A9.2) and A9.3).
In the notebooks (p. 240), Entry 19 (viii) contains two misprints.
Proof of (ix). By A9.2) and A9.3), P = t(l - t)y/2, and by A9.16),
Q V4(i - 04
_ 2 + 5A - 2tJ + A - 2tf
~ 4t(l - t)
_ 2 - 7t(l - t) + 4t2(l - tJ
t(l - t)
P
which completes the proof of (ix).

324 19. Modular Equations and Associated Theta-Function Identities
Ramanathan [10] has given a different proof of (ix).
Proof of (x). From part (i),
Dividing both sides by P, we find that
which immediately yields the desired result.
Proof of (xi). From the second formula of part (i),
= (^A + sin(/x + v)sin(/x — v)
= (|A + cosBv))I/2 = cos v,
and so
sinB/x + 2v)sinB/x — 2v) = 4A — cos vL;
that is,
cos2Bv) - cos2B/i) = 4A - cos vL.
Hence,
cos2B/x) = B cos2 v - IJ - 4A - cos vL
= 16 cos3 v - 28 cos2 v + 16 cos v - 3
= B cos v - 1JD cos v - 3),
and so the proof of (xi) is complete.
Employing the theory of modular forms, Raghavan [1], [2] and Raghavan
and Rangachari [1] have proved several results in this chapter.

CHAPTER 20
Modular Equations of Higher and Composite Degrees
In this chapter, we continue to examine Ramanujan's discoveries about modu-
modular equations. In the previous chapter, modular equations of degrees 3, 5, and
7 were derived. Modular equations of degrees 11,13,17,19,23, 31,47, and 71
are established in this chapter. Also, modular equations of composite degree,
or "mixed" modular equations, are studied. Most of the equations of the latter
type involve four distinct moduli, and so we begin by defining such a modular
equation. Let K, K\ Lu L\, L2, L'2, L3, and L3 denote complete elliptic
integrals of the first kind corresponding, in pairs, to the moduli yfa., y/p, y/y,
and у/б, and their complementary moduli, respectively. Let nl,n2, and n3 be
positive integers such that n3 = nxn2. Suppose that the equalities
И17Гь? *2K=V and п>к=Т3
hold. Then a "mixed" modular equation is a relation between the moduli yf,
y/p, y/y, and y/S that is induced by @.1). In such an instance, we say that /f,
y, and S are of degrees nu n2, and n3, respectively. Recalling that zr = q>2(qr),
we define the multipliers m and m' by
m = zi/zBl and m' = znjznb.
Amazingly, in this chapter, Ramanujan derives modular equations for
twenty distinct sets {nj, n2, n3} of degrees, namely,
3, 5,15; 5, 27,135;
з,
з,
з,
з,
з,
7,
9,
11,
13,
21,
21;
27;
33;
39;
63;
7,
7,
7,
9,
9,
9,
17,
25,
15,
23,
63;
119;
175;
135;
207;

326 20. Modular Equations of Higher and Composite Degrees
3,29,87; 11,13,143;
5, 7,35; 11,21,231;
5,11,55; 13,19,247;
5, 19, 95; 15, 17, 255.
Hardy [1, p. 220] recorded, without proof, two modular equations for the
triple {3, 5, 15} of degrees. Otherwise, none of Ramanujan's work on "mixed"
modular equations had been published until Berndt, Biagioli, and Purtilo
[l]-[3] published proofs of a small portion of Ramanujan's modular equa-
equations in Chapters 19 and 20.
In Chapter 19, we employed the theory of theta-functions and elementary,
but often complicated and tedious, algebra to prove Ramanujan's modular
equations. For many of the modular equations of this chapter, we have been
unable to establish them by these techniques. Instead, we have had to invoke
the theory of modular forms. In some ways, this approach is the best of the
three methods of attack. The first two methods become rapidly more difficult
as the degree of the modular equation increases, while the complexity of the
approach through modular forms increases only slightly as the degree in-
increases. Because a modular equation is always equivalent to an identity among
theta-functions of several arguments, the theory of modular forms provides
the theoretical backdrop explaining the raison d'etre for such identities. The
primary disadvantage of this method, as well as most elementary algebraic
approaches, is that the modular equation must be known in advance. Thus,
the proofs are perhaps more aptly called verifications.
In order to avoid a lengthy diversion later in the sequel, it seems advisable,
at this juncture, to present the theory of modular forms that will be necessary
to establish many of Ramanujan's modular equations. General references are
Rankin's book [2] and Petersson's notes [1].
Let Ж denote the upper half-plane, that is, Ж = {т: 1т(т) > 0}. Put q =
eKi\ where т е Ж. For each M = \ac bd) e M$ (R), the set of real 2x2 matrices
with positive determinant, the bilinear transformation M(x) is defined by
4.
ex + a
It is easy to see that composition of bilinear transformations is compatible
with matrix multiplication; that is, M{Sx) = (MS)x, for any M,S e M\ (R). For
M e Mj (R), it is well known that M(x) maps Ж onto Ж and R u {oo} onto
itself.
For each real number r, we define the stroke operator of weight r by
(/|(г)М)(т) = (det M)"\M: т)(Мт),
where (M:x) = (ex + d) and the power is determined by taking — n <
arg(M: t) < n. We usually suppress the index and write the left side above as
f\M. This operator satisfies the equality
/IMS = <x(M, S)f\M\S, @.2)

20. Modular Equations of Higher and Composite Degrees 327
which Knopp [1, p. 52] calls the consistency condition, where
a(M, S) = e2nirMM-s\
and where
2nw(M, S) = arg(M: Sx) + arg(S : т) - arg(MS: т).
The value of w(M, S) is either 1, 0, or — 1 and is independent of т. It is not
difficult to see that if either M or S has the form (g bd), with a and d positive,
then w(M, S) = 0. Hence,
f\MS = f\M\S,
a fact that we shall use without comment many times in the sequel.
We shall usually write/|n and/|? as abbreviations for/|(S ?) and/|(JJ °n),
respectively. Similarly, /|" \ is an abbreviation for f\(ac bd).
If A = (" bd) and n is a positive integer, we define
a nb\ fna b \
c/n d) and "A = {c d/n)> m)
which have the properties
frA\n=f\n\A=f\mA I.
и
The modular group ГA) is defined by
@.4)
ГA) = i Г b j 6 M2+ (Z): ad - be = 1
We shall be concerned with certain subgroups of the modular group, namely,
and
J e ГA): Ь = 0(mod n) ^,
Г0(п) = |r J 6 ГA): с = 0(mod n) [¦,
where n is a positive integer. The index of ГB) in ГA) is 6. The subgroups
ГоB), Г°B), and Гв form a conjugacy class in ГA), and their intersection is
ГB). Each of them has index 3 in ГA) and index 2 over ГB). Also,
n

328 20. Modular Equations of Higher and Composite Degrees
where the product is over all primes p dividing n. See the books of Rankin [2,
pp. 26, 29] or Schoeneberg [1, Chap. 4, §§3.2,4.2] for proofs of these facts.
We always denote by Г a subgroup of ГA) with finite index. Such a group
acts on Ж и Q и {oo} by the transformation V{x), for V e Г, and this induces
an equivalence relation; the equivalence classes are called orbits. We call
J^s^uQufooJa fundamental set for Г if it contains one element of each
equivalence class, and 3F n (Q и {oo}), which is always a finite set, is called a
complete set of inequivalent cusps.
A function /: Ж -> # is a modular form if there is a subgroup Г ? ГA) of
finite index, a real number r, and a function v: Г ->• {z e #: \z\ = 1} such that
the following three conditions hold:
(i) / is analytic on Ж.
(ii) f\V = v(V)f,for all Ve Г.
(iii) Let Л е ГA) and U = @ }). Define
N = min{fe > 0: ± A~xUkA e Г}
and
Then there exist an integer m0, complex numbers bm with m > m0, and a
real number к with 0 < к < 1, such that fA has the expansion
in some half-plane {т: Im т > h > 0}.
The weight of / is r and the multiplier system for / is i>. The set of all modular
forms on Г of weight r and multiplier system v is denoted by {Г, r, v). The
positive integer N — N(T; ?) is called the width of Г at the cusp С = A~lao.
The cusp parameter к = к(Г; С) is defined by
If bmo ф 0, then we write
Ordr(/; 0 = m0 + к,
which is called the order of / at С with r'espect to Г. We also write
j/r c\ W0 + K
ord(f;0 = ^-,
which is called the invariant order of/ at ?. For each ze Ж, ord(/; z) denotes
the order of/ at z, as an analytic function of z. The order of/ with respect to
Г is defined by
Ordr(/;z) = Jord(/;z),
where I e {1, 2, 3} is the order of z as a fixed point of Г.

20. Modular Equations of Higher and Composite Degrees 329
We are now in a position to state the valence formula (see Rankin's book
[2, Theorem 4.1.4, p. 98]), which is the most important fact for us as we employ
modular forms to establish theta-function identities. If / e {Г, r, v) and !F is
any fundamental set for Г, then, provided that / is not constant,
? /;z) = rPr, @.5)
where
Рг = Т2(ГA): Г). @.6)
If / e {rlt ru Vl} and g e {T2, r2, v2}, then fg e {Г\ п T2, rt + r2, vtv2}
and f/g e {Г\ пГ2, rx — r2, vl/v2}- Observe that/ + g is a modular form only
if rj = r2, in which case it is a form on the subgroup Г ? Г\ п Г2 determined
by t>i = v2.
If/ e {Г, r, v} and M e Mi (I), then by @.2),
Thus, /|M is a modular form on M~l TM with multiplier system v\M defined
by
{m){V) = «M™~lv)M\{MVM->). @.7)
Let M = C ?) and F = (J bd). Then MFM = ("'F, in the notation @.3).
If/e {Г, r, v}, then Дпт) = n~rl2f\n has multiplier system u|n(F) = v(MV),
by @.7), since a{M, *) = 1. Thus, /(т) and /(ит) are modular forms on
Г n М-1ГМ. In particular, if Г = ГA), then ГA) n МГA)М = Г0(п), and
f(z) and /(ит) are modular forms on Г0(п).
We shall need to determine the multiplier systems for the various theta-
functions that appear in the identities to be proved. To determine these
multiplier systems, it will be convenient to introduce some notation. Let
neg(c) = 1 or 0 according as с < 0 or not, respectively. We define a pair of
symbols closely related to the Legendre-Jacobi symbol (f). Let (c, d) = 1 with
с odd. Define
and for d ф 0,
щг т)
^,,-1Г««..«^. ,0.9)
Finally, define
/ 0 \
-1. @.10)
These symbols possess properties analogous to those of the Legendre-Jacobi

330
20. Modular Equations of Higher and Composite Degrees
symbol, and we summarize those that are needed in the sequel. If с and d are
odd, with (c, d) = 1, then
@.Ц)
Also,
and
сЛ*(с2"
Similar properties hold for (§)„,.
Recall that the Dedekind eta-function r\(i), т e Ж, is defined by ^(т)
e^k/i2^2. q2^ where q = enh (See section 22 of Chapter 16.) Now define
Note that if т is replaced by т + 1, then q is replaced by —q, and if 2т is
substituted for т, # is supplanted by q2. Thus, from Entry 22 of Chapter 16,
we obtain the relations
qlll2f(-q2) = Ч(т),
@.13)
We shall need to determine the multiplier systems for fjt gp and hp 0<j<2.
From the definitions of these functions, it is clear that we should employ the
multiplier system of t](x). From either the books of Knopp [1, p. 51] or
Rademacher [1, p. 163], t](x) is a modular form of weight \ on the full modular
group ГA) with multiplier system i>4 given by
-I e
с
2ni{-3c-bd(c2-l)+c(a+d)}l24.^ if С is Odd,
@.14)
(Recall that the definitions of (§)* and (^^ are given by @.8)-@.10).) Setting

20. Modular Equations of Higher and Composite Degrees
331
Mo = (i °2), Mi = (I I), and M2 = (I ?), we may readily verify that fp gp and
hj are modular forms of weight \ on Г) = ГA) n MflT(\)Mj, 0<j<2. Note
that Го = Г°B), Г\ = Te, and Г2 = ГоB). As observed earlier, Го п Г\ п Г2 =
ГB). Thus, all nine functions are modular forms of weight \ on ГB). For
A - ("c S) e ГB), their multiplier systems are given by
v (
/2r\
— I I е*М-1I4„2т{ас(<12-1)+DЬ-с)<112}124-
and
To prove these formulas, we first note that
rj\2\A = rj\{2)A\2 and ц l
a b/2
2c d
@.15)
Vf (A) = I - 1 eIti('J-1)/4e2^{ac(<J-l)+(b-cM/2}/245 @.16)
@.17)
@.18)
@.19)
@.20)
@.21)
@.22)
@.23)
Thus, @.15) and @.17) follow upon observing that 3(d2 — 1) is a multiple of
24. From the identity
(q; q)Jq4; q4)J-q; -«)» = (<z2; q2)l>
we deduce that
Thus,

332 20. Modular Equations of Higher and Composite Degrees
and @.16) easily follows. Since до(т) = f?{i)lr\(x), ho(r) = г;2(т)//0(т), and so
on, the multiplier systems for g} and hp 0 < j < 2, can readily be derived from
those of fj and r\. Hence, @.18)-@.23) readily follow.
Each of the nine multiplier systems may be written in the form vF =
?o?i ?2, where
and
? _ o2ni<Ha,b,c,d)/*8
S2 ~ e >
where Ф is a polynomial in a, b, c, and d with integral coefficients.
If F(x) is any of the functions f}, gp or /i^-, 0 < j < 2, then, when n is odd,
F(m) is a modular form on
гB)п(о 1ГГB)(о i) = rB)nr°(n)'
as seen after @.7). Since B, n) = 1, the index is multiplicative (Rankin [2,
Theorems 1.4.2,1.4.3 ff]), and so
(ГA): ГB) n Г0(п)) = 6п П f 1 + -Y @.24)
p|» \ P/
Also, from discourse after @.7), the multiplier system of F(m) is (vF\n)(A) =
pF(("U).IfF,n) = 1, then
с
- = nc(mod 48),
n
and since every (nontrivial) term of Ф has exactly one factor of b or c, it follows
that
) U&- @.25)
By @.25), the multiplier system of F(m)/F(t) is equal to
and the multiplier system for F(t)F(nt) equals
Йй+1- @-27)
We now offer a few remarks on orders and poles. The following lemma
allows us to calculate the orders at the cusps of a transformed function f\M.
The conditions permit 00 as a cusp, in the form r/s = 1/0.

20. Modular Equations of Higher and Composite Degrees
333
Lemma 0.1. Suppose r and s are relatively prime integers, M = (" *) e
m = ad — be, and g = {ar + bs, cr + ds). If f is a modular form, then
m
Proof. If A e ГA), then from the definition of order,
ord(/; A~lao) = oxd(f\A~l; oo).
If м = (g J), then Mr takes e2viz into е2«'*/*е2^вт/* ThuS;
ord(/|M;oo) = |ord(/;oo).
@.28)
More generally, if M = (ac J), we choose A = ( ', fir) e ГA) so that A~lao =
r/s. Then
oid(f\M; -J =
1; oo).
@.29)
Observe that the first column of MA*1 is (?+?)• Thus, there exists В е ГA) so
that, for some h,
Hence, by @.28),
oid(f\MA-1; oo) =
0 m/g
b oo) = ord(/|B|Mi; oo)
= -ord(/|B; oo) = ?-oid(f; Boo)
m m
= — ord(/; BM,oo) = — ord(/; MA'1 со)
m m
which, by @.29), completes the proof.
Applying Lemma 0.1 to M = (J °)> (o 2). and (I °) and recalling (e.g., see
Rankin's book [2, Theorem 4.1.2(i)]) that ordfa; ?) = & for each С e Q u {00},
we obtain the following expressions for ord(F; r/s):
F
/0
/1
/2
48 ord(F; r/s)
(r, 2J
(r + s, 2J
(s, 2J
F
do
9\
вг
Table 1
24 ord(F; r/s)
(r, 2J - 1
(r + s, 2J - 1
(s, 2J - 1
F
йо
*i
48 ord(F; r/s)
4-(r,2J
4 - (r + s, 2J
4 - (s, 2J
Thus, for each of the nine functions, ord(F; r/s) > 0 for every r/s e Q u {00}.

334
20. Modular Equations of Higher and Composite Degrees
From the definition of r), it follows that r\{x) has no zeros or poles on Ж.
This and the conclusion above show that if F = fp gp or hp 0 < j < 2, and
MeM2+(Z),
Ordr(F|M;z)>0, zeJfuQufoo},
for any group Г on which F\M is a modular form.
Suppose now that F is a modular form of weight r without poles on
Г = ГB) п Г0(п), for some odd integer n > 3. If the coefficients of q°, g\...,g^
in F are equal to zero, then, since the width of Г at oo is 2, it follows that
Ordr(F; со) > fj, + 1. Suppose further that ц + 1 > rpr. Then
Ordr(F; z) > Ordr(F; со) > ц + 1 > rpr.
@.30)
We conclude that F = 0, for otherwise we have a contradiction to the valence
formula @.5). In our applications, generally F will be a linear combination of
products of functions fp gp and hp 0 < j < 2.
We now obtain what is known as the reciprocal relation by applying a
stroke operator and show how this helps in reducing the amount of computa-
computation needed in proving a modular equation.
Let n be odd. Choose В = (ac bd}e ГA) such that d = 0 (mod n), and set
M = B(g 1). In the present situation, we assume В е ГB), and, to aid in
computation, we require that b = с = 0 (mod 16), (jj) = 1, d = 1 (mod 8), and
ac = cd = bd = 0 (mod 3).
For each modular form F of weight | on ГB) with multiplier system vF,
F(z)\M = vF(B)F\n = n1'4vF(B)F(m)
@.31)
and
F{nx)\M = n~1/4F|n|M = n
an b
с d/n
n 0
0 n
If F = r\, fp gp or fy, 0 < j < 2, then the conditions that we imposed on В
imply that vF(B) = (§)„, and also that
an b
f (с
_ ( (
where c0 = с or 2c and where we have used the facts that (f) = 1 and d/n = n
(mod 8). Hence, by @.31) and @.32),
F(t)|M = n1/4 (j\ F(m), \iF = n, fp gp hp 0
@.33)
and

20. Modular Equations of Higher and Composite Degrees 335
tiF = ti,f1,g0,g1,g2,h1,
@.34)
„-1/4/ и I „1"(и-1
" 111 e
W*
F(m)\M =
since (Ц) = 1. Combining these, we deduce that
iff
F(x)F{nx)\M =
); jf F =
@.35)
and
F(m)
F(t)
M =
@.36)
j ,
ifF=
For n = 11, 19, we shall also be considering д,Bт) and зД2пт), j = 1, 2, in
the sequel. First, by @.33),
a 2b
c/2 d
Second, by @.34),
na b
с d/n
na 2b
c/2 d/n
Hence, if n = 11 or 19 and j = 1 or 2,
= -igj\2(r)gj\2(m).
@.37)
Applying this operator M to an equality involving modular forms yields a
new equality that is either valid or invalid together with the original. This new
equality can also be obtained from the theory of modular equations and is
known as the reciprocal relation. (See Section 24(d) of Chapter 18.)

336 20. Modular Equations of Higher and Composite Degrees
Lemma 0.2. Let F and M be as above, and let Г = ГB) о Г0(п). Then
(X
(i) - = M(oo) is a cusp that is not equivalent to oo modulo Г;
с
(ii) Ordr(F|M; oo) = Ordr( F; -).
Proof of (i). If A e Г and A oo = r/s, then n\s. Since B = {ac "d)e ГA) and n\d,
then n\c. Hence, r/s Ф a/c, and the proof of (i) is complete.
Proof of (ii). Recall that U = (? \). If к е Z+,
if and only if 2n\k. Thus, N(F; а/с) = In. Since M = (™ 5). an application of
Lemma 0.1 with r/s = 1/0 and g = n yields
Ordr(F|M; oo) = N(T; oo) ord(F|M; oo)
This completes the proof of (ii).
As before, suppose that F is a modular form of weight r without poles on
Г = ГB)пГ0(п), where n is an odd integer, n > 3. If Ordr(F; a/c), Ordr(F; oo) >
H + 1 > \rpT, then
? Ordr(F;z)>Ordr(F;oo)-
> 2ц + 2 > rpr. @.38)
By the valence formula @.5), this is a contradiction unless F = 0. In conclu-
conclusion, by Lemma 0.2 and @.38), we only need to show that the coefficients of
q°, q1,..., <?" are equal to 0 in both F and F\M to conclude that F = 0.
In summary, the number of terms that must be computed without using
the reciprocal relation is generally equal to the total number of terms that
must be computed using the reciprocal relation. But in the latter approach,
half are in one identity and half are in the reciprocal identity. This produces
a considerable savings of time, as lower order terms are much easier to
compute than higher order terms. Moreover, if the modular equation, or

20. Modular Equations of Higher and Composite Degrees 337
theta-function equivalent, is self-reciprocal, then the number of terms to be
computed is actually halved.
Another approach, based on the theory of modular forms, to proving
certain identities in Chapters 19 and 20 has been devised by R. J. Evans [1].
His elegant methods are more analytical and less computational than the
methods described in the preceding pages. Instead of working on the subgroup
Г0(п), Evans employs Г°(п). Evans' ideas are especially valuable in proving
identities in which the quotients
/Y — fl2m/p _fll-2rn/p\
G(z)=G B)-=( — 1)татCт^рМ2р2)~ - - - - @39)
appear, where m is a positive integer, p is an odd positive integer (usually a
prime), and q = e2niz.
Let z e Ж and у е <€. The fundamental function employed by Evans is the
classical theta-function
SiG,z)= ? expGciz(n + \f + 2ni(n + ±)(y - ±))
n= —сзо
= -ie^+zl*\e2^qUe-2niyU4)o,, @.40)
by the Jacobi triple product identity.
So that we may eventually relate 5t to modular forms in z of arbitrary
weight, following H. M. Stark [1, Eq. A0)], we set, for u, v e <6 and ze/,
,z) e.
The function q> is analytic in each variable, for z e Ж and u, реЖ Further-
Furthermore, put
F(u,v;z):=ri(z)q>(u,v;z), @.41)
and, when v = 0,
F(«; z) := F(k, 0; z) = r,(z)<p(u, 0; z). @.42)
Combining @.40), @.41), and @.42), we find that
F(u, v; z) = e'tiB("z+t'M1(Mz + v, z)
00
= -i X (-!)" expGiiz(n + м + |J + 7Ш>Bп + « + 1))
^-«)oo(^00. @.43)
In particular, for i; = 0,
F(u; z) = е"йBг51(мг, z) = -i ? (_ ^yn+u+i/2^/2
n= —oo
21)ос(^и)ос(«)оо- @.44)

338 20. Modular Equations of Higher and Composite Degrees
From the series in @.43), it is easily seen that, for arbitrary integers r and s,
F{u + r,v + s;z) = (-enitt)s{-e-Kiv)'{- l)"F(u, »; z) @.45)
and that
F{-u, -v;z)= -F(u,v;z).
In particular, when v = 0,
F(u + l;z)= -F(u;z) @.46)
and
F(-u;z)= -F(u;z). @.47)
By @.44), for fixed ze/, the zeros of F(u; z) are the points и in the lattice
Z + Zz, and these zeros are simple. Thus, F{2u; z)/F(u; z) is an entire func-
function of u. The following lemma shows, in fact, that FBu; z)/F{u; z) is a linear
combination of F(% + u; 3z) and F(j - u; 3z).
Lemma 0.3. For z e Ж and ue%
| = F(i + u; 3z) + F(i - ti; 3z). @.48)
Proof. Replace all the functions in @.48) by their triple products, found in
@.44) above and Entry 22(iii) of Chapter 16. Setting a = q" and simplifying,
we find that
(-aq; q^-a^; q)Ja2q; q2)Ja~2q; q2)Jq; q)w
+ (a3q2; q3Ua-3q; q3)Jq3; qX- @.49)
However, @.49) is just a version of the quintuple product identity. To see this,
apply the Jacobi triple product identity to each theta-function on the left side
of C8.9) in Chapter 16 and then replace z by —am C8.9). This completes the
proof.
We focus on the quotients F{2u; z)/F(u; z) when и is a rational number m/p,
with p odd. Thus, if m and p are integers with p odd and p > 1, define
By @.46), for fixed p and z, G(m; z) depends only on the residue class of m
(mod p), since p is odd. By @.47),
m;z). @.51)
By the product representation of F(u; z) in @.44), G@; z) = 2, and so

20. Modular Equations of Higher and Composite Degrees 339
G(m;z) = 2, ifp|m. @.52)
By the Jacobi triple product identity, @.44), and @.47),
Thus,
_ n
F(u;z) Ч f(-q", -q1'") '
and so
fa a
G(m; pz) = (- i)-g-C»-rt/Brt J11^LJ. @.53)
j \ ч > ч )
This shows that
Gm(z) = G(m;z),
where Gm(z) is given by @.39) and G{m; z) is given by @.50).
We need one additional fact before we prove some theorems about Gm{z).
From @.41), @.14), and Stark's work [1, Eq. A7)],
F(u, v; Vz) = v^V) ^/cV+1 F(uv, vv; z), @.54)
where u, v e % z e Ж, V = (" hd) e ГA), and (uv, vv) is a row vector defined by
(uv, vv) = (m, v)V = (аи + cv, bu + dv). @.55)
Theorem 0.4. Let p be an odd integer exceeding 1, and let sr, /?r A < r < s) be
nonzero integers with
stpl + --- + 8SJSS2 s 0 (mod p). @.56)
Then
g(z) := X П G№; г)?- е {Г», 0,1}, @.57)
vvftere tfte sum is over all m (mod p), and the product is over all r with 1 <, r < s.
Moreover, g(z) has no poles on Ж or at the cusp 0.
Proof. Let V = (" bd)e T°(p). We first prove that g satisfies the transformation
formula
g{Vz) = g(z), z e Ж, @.58)
of a modular function. By @.42), @.50), and @.57),
и* F{2тР>/р> °; F
F(mpr/p,0;Vz)
0 59)
By @.52), the expression within braces in @.59) is to be interpreted as 2 when
p|m/?r. Applying the transformation @.54), with the notation @.55), we find that

340 20. Modular Equations of Higher and Composite Degrees
. @.60)
Since p\b, mprb/p is an integer. Thus, by @.60) and @.45), with r = 0 = v,
where
Er(m) = ( _ l)mft.(«+l+b/j»e3«tabm2#/p2i @.62)
Using @.50), we may rewrite @.61) as
0(Vz) = I П {Er(m)G(mP,a; z))\ @.63)
m r
Now, by @.62),
П Е,{тГ = exp («wife + 1 + b/p) ? erft + ^J^ E еД2) ¦ @.64)
\ P /
The sums ?r er/Jr and ?r ?Д2 clearly have the same parity, and the latter sum
is a multiple of p by @.56). Thus, if a is odd, the right side of @.64) equals 1.
If a is even, then b is odd because ad — be = 1, and so again the right side of
@.64) equals 1. Thus, @.63) reduces to the equality
@.65)
Since ad — be = I and p\b, (a,p) = 1. Thus, am runs through a complete
residue system (mod p) when m does. Thus, @.58) follows from @.65).
If m/p e Z, then G(rn; z) = 2 for all z e Jf, by @.52). If m/p ф Z, then since
m/p is not half of an integer, both FBm/p; z) and F(m/p; z) are analytic
functions of z on Ж that never vanish on Ж by @.44). It follows from the
definition dig in @.57), that g(z) is analytic on Ж. It remains to show that g(z)
is meromorphic at every cusp Loo (L e ГA)) with no pole at the cusp 0.
By @.50), @.54), and @.55), for each integer m and L = (xy g) e ГA),
where, by @.52), the right side of @.66) is to be interpreted as 2 if p\m. By @.43)
and @.66), for each pair of integers m, e, G(m; Lzf has a Fourier expansion of
the form
G(m;LzY= ? ake2ni2"lp, an&<€,
n = N
where N is finite. Thus, g(Lz) also has a Fourier expansion of this form and
therefore is meromorphic at every cusp. This completes the proof that g(z) e
{Г°(р), 0, 1}. It remains to show that g(z) has no pole at the cusp 0.
To prove this, we show that, for each integer m, Gm( — 1/z) has a Fourier

20. Modular Equations of Higher and Composite Degrees 341
expansion of the form
Gm(-Vz)=tcnq\ @.67)
n=0
where
c0 = 2( - l)m cosGim/p) Ф 0. @.68)
If p|m, then @.67) and @.68) are true because Gm{ - 1/z) = 2, by @.52). Suppose
that p \ m. Then by @.66),
Now, from @.43),
q-4*F{0,-m/p;z)=-i ? (_ij»e-««
д = — 00
I (
@.70)
From elementary trigonometry, we see that @.67) and @.68) follow from @.69)
and @.70).
Corollary 0.5. Let pbea prime such that p > 5 and p=\ (mod 4). Let R denote
the set of quadratic residues (mod p) between 1 and p/2. Then
Hz):= П W + (-I)**2-1"8 П GjHz) @.71)
/JeR /JeR
is in {Г°(р), 0, 1} and has no poles on Ж or at the cusp 0.
Proof. In Theorem 0.4, let s = (p- l)/4, let er = 1 A < r < s), and let
plt...,ps denote the members of R. (Note that {± Д.: 1 ^ r < s} is a complete
set of quadratic residues (mod p).) Set В = /?? + h ft2. If 3 is a primitive
root (mod p),
2 I ft2 + 2 X (gftJ = "f m2 = 0 (mod p),
re/S re/S m=l
and therefore S(l + g2) = 0 (mod p). Since p > 5,1 + g2 ф 0 (mod p), and so
В = 0 (mod p). Thus, @.56) holds. For #(z), as defined by @.57),
«Й = Ш G(mftz) = 2(p-1+ X nG,(z)+ X П W
() ()
by @.52), where N is the set of s quadratic nonresidues (mod p) between 0 and
p/2. Therefore,
g(z) = 2<*-^ + Up - 1) ( П W + I G^z)) • @-72)

342 20. Modular Equations of Higher and Composite Degrees
By @.50),
П ад П W - "ff см = (->>'"->» T
fieR fieN m=l
m=l
r(m/p,Z)
since F(w; z) = F(l - м; z) by @.46) and @.47). Using this calculation in @.72),
we deduce that
g{z) = 2*-1 + i(p - l)h(z),
where h(z) is defined by @.71). The result now follows from Theorem 0.4.
Corollary 0.6. Let pbea prime = 1 (mod 4), and let flbe a primitive fourth root
of unity (mod p). Then for e = 1 or — 1,
fe?(z) := ? G%m; z)G°(Pm; z)
m(mod p)
is in {Г°(р), 0, 1} and has no poles on Ж or at the cusp 0.
Proof. Apply Theorem 0.4 with s = 2, e1 = e2 = e,j81 = l, and /J2 = /J.
Corollary 0.7. Let p be odd and greater than 1. Then
g(z):= ? G*m{z)
m(mod p)
is in {Г°(р), 0, 1} and has no poles on Ж or at the cusp 0.
Proof. This follows from Theorem 0.4 with s = 1, et = p, and pi = 1.
If p is an odd prime, Theorem 0.4 provides a method for proving identities
of the type
g(z) = Ш @.73)
where g(z) is given by @.57) and E(z) is a relatively simple function in
{Г°(р), 0, 1}. The idea is to construct a function E(z) e {Г°(р), 0, 1} with no
poles, except possibly at the cusp oo, such that g{z) — ?(z) has a zero at oo.
Then since 0 and oo are the only inequivalent cusps (mod Г°(р)) when p is
prime (Schoeneberg [1, pp. 87-88]), it follows from Theorem 0.4 that g(z) —
E{z) has no poles at all. But the only entire modular functions in {Г°(р), 0,1}
are constants (Rankin [2, p. 108]), and so g(z) — E(z) is a constant, which, of
course, must be zero. Thus, @.73) follows.
To examine g(z) at oo, we need the Fourier expansion of g(z). Thus, we must
determine the Fourier expansion of Gm(z). This can be obtained by utilizing
Lemma 0.3 in conjunction with @.44) and Entry 22(iii) of Chapter 16. Thus,
by Lemma 0.3, which we saw was equivalent to the quintuple product identity,
and @.44),
rj(z){-l)mG(m;z)= f) (_i)»(q3(«+m/P-i/6)^/2 + q3i,-mi,-wn2y @.74)

20. Modular Equations of Higher and Composite Degrees 343
Since G(m; z) = G(p — m; z), by @.51), we may assume that 1 < m < (p — l)/2.
Isolating the terms in @.74) with n = 0, +1, we find that
+ т/р_ (p-2m)/p _ (p+3m)/p _ Bp-3m)/p
@.75)
By Entry 22(iii) in Chapter 16,
ф) = ?1/24A - q - q2 + 0(q5)). @.76)
Thus, by @.75) and @.76), for 1 < m < {p - l)/2,
1 _|. nmlP _ д(р-2т)/р _ _(p+3m)/p
_ _Bp-3m)/p _|_ _ _|_ q(p+m)lp _ _Bp-2m)/p _ _Cp-3m)/p
_ qCmi-mp)H2pi)^ + qm!p _ фр-2т)/р + 0(q2lt>)}. @.77)
To first illustrate the usefulness of the theory developed above, we give a
second proof of A8.2) in Entry 18(i) of Chapter 19. Recall that our first proof
was laboriously tedious. We reformulate A8.2) in terms of Gm(z) and r\(z) before
proving it.
Theorem 0.8 (A8.2) of Chapter 19). // p = 7, then
Gliz) + Gl(z) + Gl{z) = -57 - 14№ф J - №?) • @-78)
Proof. With
g{z):= t GHz),
m=0
we see that @.78) is equivalent to
@.79)
By Corollary 0.7, the left side of @.79) belongs to {Г°G), 0,1} and has no poles
on Ж or at the cusp 0.
Now, for V = (ac bd)e T°(p) and p = 1 (mod 6),
a b/p\ . 4
pc d J
+ dfn\zlp).
Thus, for p = 1 (mod 6),
In particular, for p = 7, if E(z) denotes the right side of @.79), then ?(z) is in
{Г°G), 0,1} and has no poles except at oo.

344 20. Modular Equations of Higher and Composite Degrees
Since both sides of @.79) are in {Г°G), О, 1}, by the procedure described
above, @.79) will follow if g(z) - E(z) has a zero at oo. By @.76),
E(z) = -2q~2n - 12<T1/7 + 86 + O{qw). @.80)
By @.77),
Gl(z) = -q-2n{l + q111 + Otfl1)}1, @.81)
G27(z) = <T1/7{l + Ofo2'7)}7, @.82)
and
Gl(z) = O(q3'7). @.83)
Thus, using @.81)-@.83), we find that
g(z) = -2q-2n - Uq-1'1 + 86 + 0{qin). @.84)
Hence, by @.80) and @.84), g{z) — E(z) has a zero at oo, and the proof is
complete.
We close this long introduction to Chapter 20 by proving a generalization
of Theorem 12.1 in Section 12 of Chapter 19.
Theorem 0.9. For each odd integer p > 1,
E G(m;z) = 2^)^, @.85)
mG^dp) \PJ Ф)
where (J) denotes the Legendre symbol.
Proof. By @.74),
4B) X G(m; z) = ? ? (- 1
m m n—— oo
= 2 У (-1)
7=-oo
However, from Entry 22(iii) of Chapter 16,
4(z)= f (_1)^з(л+1/бЯ/2 (Q86)
n=-oo
It therefore remains to show that
? (-l)Vo+weI/2 = f-) ? (-l)V("+1/6>2/2. @.87)
У= -oo XPJ n= -oo
This is easily verified in the cases p = +1 (mod 12) and p = ±5 (mod 12),
where I - 1 = 1 and — 1, respectively. Suppose that 3|p, so that (-1 = 0. Then

20. Modular Equations of Higher and Composite Degrees 345
the left side of @.87) equals
?
j=-a>
Replacing j by -j - 1, we see that S = - S, that is, S ~ 0. Hence, @.87) holds,
and the proof is complete.
Recall that the functions %(q), ij/(q), <p(q), and /(—q) are defined in Entry 22
of Chapter 16.
Entry 1. Define vby v:= qll3x(~4)/x3(-'43)- The following equalities are then
valid:
n _
1 + Г/3,7 зч = И + -т?7~Т\ '
and
1/3
Л1'3
cos 40° + cos 80° = cos 20°,
and
1 1 1
A L g-
cos 40° cos 80° cos 20
(iv) 3 + 4Ц

346 20. Modular Equations of Higher and Composite Degrees
and
(v)
„=o\l— q l — q
The introduction of the continued fraction in (i) appears adventitious, for
it seems to have no intrinsic connection with the remainder of the results in
Entry 1. Entry l(i) was communicated by Ramanujan [10, p. xxviii] in his
second letter to Hardy.
The first equalities of (ii) and (iii) are, respectively, B4.28) and B4.29) in
Chapter 18. A. J. Biagioli [2] and J. M. and P. B. Borwein [2, pp. 142-144]
have given proofs of these interesting identities. The proofs provided below
are in the spirit of much of the material of the present book.
We are unable to relate the trigonometric equalities in (iii) to any other
material in Entry 1.
Proof of (i). By B2.4), Entry 24(iii), Entry 19, and Corollary (ii) of Entry 31,
all in Chapter 16,
W) ql
f(q, q2) Ш'3)
qll31t(q3)
- 1. A.1)
Thus, the second equality of (i) is apparent.
In the result just proved, replace q113 by coq1'3 and w2q113 in turn, where ш
is a primitive cube root of unity. Note that v is replaced by cov and co2v,
respectively. Multiplying these three equalities together, we find that
Now using the product representation for ф given in Entry 22(ii) of Chapter
16, we find that A.2) becomes
nL (q2; q2)l (q2; q2Uq3; g6), 1 _ ФЧд)
" qiqj q )oo (?> q )aa{q ¦> q )» Ф iq ) чФ \Ч >
This establishes the third part of (i).
Lastly, Watson [4] has shown that the continued fraction of (i) is equal to
qll3^(q3)/f(q, q2). By A.1), the truth of the first equality of (i) is evinced.
This continued fraction also appears in the third notebook [9, p. 373] in
the form
f(-q,-q5) 1 q + q2 q2 + q4
f(-q3,-q3) 1+ 1 + 1 +
A.3)

20. Modular Equations of Higher and Composite Degrees 347
which, by the Jacobi triple product identity, is readily seen to be equivalent
to the formulation in (i). Proofs of A.3) have also been found by A. Selberg
[1, p. 19], [2, p. 17], Gordon [2], Andrews [1], and Hirschhorn [6]. See also
Ramanathan's papers [1], [4]. The convergence of A.3) on \q\ = 1 has been
thoroughly examined by L.-C. Zhang [1]. In fact, Ramanujan has recorded a
considerable generalization of A.3) in his "lost notebook" [11]. For proofs
and discussion of this more general theorem, see papers by Andrews [10],
Hirschhorn [3], and Ramanathan [4].
Proof of (ii). The last two equalities of part (i) yield two expressions for v, and
when these are equated and the sign of q is changed, we obtain the first equality
of(ii).
From A.1) above, B2.3) of Chapter 16, and Example (v) and Corollary (i),
both in Section 31 of Chapter 16, we find that
_ 2дх(д)Ф(д) _ 2д113П-д, -q5) _
V C) C)
(P(~q3) (P(~q3) '
and so the second part of (ii) is established.
The third equality of (ii) follows from the first equality of (ii) by elementary
algebra as follows: replace q by q3, cube both sides, cancel 1 on each side,
multiply both sides by 9qA^iA(—q9)l^iA(—q), add 1 to both sides, and finally
take the cube root of each side.
Proof of (iii). In Section 24 (vii) of Chapter 18, we showed the equivalence of
the first equalities of (ii) and (iii).
Alternatively, a proof dependent on the second equality of (ii) can be given.
In the aforementioned equality, replace qil3 by coq113 and co2q113 in turn, where
со is a primitive cube root of unity. Multiplying the three equalities together,
we deduce that
з _ qK^XaVXcoy) _ q>\-q)
q3{-q3) cp*(-q3y
where the last equality is readily ascertained from the product representation
of (p given in Entry 22(i) or B2.4) of Chapter 16. Hence,
Equating this expression for — 2v with that from part (ii) and changing the
sign of q, we deduce the first equality of (iii).
The second equality of (iii) follows from the first by very elementary algebra,
completely analogous to that outlined in the proof of (ii).
To prove the first trigonometric equality, we employ the elementary
identity
cos 30 = 4 cos30 — 3 cos в.
By taking в = 20°, 40°, and 80° in turn, we see that -cos 20°, cos 40°, and

348 20. Modular Equations of Higher and Composite Degrees
cos 80° are the three roots of
4x3 - 3x + \ = 0.
The first trigonometric equality is now evident.
Replacing x by 1/x above, we deduce that the reciprocals of —cos 20°,
cos 40°, and cos 80° are the roots of
x3 - 6x2 + 8 = 0,
and the second trigonometric equality follows at once.
Proof of (iv). To obtain one part of (iv), first observe that, by Entry 24(ii) in
Chapter 16 and parts (i) and (ii) above,
/ 1\_ 2 1
3)~ V «V v v
which gives part of (iv).
Now replace q113 by coq113 and co2q113 in turn, where со is a primitive cube
root of unity. Observe that v is replaced by cov and co2v, respectively. Multiply-
Multiplying these three equalities together and using the product representation for /
found in Entry 22(iii) of Chapter 16, we deduce that
= 4»2 + -| -27.
This easily yields the second equality of (iv).
The last equality of (iv) follows from elementary algebra. With q replaced
by q3, cube the first equality of (iv). Cancel 27 and multiply both sides by
2744/12(-<?9)//12(-4)- Add 1 to both sides and take the cube root of each
side to complete the proof.
Proof of (v). We employ the series representation for/3(—q1'3) given in Entry
24(ii) of Chapter 16 and separate the terms into three subsets according to the
residuacity of the index modulo 3. Hence,
B=l B=l
n=0
= ~3<?1/3/3(-43)+ /(-<?) +6 ? (-l)»nq»<3»+1>/2
П = 1
CO
~T О Ул \ "^/ ^Q 9
n=l
where we have utilized once again Entry 24(ii) as well as the series representa-

20. Modular Equations of Higher and Composite Degrees 349
tion for/(—q) given in Entry 22(iii) of Chapter 16. Therefore,
= 6 J (-
= 61 у{-ф,-щ2
-«/«. -<^q2)~LogД~ф, -щ2)
|{LogD/a; q3)M + Log(a«?2; )
where we have employed the Jacobi triple product identity. Identity (v) is now
obvious.
Entry 2. We have
(i) q>(q)(p(q9) - q>2(q3) =
(iii) (p(q)(p(q9) + cp2(q3) =
(v) Л-в1/3) =/(-«*, ~q5) ~ qU3fi-q\ -q1) ~ q2l3f{-4, ~q*l
(Vi) A-q, -q*)f(-q2, -f)f{-q\ -q^J-^j^l,
Я-q, -qs)
(ix) ф(^/9) - cp(q) =

350 20. Modular Equations of Higher and Composite Degrees
Proof of (i). By Corollary (i) and Example (v), both in Section 31 of Chapter
16,
= 2ql'3f{q, q5)q>(q3)
B-1)
where we have used the equality \\i(—q)q>{q) = \p{q)q>(—q2), which is easily
deducible from Entries 25(iii), (iv) in Chapter 16. Using again the same two
facts from Section 31 of Chapter 16, we arrive at
{(p(qll3)-(p(q3)}(p(q3)
2q2'3f(-q2, -q10)}X(q)
B.2)
Next, by Entries 11 (i), (iii) and 12(v), (vii) in Chapter 17, if 0 is of the third
degree in a,
W8 <P4q) t
cp\q3) \a) cp\q3) ''
by Entry 5 (iii) in Chapter 19. Substituting in B.2), we deduce that
(p(qll3)(p(q3) = 2qllMq3)(P(-q2l3)x(q) + <p2(q).
Replacing q by q3, we complete the proof of (i).
Proof of (ii). Employing Entries l(i), (ii) above and Entry 24(iii) of Chapter
16, we find that
vx3(-q3) xi-q)
where we lastly used the definition of v. Replacing q by q3, we complete the
proof of (ii).
Proof of (iii). By Entry 19, B2.3), and Entry 24(iii), all in Chapter 16,
f( 2ч _ (-<?;<?U<?3;<?3)oo _ x(-q3)A-q3) _ v(-g3) n ~
тЧ)~ (-q3;q3)« " X(-q) ~^P?' ("}
Hence, by B.1), Corollary (ii) of Section 31 in Chapter 16, and B.3),
<p(qll3)<p(q3) ~ <P2(q3) = 2qll3il/(q3M-q6Mq)
. B.4)

20. Modular Equations of Higher and Composite Degrees 351
Next, by Entries 10(i)-(iii) and 12(v), (vi) in Chapter 17 and Entries 5(i), (Hi)
in Chapter 19,
(-«)
Substituting this in B.4), we deduce that
(p(qil3Mq3) = 2Mq1'3)(p(-q6Mq) - q>2(q).
Replacing q by q3 completes the proof.
Proof of (iv). We apply Entry 31 in Chapter 16 with a = 1, b = q, and n = 9.
Using also Entry 18 (ii) in Chapter 16, we find that
Ф(Ч) = Я<736, q*5) + qf(q21, Q5*) + «W, «") + qW, Q12) + q
Replacing q by q1'9 and then utilizing Corollary (ii) in Section 31 of Chapter
16, we complete the proof of (iv).
Proof of (v). In Entry 31 of Chapter 16, set a = —q,b= —q2, and n = 3.
Hence,
f(-q) = f(-q12, -q15)~qf(-q6, ~q21) - q2f(~q\ ~q24)-
Replacing q by q1'3, we complete the proof.
Proof of (vi). Set n = 4 in B8.1) of Chapter 16.
Proofs of (vii), (viii). With A = f{-qA, -q5), В = f{-q2, -q1), and С =
/( — q, —q8), we write (v) in the form
which when cubed yields
/3(-y/3) = A3 - B3q - C3q2 + 6ABCq - 3(A2B + B2Cq - AC2q)q1/3
- 3(A2C - AB2 + BC2q)q213.
Comparing this with Entry l(v), we deduce that
A2C - AB2 + BC2q = 0
and
А2В + B2Cq - AC2q = f3(-q3),
which, with the use of (vi), immediately yield (vii) and (viii), respectively.
Alternatively, (vii) is a corollary of Theorem 0.9.
Proof of (ix). Apply Entry 31 of Chapter 16 with a = b = q and n = 9. Hence,
<p(q) = <P(qS1) + 2qf(q63, q") + 2qAf(qA\ q111)

352
20. Modular Equations of Higher and Composite Degrees
Replacing q by q1'9 and using Corollary (i) in Section 31 of Chapter 16, we
complete the proof.
Using Entries 2(v), (vi), (viii) and Theorem 0.9, Evans [1, Theorem 7.2] has
established the following beautiful identity. For p = 9,
>?(z/3){3>?3(z/3) + >?3(z/27)}
+ G2(z)Gl(z) + GA(z)G2(z) = 6 -
Entry 3. Let P and у be of the third and ninth degrees, respectively, with respect
to a. Let m = z1/z3 and tri = z3/zg. Then the following modular equations are
valid:
0)
(ii)
(iii)
(v)
(vi)
(vii)
(viii)
/mm
, V3M _ yK\ 1/24
3y3(l - aK(l - yf V'2*
_ /a3y3(l - aK(l - y
{ fP{\№
(ayI'2 + {A - a)(l - y)}1'2 + 2D^A - P)}113
{<x(l-y)}1/8 + {y(l-a)}1/8 =
f(l - P)
1-a / _ _ V a
3\l/4
(x)
(xi)
^«2\l/4
(xii) [^-| +
a(l - a)V/8 _ 3
3?)/
1/4
A - a)(l - y)J Va?(l - a)(l - y\
-'s-

20. Modular Equations of Higher and Composite Degrees 353
m>
(xiv) . n - '\;; г'\—~щ lm
{a(l - y)}1/8 - {y(l - a)}1'8
and
m'
(xvi) 1 = (ayI'2 + {A - a)(l - y)}1'2 + 2D)8A - fi)}1'3 —.
m
It is not clear why (vii)-(ix) are placed here, because they are third degree
modular equations. For several other modular equations of degree 3, see
Section 5 of Chapter 19.
Proofs of (i)-(iii). In order to prove these formulas, we first need to express
a(l — a), /?A — P), and y(l — y) as rational functions of a parameter t.
Let q be defined by E.12) in Chapter 19. Thus, by E.4) and Entry 5(xv) in
Chapter 19,
C.1)
C.2)
Analogously, let q' connect ft and y. Then
m' = У1 + Aq', C.3)
and y{l-y) = q'irT^)- C-4)
Trivially, C.2) and C.4) indicate that
Regarding this as an equation to determine q as a function of q', we see that
one solution is q = B — q')/(l + Aq'). However, because q = 0(a) and q' =
0(P) = 0(a3) as a tends to 0, this solution must be irrelevant. Dividing out
this root in the quartic equation C.5), we find that the relevant solution is a
root of
9q' 2 9q'B - q') q'B - q'J
l+Aq'4 (l+Aq1J4 A + Aq'J
Setting q = x + 3<j'/(l + Aq'), we transform this equation into
з Wit + У 2q'Bq'3 + 6q'2 + 33q' + 2) _ л
X (l+Aq'J* (l+Aq'K

354 20. Modular Equations of Higher and Composite Degrees
This cubic equation may be solved by Tartaglia's method (Hall and Knight
[1, p. 480]). Omitting the details, we find that the roots of the original cubic
equation are
q = rrv ' C6)
where со is any cube root of unity. Of course, the relevant root is the real one.
Moreover, q is a rational function of (q1I13.
Accordingly, if we set q' = 2t3, then a(l — a), /?A — /?), and y(l — y) are
expressible as rational functions of t. Thus, from C.6) and C.1)—C.4), we
deduce that
ч =
2t(l + t + t2)
1 -2t
t2
( 1 _ t \8 1 — t3
oe(l-oe)=16t —- —-3, C.7)
yl т" ZC/ 1 т ot
/J(l -P)= 16t3 ( / ~ l з ) , C.8)
\1 + 8t J
y{\ -y)= 16(91 ~13), C^)
П + 2rt4
and
m'2 = l+8t3. C.11)
C.12)
C.13)
Moreover,
and
from C
¦7)-C.9),
a3(l
-аK
-P)
?3A-
P(l-
УK
P)
% + 2t)
- 256t24.
It is now a simple matter to establish (i)-(iii). By C.10)—C.12),
M f 3 3
Ml-q)y_
4 \PAP)J l+l
P(\-P) ) 1 + 2t 1 + It
which is (i). By C.10), C.11), and C.13),
1/24
= 1 + 2t = Jmm ,

20. Modular Equations of Higher and Composite Degrees 355
which establishes (ii). By C.1O)-C.13),
i 94/3 [ a и1 ~ 44 ~ /) \ i — 2t + 4t
1 + 2t
and so (iii) is proved.
J. M. and P. B. Borwein [2, pp. 142-144] have also given a proof of (i).
Proof of (iv). Applying Entries 10(i), (iii), ll(i), and 12(v) in Chapter 17 to
Entry 2(i), we readily find that it translates into the first equality of (iv).
Similarly, the same entries in Chapter 17 can be invoked to translate Entry
2(iii) into the second equality of (iv).
Proof of (v). By E.2), E.3), and E.5) in Chapter 19,
+ 3J(m-lJrn'(m'-l) (m+lJC-mJ
+
16m2 3 + m' 16m2 3-m'
m'
16m2(9-m'2)'
m'
2m2(9-m'2)'
We now substitute C.10) and C.11) into the foregoing equality, and, after
considerable simplification, we deduce that
(ayI'2 + {A - a)(l - y)}1'2
_ 1 - At + At2 + 8t4 _ y _ 8t(l - t3)
A + 202 A + 202
_ 8t(l - t3) 32t2(l - 0A - t3)
1 + 8t3 + A + 20A + 8t3)
= 1 - 2D0A - 0)}1/3 + 8@A - 0)}1/4{ay(l - a)(l - y)}1/8, C.14)
where we have employed C.7)—C.9). Thus, (v) is evident.
Proof of (vi). Using E.1) in Chapter 19, then C.10) and C.11), and then finally
C.8), we find that
a(l - y) Y/8 , / y(l - a) Y/8 4m' 4m'
- PKJ (m - 1)C - m') T (m + 1)C + m')

356
20. Modular Equations of Higher and Composite Degrees
8m
(m2-
1 +
2t(l-
'(m'
-1H
8Г3
-t3)
1
+ 3m)
) - m'2)
C.15)
from which (vi) follows.
Proof of (vii). By E.1) in Chapter 19,
1-a
= m.
1 - {A - a)(l - Ю}1/4 l (m + 1)C - m)
4m
The proof of the second equality is similar.
Proof of (viii). By E.1) in Chapter 19,
1
while
{A - «)A -
m / C + mJ(m - IJ (m + 1JC - mJV'2 m2 + 3
16m2
Hence, (viii) follows.
Proof of (ix). This formula is simply the reciprocal of (viii), in the sense of
Entry 24(v) of Chapter 18.
Proof of (x). By (vi) and C.7)-C.11),
_ B56№ - ^Y/24
1-t 1-t v
/mm',
/yd- y)V/8
V«(i - «)/
from which (x) is apparent.
Proof of (xi). This is the reciprocal of (x).
Proof of (xii). Employing once again E.1) of Chapter 19 and C.7)—C.11), we
find that

20. Modular Equations of Higher and Composite Degrees 357
1'4 ( (i - jg>2 y/4
.ay) + V(l - a)(l - y))
1/4
- IJ m'(m + IJ
(и' - 1)C + m') (m' + 1)C - m')
Am'2(m2 + 1) - 4mm'C - w'2)
: (m'2 - 1)(9 - m'2)
l + 2t
t(i - о
A + 20A + t + t2) 3A + 202
t{\ - f)(l -2t + At2) 1 + St3
- y)J m'
which completes the proof of (xii).
Proof of (xiii). By C.16) and C.7)-C.11),
((I - «)A - y)y» _ /«y(l - a
V p (.1 ~
- fd — 0A — 2f + 4t2) J 1 + 2t _ ")
A + 20A + t + t2) jt(l - 0 ~ J
ay
_ 1 - It + At2 _ m'
~ 1 +2t ~ m'
Proof of (xiv). As in C.15),
<x(l - y) V/8 / УA - «) Y/8 Am' Am'
- m') (m + 1)C + m')
Sm'(mm' + 3)
(m2 - 1)(9 - m'2)
m'(l -2t + At2)
2t(l - t3)
1 + 8t3
2t(l -t3)\J m
1 Im'

358 20. Modular Equations of Higher and Composite Degrees
where we have utilized C.10), C.11), and lastly C.8). Inverting the last equality
and dividing both sides by {^j8(l - /?)}1/3, we complete the proof.
Proof of (xv). By E.3) in Chapter 19,
r«/»W* - (в W* = {m ~ 1)C + Ш) - K ~ 1)C + m>) = {m ~ m')[mm' + 3)
1 P> KPV Am Am' Amm'
by E.1) in Chapter 19,
m')(m'+l) C-m)(m + l)
{A _/J)(l-у)}1'*-{A-ое)A-/»)}''* =
Am' Am
(m — m')(mm' + 3)
Amm'
and by foregoing expressions,
(m — m')(mm' + 3)
Amm'
From these formulas it is evident that
- № ~ 7)}1/4 - {A - «Ш - i
Dividing both sides by /?A — /?), we obtain (xv).
Proof of (xvi). By C.14), C.10), C.11), and C.8),
(а?I/2 + {A_а)A_?)}1/2=1_8<A-'3)
~ 1 + 8tJ m
which establishes (xvi).
Entry 4. We have
W (p(-q2) + q' ¦'¦'-'* ~' -^' =
A +
8t(l
1 +
ItI
~t3)
8t3
m!
m
Pn
i^m'
m
2(№Ш) , ^(^(g),,,

20. Modular Equations of Higher and Composite Degrees 359
and
(iv) q>{q)q>{q21) - q>(-q)(p(-q21) = Mf(-q6)f(-q18) + 4<?W)</'(<754)-
Proofs of (i), (ii). If we transcribe the proposed equalities via Entries 10(iii)
and ll(i), (ii) in Chapter 17, we obtain Entries 3(x), (xi), respectively, and thus
(i) and (ii) are established.
Proof of (iii). Employing, in turn, Corollary (i), (ii) in Section 31 of Chapter
16, Entry 4(i), Example (v) in Section 31 of Chapter 16, B.3), and Entry 24(iii)
in Chapter 16, we find that
<р(-д2М-д5*) 2ШдЩд") Ф(-дЩ-д27)\
<p(-g6M-g18) q \Ф(д3Жд9) Ф(-д3Ж~д9))
_ (ф(-<?18) - 2д2П-д6, -д
<p(-g6M-q18)
2Щ{д\д6)+дФ{д9Щд21) {/(-<z3, -д6)-дФ{-д9)}Ф{-д21)\ ,
q \ Ф(д3Жд9) Ф{-д3)Ф(-д9) )
2q2f(-q6, -д*>)ср(-д") д2/(д\дбщдп) q*f(-q\
ср(-д6Ы-д18) ^ Ф(д3Жд9) ф(-д3Ж~д21)
2BФ(д1аМ-д6М-д5*) (Р(-д9Жд21) я>(д9Ж-д21)
д2 /2ф(д18М-д5*) (р(-д9Жд21) (р(д9Ж~д21)
(
П-д6А ф(-?18) Ф(д9) Ф(-д9)
Now by Entries 10(i)-(iii) and 11 (i)—(iii) in Chapter 17,
2ф(д2М-д6) д>{-дЖд3)
(p(-g2) Ф(д) Ф(-д)
/d-aKV'8
by Entry 5(i) of Chapter 19. Hence, the far right side of D.1) is also equal to
0, and so the proof of (iii) is complete.
Proof of (iv). We employ in turn the following results from Chapter 16:
Corollary (i) in Section 31; C6.2) with A = В = I, fi = 2, v = l, and g replaced
by q9; C6.1) with A = 1, В = д6, ц = 2, v = 1, and g replaced by g9; Entry
18(iv) three times with n = 1; Entry 31 with a = ±q, b = +q2, and n = 2;
C6.2) with A = q9, В = q~3, ц = 2, v = 1, and q replaced by q9; Entry 18(iv)
three times with и = 1; and Corollary (ii) in Section 31. Hence,
(p(gMq21)-(p(-g)(p(-g27)
Mg27) - (p(-g9M-g21) + 2q{f(q\ q15)q>{q27)
f(-g\-gl5M-g21)}

360 20. Modular Equations of Higher and Composite Degrees
= 2{q27f(q5\ q162)f(q90, q~18) + qll"f{q210,q-^)f(q126, <T54)}
+ 4q{f(q90, q126)f(q30, q*2) + q24f(q18, q198)f(q6, q66)}
0, qi26) + q18f(q18, q198)} {f(q30, q42) + q6f(q6, q66)}
+ {f(q90, q126) ~ q18f(q18, 4198)} {f(q30, ?2) ~ q6f(q6, <?66)})
4q9t(q5*Wq18) + 2q{f(q18, q36)f(q6, q12)
+ A-q18,-q36)A-q6,-q12)}
4q9f(q5\ q162)f(q18, q54) + 4qf(-q18, -q36)f(-q6, -q12)
+ 2q{f{qlb,q36)f{q6,q12)- f(~q18, ~q36)f(-q6, ~q12)}
2, qS*)f(q102, q~3°) + q123f(q210, q^fiq138, q~66)}
q162){f(q30, q*2) + q6f(q6, <z66)}
) + 4qf(-q18)f(-q6) +
This completes the proof of Entry 4(iv).
Entry 5. Let a, /?, y, and д be of the first, third, ninth, and twenty-seventh degrees,
respectively. Let m be the multiplier connecting a and ft, and let m" denote the
multiplier associated with у and д. Then
n W A1 - «m - *)V/8 + HQ - «)(i - flV/8 _ К
W
W va - «a - ?)/ w-w-
| /j8yV/8 /A-/?)A-у)У/81 _ m
1 W +\A-*)A-S)) J m"'
and
{16jgy(l -
m
Proof of (i). If we utilize Entries 10(iii) and ll(i), (ii) of Chapter 17 in Entry
4(iii), we obtain the desired result immediately.

20. Modular Equations of Higher and Composite Degrees 361
Proof of (iii). Employing Entries 10(i),(ii), 11 (iii), and 12(iii) of Chapter 17in
Entry 4(iv), we obtain (iii) with no difficulty.
Proof of (ii). Since 5 has degree 3 over y, we proceed as in Section 3 and (see
C.11)) define a parameter u by
m" = 1 + 8u3. E.1)
Thus, we deduce analogues of C.7)—C.9), namely,
i^, E.2)
and
<5A — д) = 16u91 з I. E.4)
Let m! be as in Section 3. It follows from C.10) and C.11) that
1 + 8t3 = m'2 = " , E.5)
1 + 8u
from which we readily deduce that
f3(l - 2u + 4u2) = u(l + и + и2). E.6)
From C.7), C.9), E.2), and E.4),
E.8)
1 - y) P(l -
while from C.10), C.11), E.1), and E.5),
Im" Im' m" 1 + 2u
\j m \j m m! 1 + 2t'
Thus, from Entry 5(i), E.7), and E.8),
\Py) + \A - P)(l - y)) = l + 2t ~ t(l + 2t)(l - u)
= lll++22%~-UuY E9)
Next, by E.7)-E.9),
/ CCO \ ffl I CtO A — Oi) A
\гУ/ wi Vpy(i — p)i\ — y)j
_ (l + 2MJ(t-uJ ^u(l + 2u)(l-t) ^/1 + 2гуц2A+2цJA-р2
t2(l + 2f) A-й) t(H-2t)(l-u) \l+2u/ t2(l + 2t) A-й)

362 20. Modular Equations of Higher and Composite Degrees
+ 4(t - u){t3(l -2u + Аи2) -{u + u2 + u3)})
= 2(l+2u) Ш_
1 + It V m
where we have omitted a heavy amount of tedious elementary algebra to
obtain the penultimate line, where E.6) was used in the penultimate step, and
where E.8) was employed at the end. Multiplying the extremal sides of E.10)
by (/Jy(l - /J)(l - y)/aS(l - a)(l - <5)I/4, we arrive at
- 7)
\ - d)J \aS{l - a)(l - 5\
_ p(Py
1/4
( ъ
m \aS(l - a)(l - <5)/ и""
Lastly, if we substitute the formula for yjm"/m, given in (i), into the left side
above, we obtain (ii).
Proof of (iv). Using C.7)-C.9) and E.2)-E.4), we observe that
- y)}1/24
1+256
УA ~ У)
Ki-y) PiX-P)
1 -t
m
1-й 1 + It V m '
1 + 2ТТ2и*
where we have used E.8). Thus, the truth of (iv) is manifest.
Entry 6. We have
(i) *{qmi) ~ q15n W1) = AS, q6) + qinif(q\ q1) + <?3/17(<Z3,«
(ii)
+ 2^11/(95,917)
+ 2q25lllf(q,q21),

20. Modular Equations of Higher and Composite Degrees 363
and
f(~q, ~
Proof of (i). In Entry 31 of Chapter 16, set a = 1, b = q, and и = 11. Using
Entry 18 (ii) of Chapter 16, we obtain the equality
№ = f(q55, q66) + qf(q", q11) + q3f(q33, <z88)
+ q6f(q22, q") + «10/(«n, q110) + ql*W21\
If we replace q by q1/l \ we complete the proof.
Proof of (ii). Putting a = b = q and n = 11 in Entry 31 of Chapter 16, we find
that
cp(q) = cp(q121) + 2qf(q", q1*3) + V/(«". Я™5) + 2q9f(q55, q181)
Replacing q by q1111, we achieve the desired result.
Proof of (Hi). Set n = 11 in A2.26) of Chapter 19.
Entry 7. The following are modular equations of degree 11:
(i) (a/?I'4 + {A - a)(l - P)YIA + 2{16aj?(l - «)A - P)V<12 = U
(ii) m - H = Щару* - {A _ a)(l -
- = 2,/2B + (a^I'4 + {A - a)(l -
m
(iv) Д1-/?K\1/8 /П1/8
1 - a / \ a У V a(l - a)
— wVX1/8
11

364 20. Modular Equations of Higher and Composite Degrees
a(l-a)
= 2(WI'2 - {A -
and
{A - a)(l -
These modular equations are followed by two further modular equations
of degree 11, designated by (viii) and (ix). However, Ramanujan (p. 244) has
crossed them out.
The first modular equation of degree 11 to appear in the literature was
established by Sohncke [1], [2]. The modular equation (i) is due, in 1858, to
Schroter [3], [4], who earlier [1], [2] had established slightly more compli-
complicated modular equations of degree 11. The remaining six modular equations
in Entry 7 are due to Ramanujan. More complex modular equations of degree
11 were discovered by Schlafli [1], Fiedler [1], Fricke [1], and Russell [1], [2].
Because of its extensive use throughout the sequel, we record here the
equality
f(a, b) = f(a3b, ab3) + af(^, ?a4b4j, G.1)
which can be deduced either by adding Entries 30(ii), (iii) or by applying Entry
31 with и = 2, where the cited entries are in Chapter 16.
Proof of (i). Setting A = В = 1, fi = 6, and v = 5 in C6.2) of Chapter 16, we
find that
<p(Q)(p(q)-<p(-QM-q)
5
в=0
where Q = q11. Next, in C6.10) of Chapter 16, replace q by q2 and set /i = 6
and v = 5. Thus,
t
в=0
\ t 22+20n
\ t 9/ e20n, q22+20n). G.3)
The last equality can be demonstrated by showing that the terms of index n
and 5 — n, 0 < n < 2, are equal. In order to do this, we apply Entry 18(iv) of

20. Modular Equations of Higher and Composite Degrees 365
Chapter 16 twice to deduce that
-4И> Ql4+4B)
and
/Y_-98 + 20b 122-20n\ _ 15(-98+20b)+10A22-20b)/Y 2-20я 22 + 20вч
Multiplying G.3) by 4q3 and subtracting the result from G.2), we deduce that
<P(QMQ) ~ <p(-QM~q) - 4«V
_2 V д12п2+22в+11/-/л10-4в 0l4+4n\ I f/g-lO-lOn _34+20я\
л=0
We next apply G.1) with a = -q'8'10- and b = -<?14+1Ои. The expression
within curly brackets above is thus found to equal /(—q~s~iOn, — q14+1On).
Hence,
= 2 X (?12п* + 22Я+11дд10-4»)?>14+4п)Д_4-8-10») _914+10«) G4)
в=0
Replacing n by n + 3 in the last three summands above, we find, after an
application of Entry 18(iv) in Chapter 16, that
_12<n+3J+22(n + 3)+ll/Y0lO-4(n+3) /}14+4(в+3)\ /Y_ --8-10(в+3) _ 14+10(n+3)\
_ _12п2 + 94и+185/-/л-2-4я q26+4b\/_ -38-10и\15/_ 44+10и\10
x/(-<r8-10",-<z14+10B)
_ _„12я2+22в+11л4+2в/-/л-2-4в q26+4b\ ft _ _-8-10в _„14+10в\
Thus, the right side of G.4) may be rewritten in the form
2
2 у „12и2 + 22в+11 f г/лЮ-4я q14+4b\ _ Q4-+2nfiQ-2-4-n /}26+4в\1
0
в=0
Applying G.1) with a = - Q4+2n and b = — Q2", we find that the expression
in curly brackets above equals /(- Q4+2b, - Q2~2n). Thus, from G.4) and the
observations just made, we deduce that
(p(Q)<p(q) - v(-
= 2 ? g1
я=0
G.5)

366 20. Modular Equations of Higher and Composite Degrees
where, to obtain the penultimate line, we used the fact that /( — 1, — Q6) = 0
(Entry 18(iii) of Chapter 16), and, to obtain the last line, we employed Entry
18(iv) of Chapter 16 three times.
Lastly, we utilize Entries 10(i), (ii), ll(iii), and 12(iii) in Chapter 17 to
transcribe G.5) into the proposed modular equation.
The theta-function identity G.5) has also been proved by Kondo and
Tasaka [1, Eq. (T24)].
Before proceeding with the remaining proofs, which utilize the theory of
modular forms, we observe that (v) is the reciprocal of (iv). Thus, it remains
to prove (ii), (iii), (iv), (vi), and (vii). Transcribing these modular equations
via Entries 10(i)-(iv), 11 (i)—(iv), and 12(i) of Chapter 17, we deduce that,
respectively,
G.6)
'
G.8)
2A6ч6ф2(д2)ф2(д22)-(р2(-ч)(р2(-д11))
, G-9)
\ JK4)V"K4")/ ll \ JW'WW)/
and
G.10)
Next, we convert G.6)-G.10) into equalities relating the modular forms fu
9o> 9i, 9г> K> hlt and h2, defined by @.12). Thus, we deduce from @.13) that
) = 2Dff2(T)e2(llT)
хDд1(х)д1(П<
= 4(в1Bт)в1B2т) Н
x Bд1(х)д1(Пх)
-во(т)воA1т))
) + 4з2(тH2A1т) + 0о(тЫ
h 4з2BтH2B2т))
+ 4йB(т)йBA1т) + й(о(т)й((
11т)), G.11)
0A1т)), G.12)

20. Modular Equations of Higher and Composite Degrees 367
2^, С7.1Э»
and
flf1B2T) + 4<,2Bт)<,2B2т))
. G.15)
We next determine the multiplier system of each term in G.11)-G.15). From
the theory at the beginning of this chapter, fi(x), ft(llx), gj(x), йгДПт), йДт),
and йДПт), 0 < _/ < 2, are modular forms on the subgroup ГB) n ГоA1). Let
A = ("c 5N ГB)пГ0A1). From @.16), @.19), and @.25), we find that g\(x),
have a multiplier system identically equal to 1. Also, <^(т)йгД11т), О < j < 2,
has a multiplier system equal to (¦&), by @.18)-@.20), @.27), and @.11), since
b and с are even, where (") denotes the Legendre-Jacobi symbol. By @.21)-
@.23), @.25), and @.11), for 0 < ; < 2, hf(lU)/hj(x) has a multiplier system
equal to
It remains to find the multiplier system for g1Bx)glB2x) + 4g2Bx)g2B2x).
Lemma 7.1. The function g1Bx)g1B2x) + 4g2{x)g2{22x) is a modular form on
ГB) n ГоA1) with multiplier system (^-).
By multiplying out, if necessary, in G.11)—G.15), we now see that each side
is a modular form on ГB) n ГоA1). Furthermore, the modular forms on each
side of each equality sign have the same multiplier system. Except for G.13),
each of these multiplier systems is identically equal to 1; the modular forms
on each side of G.13) have a multiplier system equal to (yy).
Proof of Lemma 7.1. Note that gjBx), j = 1, 2, is a modular form on Г' =
ГB) п ГоD). Since
it follows that (ГB): Г') = 2. Now йг,B2т), ; = 1, 2, is a modular form on
Г' n ГоA1). The index of Г' n ГоA1) in ГB) n ГоA1) equals 2, for by a lemma

368
20. Modular Equations of Higher and Composite Degrees
in Schoeneberg's book [1, p. 74], it can be shown that
ГB)пГ0A1) = (Г'пГ0A1))иЛ(Г'пГ0A1)),
where A = ("c bd), with a = d = b + \ = \ (mod 2), с = 2 (mod 4), and c = 0
(mod 11). (The indices of these two subgroups will not be explicitly used in
the sequel.)
Suppose first that A = (? bd)e ГA), with a, c, and d odd. Then
с d
=: v',(AJg2,
where, by @.14), with с odd,
Similarly,
2a b — a
d-c
1 1
0 2
a b — a
с d — с
1 1
0 1
12a b-aY
d-c
a b — a
с d — с
12вг
¦er-
where again we have employed @.14).
Now let A e ГB) n ГоA1) with с = 2 (mod 4). Then, by G.17),
G.17)
G.18)

20. Modular Equations of Higher and Composite Degrees 369
(gi\2-gi\22)\A = v'g(^A)v'ge2)ALg2\2-g2\22
) G19)
by @.8). Similarly, by G.18),
{Ag2\2-g2\22)\A = (^\g,\2-gi\22. G.20)
Let j = 1 or 2 and A e ГB) n ГоA1) with c = 0 (mod 4). Then c/2 is even.
By @.4), @.19), @.20), @.25), and @.11), it follows that
Equalities G.19)-G.21) imply the truth of Lemma 7.1.
We now are in a position to prove G.11)—G.13). Clearing denominators (if
necessary) and collecting terms on one side, we can write each proposed
equality in the form
where F is a modular form of weight r on the group ГB) n ГоA1). From @.6),
@.24), and @.30), if we can show that the coefficients of q°, q1,..., q" in F are
equal to zero, where ц + 1 > 6r, then, in each case, G.11)—G.13) are estab-
established. For 0 < j < 2, gj and hj each have weight \. Thus, we obtain the
following table:
G.11)
G.12)
G.13)
г
2
2
5/2
fi
12
12
15.
By employing the computer algebra system MACSYMA, we have, indeed,
verified that all of the required coefficients are equal to 0. Thus, the proofs of
G.11)-G.13) are completed.
In principle, the same procedure can be used to verify G.14) and G.15).
However, in each case, r = 10, and so fi = 60. Since the amount of computa-
computation is considerably greater, we show how to decrease the value of fi by
deriving more information about the orders at the cusps.

370 20. Modular Equations of Higher and Composite Degrees
Let Г := ГB) о ГоA1), and recall that N(Г; {) denotes the width of Г at the
cusp (eQu {^}, where ^ denotes the point at oo.
Lemma 7.2. // r and s are coprime integers, then
(ll,s)
Proof. If we choose В е ГA) so that
we find that
^0 1/ ~ \ -ks2 1 + krs
Thus, P e Г if and only if 2\k and 11 \ks2. The smallest positive integer к with
these properties is к = 2 ¦ 11/A1, s).
Lemma 7.3. // we set
1 2 1 1 0 _ 1
Cl~o' 42~TT' Сз~ТТ' U~T Cs~I' Сб~Т'
t/ien
(i) ?lt ...,?6isa complete set of inequivalent cusps for Г, and
(ii) if rx, r2, s1; and s2 are integers such that (r1,s1) = (r2, s2) = 1, then r1/s1
and r2fs2 are equivalent cusps modulo Г if and only if
rj = r2 and Sj = s2 (mod 2) and A1, st) = A1, s2). G.22)
Proof. If r1/s1 and r2/s2 are equivalent cusps modulo Г, we can choose В еГ
so that
Then the conditions G.22) follow, which shows that they are necessary.
Using G.22), we easily check that no two of ?x,..., ?6 are equivalent cusps
modulo Г. Then an application of Lemma 7.2 shows that
t N(T; Q = 12 = (ГA): Г).
By a theorem in Rankin's book [2, Eq. B.4.10)], this shows that ?x,..., ?6 is
a complete set of inequivalent cusps for Г.
Now suppose that the conditions G.22) hold. Choose i and j so that
ri/si ~ Ci and r2/s2 ~ ?j modulo Г. But then by G.22) and the definitions of
Ci,..., Сб> it follows that i = j. Thus, r^ ~ r2fs2, and the proof is complete.

20. Modular Equations of Higher and Composite Degrees 371
Recall that for a cusp ? and a modular form /, the order of / with respect
to Г at ?, Ordr(/; ?), and the invariant order of/ at ?, ord(/; ?), are related
by the equality
Ordr(/; О = N(T; C)ord(/; 0, G.23)
where, as above, N(T; ?) is the width of Г at ?. We also recall from Lemma
0.1 that if M = (" Jj) is an integral matrix with determinant m = ad — bo 0,
and if (r, s) = 1, then
ord(/|M;-) = ^-ord(/;M-), G.24)
where g = (ar + bs, cr + ds).
Lemma 7.4. Let Г = ГB) n ГоA1) and M = (xv ?)(У ?), wfcere 8 = 0 (mod
11) and (" ^) e ГB). Let f denote any modular form on Г, and let С denote any
cusp of Г. Then
Proof. Set
r
M =
s llyr + 8s s1'
where (r', s') = 1. Then, by G.23), G.24), and Lemma 7.2, if С = r/s,
( r\q2 ( r\
Ordr(/|M; r/s) = N Г; - Г-Ord /; M
V s/ m V s
It thus suffices to show that, for each cusp r/s,
(ll,SO(ll«r+^,llyr + ^ = i (?25)
By examining each of the inequivalent cusps in Lemma 7.3, we may easily
verify that G.25) holds in each case. This completes the proof.
Lemma 7.5. Let M and Г be as given in Lemma 1A, and let ?x,..., ?6 be as
defined in lemma 7.3. Then
MCt ~ C,+3, 1 < i < 3,
where, of course, the symbol ~ denotes equivalence.

372 20. Modular Equations of Higher and Composite Degrees
Proof. Lemma 7.5 follows by direct computation with the aid of Lemma 7.3.
We are now in a position to find improved values of \i in order to prove
G.14) and G.15).
By @.33), @.34), @.35), and @.37), we find that both G.14) and G.15) are
self-reciprocal. In particular, if we set
then F2 = Ft\ M. We use Table 1 in the introduction of this chapter to calculate
Ordr(F1;0andOrdr(F2; Q, when ? = d,^,^. We next employ Lemmas 7.4
and 7.5 to calculate Ordr(Fx; Q and Ordr(F2; Q, when ? = ?4, Cs, U- Each of
the remaining terms in G.14) and G.15) does not have any poles, and so we
just use 0 as a lower bound for the order at each cusp of each such term. Now
write each of the proposed identities G.14) and G.15) in the form F(x) — 0.
Suppose that the coefficients of q°, q1,..., q" in F(t) are each equal to 0. We
now summarize our calculations in the following table:
С
Oidr(Fi;C)
Ordr(F2; 0
Lower bound for Ordr(F; ?)
Ci
5
0
fi+l
5
0
0
-10
0
-10
и
0
5
0
5
0
c6
0
-10
-10
By @.30), we may conclude that F(t) = 0, provided that
2(ц + 1) - 20 > ^г(ГA): Г) = 12,
where we have used @.6), @.24), and the fact that the weight r equals 2. Thus,
ц> 16.
Using the computer algebra system MACSYMA, we have, indeed, verified
that the coefficients of q°, q1,..., q16 are each equal to 0 for each of G.14) and
G.15). This finally completes the proofs of Ramanujan's modular equations
of degree 11.
Entry 8(i). Define
f(-q\-q9)
f(-q2, -
q5ll3f(-q3, -q1
fi-q5, ~q*)
-, and /z6 =
fi-q6, -q1)
q6ll3f(-q3, -q10)'
f(-q5,-qS)^
q15ll3f{-q, -q12)
u-q^-q1) '

20. Modular Equations of Higher and Composite Degrees 373
Then
fi-q1'13)
- Pi - Мз + M4 + 1 - Ms + Me, (8-1)
7/i3^_ 13ч = /*i
1 +
f2(_
/-2/ 13ч М2М3М4 MiMsMe. (8-4)
(8-5)
Proof of (8.1). Put и = 13 in A2.25) of Chapter 19.
Proof of (8.2). Using @.39), @.51), and @.52), we first translate (8.2) into the
equivalent form
? G(m; z)GEm; z) = -4' v' ' (8.6)
m(modl3) f/ (z)
This, in turn, is a special case of the following theorem of R. J. Evans [1,
Theorem 6.2].
Theorem 8.1. For each prime p = 1 (mod 4),
(8.7)
X ()(i) p4/^,
m(modp) f/ (Z)
where /? is any primitive fourth root of unity (mod p), and where
ap= X (-1Г+И. (8.8)
ГП,П€ Z
Fm-lJ+Fn-lJ = 2p
To see that (8.6) follows from Theorem 8.1, let p = 13and/? = 5 and observe
that a13 = — 2.
Proof of Theorem 8.1. By a general theorem on Hecke operators (Rankin [2,
pp. 289-290, Theorem 9.2.1]), the space of cusp forms {ГA2), 1, l}0 is in-
invariant under the Hecke operator Tp defined for /e {ГA2), 1, l}0 by
Evans [1, Lemma 3.1] has shown that the dimension of {ГA2), 1, l}0 is 1.
Moreover, since r\{z) is a modular form of weight \ on ГA) with multiplier

374 20. Modular Equations of Higher and Composite Degrees
system given by @.14), we easily see that r\2{z) e {ГA2), 1, l}0. It follows that
for some complex number <xp,
1 Ькв% (8.10)
fceZ
f г,2 (гЛ^Л = e ,aD (8.9)
P n=0 V P J
Since by @.76), r\2(z) = q1/12(l — 2q + •••), a comparison of the coefficients of
q1112 in (8.9) shows that <xp is the coefficient of qp/l2 in the Fourier expansion
of r\2{z). Squaring the Fourier series of r\(z) given in @.86), we see that <xp equals
the expression for ap given by (8.8).
For a modular form h(z) with a Fourier expansion of the form
define
p\k P n=O
Thus, I(h) is the sum of those terms in (8.10) that are integral powers of qi/i2.
Therefore, (8.9) can be written in the form
r,2(pz) + I(r,2(z/p)) = apr,2(z). (8.11)
Squaring both sides of @.86), we arrive at
г\2Ш = \ Z {r,(pz)Gm(pz)}{r,(pz)Gn(pz)}. (8.12)
m,n(modp)
By @.74),
*=-00
Thus, either all or none of the terms in the Fourier expansion of the product
{t\{pz)Gm(pz)} {n(pz)Gn(pz)} will contain integral powers of qili2 according as
m2 + n2 is divisible by p or not. Now m2 + n2 is divisible by p if and only if
n=± mfl (mod p), for /? as defined above, and then there are two such values
for each nonzero m(mod p). Thus, by (8.12), @.51), and @.52),
1(г\2Ш) = -r\\pz) + 1 I {r\(pz)G{m; pz)} {r,(pz)G(mp; pz)}. (8.13)
m(mod p)
In conclusion, (8.11) and (8.13) yield
X G(m; pz)G{mp; pz) = 2(r,2(pz) + I(r,2(z/p)))/r,2(pz)
m(mod p)
= 2apr,2(z)/r,2(pz),
and (8.7) follows.
Proofs of (8.3), (8.4). Transcribing (8.3) and (8.4) by means of @.39), @.51),
and @.52), we find that, respectively,

20. Modular Equations of Higher and Composite Degrees 375
? ?? \{z) = 4 *^М
r\\z)
and
Gx(z)G3(z)G4(z) - Gr^zJGj-^^G;1^) = 3 + TL^U, (8.15)
where p = 13. By Corollaries 0.6 and 0.5, respectively, the left sides of (8.14)
and (8.15) are in {Г°A3), О, 1} and have no poles on Ж or at the cusp 0.
We now follow the procedure outlined at the end of the introductory
material in this chapter. By @.77), for p = 13,
(8.16)
(8-17)
(8.18)
(8.19)
G5(z) = -^{1 + Oiq2'")}, (8.20)
and
G6B) = q^{\ - qw + O(q2")}. (8.21)
For p s 1 (mod 12) and V = {"c bd)e T°(p), by @.14),
t,2(Vz/p) = v2^ »>f)(cz + dW{zlp) = v2(V)(cz + d)n\zlp).
Hence, for p s 1 (mod 12), r\2{zjp)lr\\z) e (Г°(р), 0, 1}. Thus, both sides of
(8.14) and (8.15) belong to (Г°A3), 0, 1} and have no poles except at oo. By
@.76),
r,(z/p)/r,(z) = e
Thus, for p = 13,
r,2(z/p)/r,2(z) = q-1'" - 2 + O^1'"). (8.22)
Finally, from (8.16)-(8.21) and (8.22), both sides of (8.14) equal
q-llp + 2 + O(qllp),
while both sides of (8.15) equal
q'11" + 1 + O{qllp).
This completes the proof of (8.14) and (8.15).
Proof of (8.5). The desired result is an immediate consequence of the defini-
definitions of ци..., ц6.
Evans [1, Theorem 7.1] has proved another beautiful identity in the spirit
of(8.2)-(8.4). For t = q1113,

376 20. Modular Equations of Higher and Composite Degrees
1 t 1
which is equivalent to, for p — 13,
G:1(z)G;\z) + G4(z)G6(z) = 1.
Entry 8(H). We have
f(-q, ~q12)f(~q2, -qll)f(-q\ -qlo)f(-q\ ~q9)f(-q5, -qs)
x/(-q6,-q7)
= f(-q)f4-q13).
Proof. This identity is just the special case n = 6 of B8.1) in Chapter 16.
Entries 8(iii), (iv). If ft is of the 13th degree,
and
13
Proof. The modular equation (8.24) is simply the reciprocal of (8.23), in the
sense of Entry 24(v) of Chapter 18, and so it suffices to prove (8.23).
By Entries 10(i), (ii), ll(i), (ii), and 12(i) in Chapter 17, (8.23) is equivalent
to the theta-function identity
2613 <p(-q13)<p(q13)
<p(-qMq)
Employing @.13), we translate this identity into an identity involving modular
forms,
g2A3T)glA3T) | goA3T)glA3T) g2A3T)goA3T)
в 2^)91A:) fi'oWflfiW fif2Wflf0W
-1- (826)
Since b and с are even, it follows from @.26), @.16), and @.18)-@.20) that the
multiplier system of each term of (8.26) is trivial; that is, v(A) = 1 for each
A e ГB) п ГоA3). Clearing denominators in (8.26) and collecting terms on one
side, we can write the transformed equation in the form

20. Modular Equations of Higher and Composite Degrees 377
F := Fx + • • ¦ + Fs = 0,
where F is a modular form of weight § on Г = ГB) п ГоA3). By @.6) and
@.24), pr = 7. From @.30), we then see that ц = 31.
To decrease computation, it seems advisable to use the reciprocal relation.
Applying @.36) in (8.26) and then converting the new equality back to an
equality involving q -series, we find that
f4(q)(p2(q13)
q2fA(q13)(p4q)
(Of course, we can also obtain (8.27) directly from (8.24) by using Entries 10(i),
(ii), ll(i), (ii), and 12(i) in Chapter 17.) Thus, by @.38), ц = 15. Clearing
denominators in both (8.25) and (8.27), transforming all terms to one side of
the equation in each case, and employing the computer algebra system MAC-
SYMA, we have indeed verified that the coefficients of q°, q1,..., qls are equal
to 0 for each of the two proposed identities. This then completes the proof.
Modular equations of degree 13 have been developed by Sohncke [1], [2],
Schlafli [1], Klein [1], and Russell [2], but all of these modular equations are
considerably more complicated than those of Ramanujan established above.
Entry 9. We have
(i)
(ii)
(Hi) <p( — q2)<p( — q30) + 2q2\l/(q)ij/(q15) = (p(q3)(p(q5),
(v) cp(q)cp(q15) - cp(q3)<p(q5) = 2qf(-q2)f(-q30)X(q3)x(qS),
(vi) (p(q)(p(q15) + (p(q3)cp(q5) = 2f(-t
and
(vii)
= f(-q)f(-q3)f(-q5)f(-q15).
Proof of (i). In C6.8) of Chapter 16, let ц — 4 and v = 1 to obtain the identity
Now replace q by — q and subtract the result from the equality above. This
yields (i) at once.

378 20. Modular Equations of Higher and Composite Degrees
Proof of (ii). As in Entry 11 below, let a, /?, y, and S be of the first, third, fifth,
and fifteenth degrees. As in previous work, we set cp(q") = -Jzn. Then, trans-
translating Entry 9(i) via Entries ll(i)-(iii) in Chapter 17, we find that
\1/2
\Z3Z5 /
(9-1)
Inverting the roles of <x and 8 and also of /? and y, we derive the reciprocal
modular equation
1/2
z
\Z5Z3/
(9.2)
By Entries 10(ii), (iii) and 11 (ii) in Chapter 17, the translation of this equality
is the identity
<p(-q6M-q10) - 2#(-<г3Ж-<?5) = cp(-q)cp(-q15).
Changing the sign of q gives (ii).
PROOFOF(iv). In C6.10) of Chapter 16, replace q by q2 and set fi = 4andv = 1.
We then apply Entries 30(ii), (iii) in Chapter 16 twice apiece. Thus,
5, qlo)f(q90, q150) + 2q16f(q2, q")f(q30, q210)
-q, -Q3)}{f(q15,q*5)+f(-q15, ~q*5)}
Thus, (iv) has been established.
Proof of (iii). Translating Entry 9(iv) via Entries 1 l(i)-(iii) in Chapter 17, we
deduce that
1/2
)
)
\Z1Z15/
The reciprocal of this formula is
G7 \1/2
{A - a)(l - 8)У* + {x8(l - «)A - 6)}* = {A - P)(l - у)}1'4М-Ч .
\Z1Z15/
Employing Entries 10(ii), (iii) and Entry 11 (ii) in Chapter 17, we find that the
translation of this formula is the identity
cp(-q2M-q30) + 2q2ф(-qЖ-q15) = cp(-q3)(p(-q5).
Replacing q by —q, we deduce (iii).
We postpone the proofs of Entries 9(v)-(vii) until Section 11 where it will
be convenient to use the theory developed there.

20. Modular Equations of Higher and Composite Degrees 379
Entry 10. We have
(i) f(-q\ -q») + qf(-qi, -q13) = ff~J^'~^f(-q5\
(ii) f(-q\ -qu)-qf(-q, -q1*) = ji(~yqAJ)f(-q5),
(iii)/(-<?7, -q8) - qf(-q2, -q13) =f(-q2'3, -q) + q2l3f(-q3, -q12),
(iv) {f(-q\ -q11) + qf(-q, -q14)}qlfi =f(-q6, -q9) -f(-qm, -
(v) q ^ ^ ^
\-q l-q1 1-q11 l-q
, q11 , q19 ,
13
(vi) <p(q3)<p(q5) + <P(<ZM<Z15) = 2A
-q17 l-q19 l-q23
1 — Q
. q q2 . q4
q13 ,
q13
^~6 + 1 - q11 + 1 - q19 + 1 - q22
д2Ъ q26 q28 q29
+
+
1 - q26 l-q28 1 - q29
q32 q3* q31
34 1 _ „37
1 - q31 ' 1 - q32 l-q34 1 -
q3* q41 q43 q44
q46 , q41 , q49
1 - q46 1 - qA1 1 - q49
g" g56 q5S
1 - q53 l-q56 + 1- q58
a59
55
1-Я
In (v), the cycle of coefficients is of length 30, while in (vi), the cycle of
coefficients is of length 60. These rules of formation were not made explicit by

380 20. Modular Equations of Higher and Composite Degrees
Ramanujan (p. 245), since he recorded only six and four terms, respectively,
in the two series.
Proofs of (i), (ii). For the derivation of (i), first use the identity f( — q~2, — q11)
= -q~2f(-q13, -q2), which is deducible from Entry 18(iv) in Chapter 16.
In the quintuple product identity C8.2) of Chapter 16, replace q by q5'2 and
let В = — q3'2, —q112, in turn. Formulas (i) and (ii), respectively, now follow.
Proofs of (iii), (iv). In Entry 31 of Chapter 16, set n = 3, a = -q2'3, and
b = — q to achieve (iii), and let n = 3, a = — q1'3, and b = — q*13 to obtain (iv).
The proofs of (v) and (vi) are considerably more difficult.
Proof of (v). Let S denote the series on the right side of (v). Taking the
summands of S, expanding them into geometric series, and then inverting the
order of summation, we find that
» ?»_ qln - q11» _ ql3n + q17» + q19n + q23n - q29n
n=i 1 q
oo qn _ qS« _
— 1 _ я30"
я=1 * Ч
Applying next the addition theorem, A7.1) of Chapter 19, with q replaced by
q15, a = q*, q2, and b = q10, q8, respectively, we deduce that
f4~q30)(qf(-q\ -<?26)/(-<?10, -q20)f(-q14, ~q16)
<p(-qi5) \ f(-q19, ~qll)f(-q2\ ~q5)f(-q29, -q)
, q5f(-q2, -q2s)f(-q\ -q22)f(-q10,
1
q" - qSn - qlin + qi9n
1 - q3On
-q3
On
5n-q
-q13n
29n
-q25n
f(-q", -q13)/(-q"; _qi)f(_q25y _q;
f3(-q3O)f(-q10) fqf(-q\ -q26)f(~q14, ~q16)f(-q2\ -i
cp(-qi5)f(-q25, -q5)\ f(-q, -q29)f{-q", -q19)f(~q21, ~q9)
f(-q6, ~q24)
q5f(-q2, -q28)f(-q12, -q18)f(-q22, ~qs) f(-q3, -q
f(-q\ -q23)f(-q17, -q13)f(~q2\ ~q3) f(-q12, -i
f3(-q3O)f(-q10) fqf(-q\ ~q6)f(-q9, ~q21)
,
f(-q, ~
q5f(-q\ -д'Ж-д3, -q21)
where we have employed the Jacobi triple product identity and Example (v)

20. Modular Equations of Higher and Composite Degrees 381
in Section 31 of Chapter 16. Putting everything under a common denominator
and then using the Jacobi triple product identity to transform the denomina-
denominator, we arrive at
s = д/Ч-дзо)Я-д1ОМ-д5)
cp(-q15№(g15M-g5M-g)f2(-gio)f{-q6)f(-q30)
x (f(-g\ -д'Ж-д4, -g6)f(-g9, -g21)f(-q12, -gls)
+ g4f(-g, -g9)f(-g\ -g8)f(-g\ -g21)f(-g6, -g24))
= g
2x(-q15)f(-qlo)f(-q6M-q)
x ({f(-q3, -gJ)f(-g\ -g6) + qf(-g, -gg)f(-g\ -g*)}
x {f(-g9, -g21)f(-g12, -g18) + g3f(-g\ -g21)f(-g6, -g24)}
+ {f(-g\ -д'Ж-д4, -g6) - gf(-g, -g9)f(-g2, -g*)}
x {f(-g9, -g21)f(-g12, -gia)-g3f(-g\ -gln)f{-g\ -g24)}),
A0.1)
where we have used two of the equalities in Entry 24(iii) of Chapter 16.
Next, applying Entries 29(i), (ii) of Chapter 16, we find that, respectively,
f(g, g4)f(-g\ -g3) +/(-«, -g4)f(g2,g3) = Ж-д3, -q7)f(-g4, -g6)
and
f(g,д4Ж-д2, -д3) -Л-9, -g4)f(g2, g3) = 2gf(-g2, -g*)f(-g, -g9).
The results gotten by adding these two equalities and then by subtracting
them are used in A0.1) to give us
n
S =
2x(-gi5)f(-glom-g6)x(-g)
x (Яд,д4Ж-д2, -g3)f(g3,g12)f(-g6, -g9)
+f(-g, -д4Жд2,д3)Я-д3, -gi2)f(g6,g9))
gfH-g5)
2f4-gi5M-giS)f(-glom-g6m-g)
x №,gl4)f(.g6,g9№l\g4)f{-g2, -g13)f(-g\ -g*)f(-g12, -g3)
xf(g\g12)f(-g6,-g9)
+f(-g, -q14)f(-g6, -д9)Я-д'\ -g4)f(g2,qi3)f(g\gs)f(gi2,g3)
xf(-q3,-g12)f(g6,q9))

382 20. Modular Equations of Higher and Composite Degrees
_qf4-q5)f(q3,q12)f(q6,q9)f(-q3, -q12)f(~q6, ~q9)
2f6(-q15)x(-q15)f(-ql0)f(-q6)x(-q)
x (f(q,q14)f(q\qiX)f{-q2, -«")/(-«7, ~q8)
+ f(-q, -qi4)f(-q\ -q\*
where in the penultimate equality we utilized the Jacobi triple product identity
several times. Employing the Jacobi triple product identity as well as Entry
18(iv) in Chapter 16, using the corollary of Section 30 in Chapter 16, utilizing
the Jacobi triple product identity three more times, and invoking results in
Entries 24(ii), (iii) of Chapter 16, we deduce that
q2f4-q5)<p2(-qi5)f(-q6)f(-q30)
S =
2f6(-q15M-q15)f(-q10m-q6M-q)
x (f(q-\q16)f(-q2, -q13)f(q\ qll)f(-q\
(_^ _q6)
' ГЧ-д15М-д15Ж-д10М-дУ
д/2(-д5)<Р2(-д15Ж-д30Ж-д3)ГЧ-д15Ж-д10)Ф(д15)
/Ч-д15М-я15Ж-д1ОМ-дЖ-ч5Ж-<130)
дЯ-д5)Я-д3)срЧ-д15)Ф(д15)
f4-gi5)x(-q15)x(-g)
дЯ-дЖ-д3Ж-д5)
cp{-g) f4-gi5)x(-g15)
_дЯ-дЩ-д3Ж-д5Ж-д15)
<p(-gM-gi5)
Invoking Entry 9(vii), we complete the proof.
Proof of (vi). Combining Entries 9(ii), (iii) and 10(v), we find that
q11 q19
+. ,,+
l+q l + q1 l+q11 l + q13 l+q17 l+q1
'l+q23 l+q29
Replacing q by q2" and summing on n, 0 < n < oo, we deduce that

20. Modular Equations of Higher and Composite Degrees 383
<p(-q3)q>(-q5) + (p(-q)cp(-q15) - 2
1 + q 1 + q2 1 + q4 1 + q1 1 + q8 1 + q1
l + qi3 l + qi*^
where the cycle of coefficients is of length 15. If we change the sign of q, we
find that
<p(q3)<p(q5) + <p(q)<p(q15)
l-q l+q2 l+q4 1 - q1 1 + q8 l-q
11
q23 q26 q28 q29
+ +
where now the cycle of coefficients is of length 30. For each even value of n,
use the trivial identity
q" _ q" _ 2q2n
1 + qn ~ 1 - q" ~ 1 - q2n
in the foregoing series. We then obtain the proffered identity, with a cycle of
coefficients of length 60.
Entry 11. Let a, /?, y, and S be of the first, third, fifth, and fifteenth degrees,
respectively. Let m denote the multiplier connecting a and B, and let m' be the
multiplier relating у and S. Then,
(i)
(ii) (ByI* + {A - /?)A - y)}118 = J^,
_(ВуI<8-{ВуA-В)A-у)У'8
{A - /?)(! - у)}1'8 - {By(l - /?)(! - у)}1/8
(iii) ИI/8 - {A - a)(l - <5)}1/8 = (ByI18 - {A - № ~ У)}11*,
/B2y2(l - BJ(l - vJ\1/24
(iv) 1 + (ByI18 + {A - № - y)}118 = ^ ( ^A _,)A_ д) ) '

384
20. Modular Equations of Higher and Composite Degrees
{A - a)
{(l - У
Уу)}1/4 - {(i - v^)(i - Уу)}1/4)
m
(x)
A - a)(l -
A-
- a)(l - 8)
/m
1 m
- «)A - y)
- y)
= mm ,
(xi)
fid)
-4
ay(l - a)(l - y)
mm
(хш) U
_
«/?(!-a)(l -
'«/?(! ~ «)A ~ ,
удA-
A-
A-
= 25
Z5Z15
and

20. Modular Equations of Higher and Composite Degrees 385
(xiv) (фдI* + {A - «)A - /?)A - y)(l - б)}1'6
- a)(l - /?)A - y)(l - *)}«" = 1.
(xv) //
P = {256фдA - a)(l - j3)(l - y)(l - <5)}1/48
and
^(i - «Ш -
then
If we multiply (i) by the first equality in (ii), we obtain a modular equation
sent by Ramanujan [10, p. xxix] in his second letter to Hardy. Entry ll(xiv)
also appears in the same letter [10, p. xxix]. Both (iii) and (vi) were recorded
by Hardy [3, p. 220] in his brief description of some of Ramanujan's work on
modular equations. (In his statement of (vi), Hardy made two sign errors.)
If we multiply (viii) and (ix) together, we obtain a modular equation
established by Weber [1]. Weber [1] also established (xiv).
Proof of (i). If we translate Entry 9(iii) via Entries 10(i), (iii) and ll(i) in
Chapter 17, we obtain Entry ll(i) at once.
Proof of(H). Transcribing Entry 9(ii) by means of Entries 10(i), (iii) and ll(i)
in Chapter 17, we obtain the first part of Entry 11 (ii) immediately. The second
and third equalities of (ii) are (9.1) and (9.2), respectively.
Proof of (iii). Let
A'= {A -<x)(l -<5)}1/8, B =
- y)}1'8, and M =
where the notation of Section 9 is used. Then in this abbreviated notation,
Entries ll(i) and (ii) yield the equalities
_L^ = B + F = ^ = ^ = M. (iu)
The last three expressions yield
B-BB- (B' - BB) = MA2 - MA'2,
or
B-B'
(A - A')(A + A')
= M.

386 20. Modular Equations of Higher and Composite Degrees
Using the extremal parts of A1.1), we conclude that
B-B' = A-A', A1.2)
which is (iii).
Proofs of (iv), (v). According to A1.1) and A1.2), we may set
A = ^M - p), A' = fcM + p),
and B' i(M )
where p is positive when <x is small. Taking the equality В — BB' = AM2 from
A1.1), substituting for В, В', and A from A1.3), and solving for p2, we deduce
that
Thus, from A1.4) and A1.1),
72A - j3J(l - yJV/8
M -
Part (v) is now apparent.
Proof of (vi). First, by A1.3) and A1.4),
= 2A
= 2A + A* + Art)
= 2A + ^(M-1 - pL + UM-1 + PL)
(M3 - M2 + 3M + IJ
after some algebraic manipulation and simplification.
Hence, from the last calculation, A1.3), and A1.4),
{A - v?)(l -
Taking the cube root of both sides, we deduce (iv).
Similarly, from A1.4) and A1.1),
/<x2<52(l - <xJ(l - <5J\1/8 _ A2A'2 _ (M'2 - p2J _ / _ 1
V №№) ) ~ ~вв' M2-p2 ~\ ~м
- {A - a)(l - ^)}1/8K. A1.6)

20. Modular Equations of Higher and Composite Degrees 387
_/ su/8/m3-m2 + 3m + 1 1 + 2Mp + M + M2-M3
~(CCd) { 2M~2 + 2M5
- pJ
im2 ¦
Thus, we have shown that the first expression on the left side of (vi) is equal
to (M - p)/(My/2).
Suppose now that we repeat the analysis above, but with a and <5 replaced
by 1 — a and 1 — <5, respectively. The calculations are seen to be exactly
the same, except that p is replaced by —p. Hence, the second expression
on the left side of (vi) is equal to (M + p)/(MN/2). The truth of (vi) is now
apparent.
Proof of (vii). The proof is analogous to that above. First, by A1.3) and A1.4),
({A + v^)(l + fy)}1'2 + {A -
= 2A + Jfy + 7A - /?)A - y))
= 2A + им -рГ + Ым + рГ
_ A + M + ЗМ2 - М3J
4М2
Thus,
1/8
l +M + ЗМ2 - М3 М3 + 2М2р + 1 + М - М2
2M
_ (M - p)BM2 - M3 - M2p)
~ 2M
= \{\ - MpJ.
Hence, the first expression on the left side of (vii) is equal to A —
We now repeat the procedure above but with /? and у replaced by 1 —
and 1 — y, respectively. As in the proof of (vi), we see that the calculations are
the same except that p is replaced by —p. Hence, the second expression on
the left side of (vii) is found to equal A + Мр)/^/2. The truth of (vii) is now
manifest.
Proof of (viii). Observe that, by A1.3),

388 20. Modular Equations of Higher and Composite Degrees
W/ + U - № - i)) ~ V/W - /
_ АГ1 - p M + p M~2 - p2
М-р + M + p M2 - p2
_ 2 - 2p2 _ M - p2 _ 1
~ M2-p2 ~ M2-p2 ~ M'
upon the use of A1.4). Thus, (viii) is established.
Proof of (ix). The proof is analogous to that of (viii). Thus, by A1.3) and A1.4),
f{\ - /»)A - y)V/8 /j8y(l - /»)A - y)\118
_
M - p M + p M2 - p2 2-M2 - p2
M x — p M l + p M 2 — p2 M 2 — p2
which verifies (ix).
The proofs of (x)-(xiii) are somewhat difficult. It seems necessary to express
a, /?, y, and <5 as algebraic functions of a single parameter. To that end, we set
t = (hh = 1
\ziZlsJ M'
or
m' = mt2. A1.7)
Thus, from (ii) and (iv),
and from (i) and (v),
/a.(l-a)(l-^
Since j8 and <5 are of the third degree in a and y, respectively, it follows from
E.2) and E.5) of Chapter 19 that
_ (m - 1)C + mf a _ (m - 1KC + m)
16w3 ' P 16m
_ (m - 1)C + m) (m' - 1KC + m')
У 16m'3 ' 16m'
A1-10)
(m + 1KC - m)
1Р
(m
(m
(m
f
+
' +
16m3
1)C
16m'
1)C-
1)C
+
3
m
m)
m
'
3
'K
(m' + 1KC - m')

20. Modular Equations of Higher and Composite Degrees 389
Put
ц = — and ц' = —
and note that, from A1.7),
H' = Ht2. A1.11)
Since у and <5 are of the fifth degree in a and /?, respectively, it follows from
A4.2) and A4.4) of Chapter 19 that
(fi - 1)E - ЦM (fi' - 1)E - n'f
a(l-a) = 2V ' P{1~P) = 2V '
A1.12)
(fi - 1ME - fi) _ {Ц' - 1ME - ц')
У\ У) 28ц ' 28ц'
Substituting these values in A1.8) and A1.9), we find that, respectively,
(m2 - lM(9m'~2 ~ 1MV/24 _ 1 _ /(^ - lK^'-1 ~ 1KV/8
t'2 — l)(9m~2 — 1) / t \ (ц' — 1)E^-1 — 1) /
and
m'2 - lM(9m-2 - 1MV'2* _, _ (W ~ 1KEm"' - 1KV/8
(m2 - l)(9m'-2 - 1) ) ~ 1~\(ц- Щ5Ц - 1) ) '
If follows that
/ 1\5
1 - t) = (m2 - 1)(9т'~2 - 1), A1.13)
1 + -J(l - tM = (m'2 - 1){9т~2 - 1), A1.14)
1 + -J(l - t) = (ц - Щ5Ц - 1), A1.15)
and
From either A1.13) or A1.14) and A1.7), it readily follows that
m +^ ? . A1.17)
Also, from either A1.15) or A1.16) and A1.11), it follows easily that
5 t4 + 3t3 + 3t - 1

390 20. Modular Equations of Higher and Composite Degrees
Solving A1.17) for m2 and A1.18) for ц, we find after a considerable amount
of elementary algebra that
2t5m2 = t6 + 5t5 + 5t4 - 5t2 + 5t - 1 - 4t2{t2 + 2t - l)RS A1.19)
and
2t3n = tA + 3f3 + 3t - 1 - 4t2RS, A1.20)
where
4t2R2 = t4 + t3 + It2 - t + 1 A1.21)
and
4t2S2 = tA + 5t3 + 2t2-5t + l= (t2 + 4t- l)(f2 + t - 1). A1.22)
The negative signs on the two radicals were chosen in order to be consistent
with the fact that t tends to 1 as m and m' tend to 1. Since a, /?, y, and <5 can
be expressed in terms of m and m', it follows from A1.7) and A1.19) that a, /?,
y, and <5 can be expressed in terms of a single parameter t.
Proof of (x). By A1.12), A1.15), and A1.16),
(Pd(l - /?)(! - S)V'* = /V -
( =
V ay(l - a)(l - у) У V (M - lN^1 - lN У
(^' - lKE/x-1 - IK (^' - 1KE^'-1 - IK
A + rxN(l - tN (Г1 - tN
On the other hand, by A1.10), A1.13), and A1.14),
¦j8<5A - j8)(l - <5)\1/4 / (m2 - lJ(m'2 - lJ У'4 (m2 - l)(m'2 - 1)
- а)A -у)) \(9т-2 - lJ(9m'-2 - lJ/ (Г1 - tK
A1.23)
Equating the right sides above, taking cube roots, and using A1.11), A1.18),
A1.20), A1.21), and A1.22), we find that
_ t4 + 3t3 + 2t2 - 3t + 1 - 4t2RS
2T2
= (/?-SJ. A1.24)
Thus, by A1.23),
(P8(l - Д)A - ^)V/4 _ (Я - SN
-y)/ (Г1-!K'
Next, by A1.10), A1.23), A1.13), A1.14), A1.7), and A1.17),

20. Modular Equations of Higher and Composite Degrees 391
ay
(m - l)(m' - 1) (m + \)(m' + 1) 2(m2 - l)(m'2 - 1)
* /^ — 1 4 \ S ^ / — 1 44 '
Cm-1 + l)Cm'-x + 1) (ЗпГ1 - ЩЗт' - 1) (Г1 - tK
20 + 2mm' + Щтт') + 6(т + m'J/(mm') 2(m2 2
+
(9m - l)(9m'-2 - 1) (Г1 - tf
!-l)(m'2-l)/ ft 9 3(m + m'J , _, ,.
——j -? 10 + mm -\ H \-(t — ty
(t l — t) \ mm mm
2(m2 - l){m'2 - 1)/ t6 + 5t5 + 5t4 - 5f2 + 5t - 1
3A + t2J A - t2)
t3
_ Щт2 - l)(m'2 - l)(t4 + t3 + 2t2 - t + 1)
So, taking square roots and utilizing A1.24) and A1.21), we deduce that
Because of the pervasiveness of t 1 — t, it will be convenient to introduce
a new parameter
u:=Cl-t. A1.27)
Thus, by A1.19), A1.21), and A1.22), respectively,
2m2t2 = 10 - 8u + 5u2 - u3 - 4RSB - u), A1.28)
4R2 = u2-u + 4, A1.29)
and
4S2 = u2-5u + 4. A1.30)
We are now prepared for the final calculations necessary to complete the
proof of (x). Employing A1.26), A1.25), A1.28), A1.29), and A1.30), we deduce
that
¦ ¦ — mm
= SR(R - SK -(R- SN - u3m2t2
= 8Я4 + 24R2S2 - (R2 + S2)(R* + HR2S2 + S4)
- |u3A0 - 8u + 5u2 - u3)
- RS{24R2 + 8S2 - 6Я4 - 20R2S2 - 6S4 - 2u3B - u)}

392 20. Modular Equations of Higher and Composite Degrees
= i(u2 - и + 4J + f(u2 - и + 4)(u2 - 5u + 4)
- |u3(l0 - 8u + 5u2 - u3)
- ?(u2 - 3u + 4){(u2 - u + 4J + 14(u2 - и + Щи2 - 5u + 4)
+ (и2 -5u + 4J} - RS{6(u2 - и + 4)
+ 2(u2 - 5м + 4) - |(u2 - и + 4J - f(u2 - м + 4)(u2 - 5u + 4)
- |(u2 - 5u + 4J - 2u3B - и)}
= 2u5 - 12u4 + 30u3 - 48u2 + 32u - SRS(u3 - 3u2 + 4u)
= 4u(R4 + 6R2S2 + S4) - 16uRS(R2 + S2)
= 4u(R - SL
— dii3
by A1.25) again. Hence, at last, (x) is established.
Proof of (xi). Formula (xi) is the reciprocal of (x).
Proof of (xii). First, by A1.10), A1.13), and A1.14),
fy5(l - y)(l - S) V/4 (m12 - l)(9m'~2 - 1) (r1 - tN
(m2 - l)(9m~2 - 1) (m2 - lJ(9m~2 - IJ'
A1.31)
Second, by A1.10), A1.12), A1.15), and A1.16),
{(m2 - l)(9nT2 - I)}4 = 216a(l - a)j8(l - Д)
= (/x - 1H*' - 1)E^х - 1П5Ц'-1 ~ IM
= (Г1 - 1П5Ц-1 - 1LE^'-! - IL;
that is
(m2 - 1)(9иГ2 - 1) = (Г1 - ^(S^-1 - Щ5Ц'-1 - 1). A1.32)
We want to express (m2 - l)(9m 2 - 1) entirely in terms of t. By A1.17),
A1.19), A1.21), and A1.22),
(m2 - l)(9m - 1) = A~/ \t8 + 5f7 + 6t6 + 5t5 + 2t4 - 5t3
+ 6t2 - St + 1 + 4t2(t2 + l)(r2 + 2t- l)RS)
= (Г1 -t){B- Г1 + t)R + (Г1 + t)S}2. A1.33)
We have omitted some rather tedious, but straightforward, algebraic compu-
computations. We now claim that

20. Modular Equations of Higher and Composite Degrees 393
To verify this equality, cross-multiply and express everything in terms of t via
A1.21) and A1.22). Putting A1.34) in A1.33), we see that
(m2 - l)(9nT2 - 1) = (Г1 - tM{B - Г1 + t)R - (Г1 + t)S}~2. A1.35)
Hence, by A1.10), A1.31), and A1.17),
(m' - ЩЗт + 1) jm' + ЩЗт'-1 - 1) 2(t~l - tf
(m - ЩЗт-1 + 1) (m + ЩЗт'1 - 1) (m2 - Щ9т~2 - 1)
т \т
2f
2A-t2
4A - t2)(l - t + It2 + t3 + t4)
A1.36)
[m2 - Щ9т-2 - Щ3
16A - t2)R2
t(m2 - Щ9т~2 - 1)"
Thus, by A1.36), A1.35), A1.27), and A1.29),
A-у)A-<5)У8 , ,
= tT2{4Я2B - Г1 + t)
- ARSir1 + t) + u2}
= tT2{8 - 6u + 4u2 - u3
+ t)}- A1-37)
Using our calculations in A1.36), as well as A1.32), A1.31), A1.17), A1.27),
and A1.35), we find that
x (9m-2 - 1)
m2t2/ t2 (m2 - 1)(9»j-2 - 1)
- (Г1 - t)E - Ж5 - /x')

394 20. Modular Equations of Higher and Composite Degrees
2-4t + 10t2 + 8t3 - 10t4 - 4t5 - 2t6
t3
+ (т2-Щ9т-2-1)-^-^5'^
= иBи2 - 4u + 16) + м{B - u)R - (Г1 + t)S}2
- иB5 - 5цA + t2) + n2t2). A1.38)
The expressions in ц must be converted to terms in u. By A1.18) and A1.20),
5A + t2)n - tV
+ 4(t4 + 3t3 + 3t - 1 - 5t(l + t2))RS
= i( + 4)(м + 2)C - u) - 5 - 2B + u)(t + r
Using this in A1.38) and utilizing A1.29) and A1.30), we find that
A - y)(l - S)V'4 + fyd(l -
- P)J
x (m2 - l)(9m'2 - 1)
= u{2u2 - 4u + 16 + {B - u)R - (Г1 + t)S}2 - 20
+ i( + 4)(м + 2)C - u) - 2B + u){t~l + t)RS}
= u{16 - 12m + 8u2 - 2u3 - ЦГ1 + t)RS}. A1.39)
By comparing A1.37) and A1.39), we conclude that
yS\ /A - y)(l S)y ,
a/»/ Vd )A W
W(l)AW 1 W/
by A1.31). This completes the proof of (xii).
Proof of (xiii). This formula is the reciprocal of (xii).
After the proofs of (x) and (xii), the proofs of (xiv) and (xv) are comparatively
simple.
Proof of (xiv). By A1.3) and A1.4),

20. Modular Equations of Higher and Composite Degrees 395
4M(aj8y<5I/8 = 4MAB = MiM'1 - p)(M -p)=l+2m-M2- p{M2 + 1).
Similarly,
4M{A - a)(l - j8)(l - y)(l - <5)}1/8 = 1 + 2M - M2 + p(M2 + 1).
Thus, by A1.4),
- S)}1*8
^1 +2М~м2- PW2 + Ш {1 + 2M - M2 + p(M2 + I),
\{(l+2M- M2J - 1A + M - M2)(M2 + IJ}
16M2lv ' M
=Ii(M2-1K-
It is now clear that by combining these last three calculations, we achieve (xiv).
Proof of (xv). From A1.6),
and from A1.5),
Hence, upon multiplication,
Adding A1.40) and A1.41) and using the formula above, we deduce that
Dividing both sides by P, we complete the proof.
With all the groundwork developed in Section 11, it will now be a relatively
easy task to prove Entries 9(v)-(vii), which we left unproved in Section 9.
Proof of Entry 9(v). By Entries 12(iii), (v) in Chapter 17, as well as A1.40)
above,
M

396
20. Modular Equations of Higher and Composite Degrees
which completes the proof.
Proof of Entry 9(vi). Again, by Entries 12(iii), (v) in Chapter 17, as well as
A1.41) above,
Vd - BJ(l - vJY/24
= \/z1z15
Proof of Entry 9(vii). Employing Entries ll(i), 10(ii), and 12(ii) of Chapter
17, A1.3), A1.40), A1.41), and A1.4), we find that
{A - a)(l - «5)}1'4
2M
(M-1 + pJM2
x {A + pM){M + 1) - M(M + p)(M - 1)}
This concludes the proof of part of (vii).
The second equality of (vii) is established by completely analogous reason-
reasoning. Thus,

20. Modular Equations of Higher and Composite Degrees 397
Xi 2 + 2 ' ^
= qf(-q)f(-q3)f(-q5)f(-q15),
which completes the proof of Entry 9(vii).
Entry 12.
(i) Let
Я-46, -q11) /(
-12/17/-1- -3 rt14\' f*Z
qmif{-q5,-q12)
-q*, -q") q^R-q, -q16)
_ 17) = Ml - ^2 - ^3 + ^4 + ^5 - 1 - Мб + ^7 - /*8. A2.1)
M1M5M6M7 = М2М8М3М4 = 1. A2-2)
-1- A2.3)
(H) n-q)f4-q")
= f(-q, -q16)f(-q2, ~q15)f(-q3, -914)Л-«*, -<?13)
x Л-95. -q12)f(-q6, -qll)f(-q\ -qlo)f(-qs, -q9)-
The following are modular equations of degree 17:
i - a/ v«(i -«:
a(l - a)J j + W + Vl - a

398 20. Modular Equations of Higher and Composite Degrees
and
av) 17 - (*Y+(l- *Y+(*{l ~ a)Y/4
/8
Proof of (i). We first observe that A2.1) follows from A2.26) of Chapter 19
with n = 17.
The equalities of A2.2) follow immediately from the definitions of fi1 ,...,fi8.
A proof of A2.3) depending on Entry 22(iii) in Chapter 16 could be con-
constructed. However, it seems easier to use the identity
oo
f2(-q)= ? (_1)m+Y,0.+ l)/2-pnCiii-iy2j ^2A)
m,n=—oo
n>2\m\
first proved by L. J. Rogers [1] in 1894, rediscovered by Hecke [1], [2,
pp. 418-427] in 1925, and more recently proved by Andrews [16], [17],
Bressoud [3], and Kac and Peterson [1].
First, square both sides of A2.1) and equate rational parts on each side to
deduce that
/•2/ 1/17\\
~ /2(V7)J=2{flitl5
where Щ ¦) denotes the rational part of (•). From A2.3), it thus suffices to show
that
q- 24Д i->_
Now, by A2.4),
-24/17г2/__1/17\ _ у l_iyn+n n(n+l)/34-mCm-l)/34-24/l 7
д-24/17г2/__1/17\ _ у l_iyn+n
m,n=—oo
n>2\m\
We thus want to isolate those terms where
n(n + l)/2 - mCm - l)/2 = 7 (mod 17). A2.6)
Examining complete residue systems of m and и (mod 17), we see that the only
solutions of A2.6) are
и = 8 (mod 17) and m = 3 (mod 17).
Thus, setting m = 3 + 17fe and n = 8 + 17/, we discover that

20. Modular Equations of Higher and Composite Degrees 399
у 1
(>2\k\
k,f=-oo
г2\к\
by A2.4). Hence, A2.5) is established, and the proof of A2.3) is complete.
We offer a second proof of A2.3) that depends on Theorem 8.1. By @.53),
@.51), and @.52), A2.3) is equivalent to the identity
? G(m;z)GDm;z) = 0. A2.7)
m(mod 1 7)
However, A2.7) follows immediately from (8.7) for p = 17, since a17 = 0.
Proof of (ii). This identity is merely the special case и = 8 of B8.1) of Chapter
16.
Proofs of (iii), (iv). First, observe that (iv) is the reciprocal of (iii).
Using Entries 10(i)-(iii) and ll(i)-(iii) of Chapter 17, we find that (iii) and
(iv) are equivalent to, respectively,
1= 4>%X<7) vigMg)
q t(q2Mq) <p(qMq) 9
2 2<HqMq) 2 4*(gXg) 2
q Ф(дЫд) q Ф(д2Мд2) q
Ф(д2М-д2) q Ф(дМ-д)
A2.8)
and
^ ' A. i / ЛА.\ ~i 1 1\ I ~/ 1 1\ ~/ 1 *7\ I
И-д)<р(д) ? Ф(д2Ы-д2)
A2.9)
Using @.13), we transform A2.8) and A2.9) into equalities involving modular
forms. Thus, respectively,
t _ g2A7T)glA7T) | goA7T)glA7T) { д2A7т)д0A7т)
()()
к2(Пх)д2(Пт) /10A7т)д0A7т)
Л(т)»(т) hQ(x)g0(x) ( • ^
and

400 20. Modular Equations of Higher and Composite Degrees
3iA7t) в0A7т)в1A7т)
й2(т)а2(т) Мт)д
Ь(ПфA7)
,
Since b and с are even, we see from @.18)-@.23) and @.26) that the multiplier
system of each term in A2.10) and A2.11) is trivial; that is, v(A) = 1 for each
ЛеГB)пГ0A7).
By @.6) and @.24), ft. = 9 for Г = ГB) n ГоA7). By clearing denominators
in A2.10) and A2.11), we may write each of the proposed identities in the form
F := Fxl + F7 = 0, where FJt 1 < j: <, 1, is a modular form of weight 3 on
Г. Thus, by @.38), it suffices to show that the coefficients of q°, ql,..., q13 are
equal to 0 for the functions F arising from A2.10) and A2.11). Using the
computer algebra system MACSYMA, we have, indeed, shown that all of
these coefficients are equal to 0. Thus, A2.10) and A2.11) are established, and
the proofs of (Hi) and (iv) are complete.
Modular equations of degree 17 have previously been established by
Sohncke [1], [2], Schlafli [1], Russell [1], [2], and Greenhill [2], but none of
these has the simplicity of the two modular equations of Ramanujan that we
have just proved.
At the end of Section 12, Ramanujan remarks:
N.B, Thus we see that <р(х1/л), ф{х1/1!} or /(—x1/n) n being any prime number can
be expressed as the sum of \{n — 1), nth roots of several functions and <p(x"),
ф(х") or /(—x"). In finding the values of these functions, quadratics only appear
in the case of the 5th, 17th, 257th, etc. degrees, and cubics in case of the 7th,
13th, 19th, 37th, 73rd, 97th, 109th, 163rd, 193rd etc. degrees not as cube roots
but as sin(^ suT'fl) and quintics in case of the 11th, 41st, 101st etc. degrees.
/3( — x11") can also be similarly expressed.
We are unable to provide a proper interpretation for most of this statement.
As we have seen in past entries, for example, Entries 12(i) and 18(i) in Chapter
19 and Entries 6(i), 8(i), and 12(i) in Chapter 20, by employing Entry 31 in
Chapter 16, we may express q>{qlln), 4i{qlln), and /(-Vя) in terms of <p(q"),
ф(а"), and /(— q"), respectively. Moreover, in each case, there appear %(n — 1)
additional expressions that involve nth roots of q. The words "these functions"
of Ramanujan evidently refer to what he calls "nth roots of several functions."
We cannot identify the "quadratics," "cubics," and "quintics" to which
Ramanujan refers, nor do we know what в denotes. It does not seem possible
to find the values of "these functions" by purely algebraical means, which is
what is seemingly indicated; Ramanujan evidently has something else in mind.
In the last sentence, Ramanujan possibly is employing the identity /3(—q) =
(Р2( — ч)Ф(ч\ which is part of Entry 24(ii) of Chapter 16.
Entry 13. // P, y, and S are of degrees 3, 7, and 21, respectively, m = гх1гг, and
m' = z7/z21, then

1 j
am/
20. Modular Equations of Higher and Composite Degrees 401
- S)
a<5(l - a)(l - S)J m"
- «Ш -
m '
yd(i - y)(i - г)У'8
W
_ /a/Kla)(l/?)Y
V5(l)(l5);
_ /ay(l-a)(l-y)Y/8 f j /ayY/8 + Л1 ~ a)A ~
V^(l - j8)(l - 5)/ 1 \ps) +\(l-m-S)J J mm'
Observe that (ii), (iv), and (vi) are the reciprocals, respectively, of (i), (iii),
and (v).
Proofs. Our proofs utilize the theory of modular forms.
Using Entries 10(i)-(iii), ll(i)-(iii), and 12(iii) in Chapter 17, we convert (i),
(iii), and (v) above into the equivalent theta-function identities
P(g3)p(g7)p(-g3)p(-g7)
) cpiqMq^M-qM-q21)
3)<p{-cf) , A /2(-<76)/2(-<714) _, „,„

402 20. Modular Equations of Higher and Composite Degrees
д3Ф(д7Жд21) <р(-дыМ-д42) <?У(-<?7Ж-
<p(-q2)<p{-q6) И
f(-g2)f(-g6)
and
ае}ф(дА2)(р{дъ)(р(д21)
TZ / T~i\ 1"
ф(д )(p(g)(p(g )
д4Ф(д6Жд*2М-д3М-д21) _ -«Vf-VW-e21)^3)^*21)
Ф(д2ЖдыЫ-дЫ-д1)
дФ(дЖдМ-д6М-д*2) 2д2Ф(д3Жд21М-д3М-д21)
ФBЖ1АЫ2Ы1А)
A3.3)
respectively. Because the theta-function identities equivalent to (ii), (iv), and
(vi) are similar to A3.1)—A3.3), respectively, we do not record them. Next,
we transcribe, via @.13), the identities A3.1)—A3.3) into the modular form
identities
go(z)goBU)g2{-:)g2BU)+ r,2(*)t,2BU)
=
ho(lz)hQBU) h2Gz)h2BU) /»1Gт)й1B1т) >?GфB1т) _
h(z)hCz) Л(т)Л(Зт) h^htfx) Г1{х)фх) '
and
T)giBlT)g2CT)g2BlT) |
| доCт)доB1т)д2Cт)д2B1т)
gQQx)gQ{2U)h0{lx)h0{2U)
respectively.
In the introduction to this chapter, we showed that if F = gp hp or r\,
0 < j < 2, then F has a multiplier system uF of the form
° ьН$.и- (Ш)
Furthermore, if и is a positive odd integer, then F(m) is a modular form on

20. Modular Equations of Higher and Composite Degrees 403
ГB) п Г0(п) with a multiplier system vF\n that has the values
provided that (и, 6) = 1. Here, (" bd) e ГB) n Г0(и) and c0 = с if F = r\, g0, gx,
g2, ht, while c0 = 2c if F = h0, h2. If и is odd and 3|n, A3.8) is still valid for
F = gj or /i,-, 0 < j <2, but is not valid when F = r\. Moreover, from @.14)
and @.18)-@.23) we note that on ГB), Q = 1 for F = gs and ?| = 1 for F = h},
where 0 < j < 2. Using A3.7), A3.8), and the foregoing observations, we may
easily verify that each expression in A3.4)—A3.6) has a multiplier system that
is identically equal to 1, with the possible exceptions of the two expressions
involving eta-functions. Now observe that, in the instance at hand, 31 d, so
that d2 - 1=0 (mod 24). Thus, from @.14), the multiplier system of r\{m) is
equal to
Using A3.9) for n = 1, 3, 7, and 21, we readily verify that the last expressions
on the left sides of A3.4) and A3.5) each have a trivial multiplier system. In
conclusion, for each proposed identity, A3.4)—A3.6), each expression has a
trivial multiplier system.
If we were now to use @.30) or @.38), we would find that the amount of
computation is fairly high because the weights r are relatively large. Thus, as
in Section 7, we derive additional information about the orders at the cusps
in order to decrease the amount of computation. Because similar knowledge
is needed in Sections 18 and 19, we proceed more generally than is necessary
here. Furthermore, we discuss the function fu which appears in Sections 18
and 19 but which is irrelevant in this section.
Let n = pq, where p and q are odd primes, let Г = ГB) п Г0(и), and let
N = N(T; 0 denote the width of Г at the cusp С ? Q и {и}, where ? denotes
the point at oo.
Lemma 13.1. // r and s are coprime integers, then
/ {n, s)
Proof. The proof is exactly like that of Lemma 7.2; just replace 11 by и
throughout the proof.
Lemma 13.2. // we set
t 1 t 2 i 1 i l с 2 : 1
1 , 2 r 1 r 1 г 2
^-> Ce = -> ?9 = ~> Сю= у 4n—т
ip p p I 1

404 20. Modular Equations of Higher and Composite Degrees
then
(i) Ci, • ¦ •, C12 's a complete set of inequivalent cusps for Г, and
(ii) if ru r2, slt and s2 are integers such that (ru sj = (r2, s2) = 1, then r1/sl
and r2fs2 are equivalent cusps modulo Г if and only if
r^ = r2 and s1=s2 (mod 2) and {n, s^ = {n, s2). A3.10)
Proof. If rl/s1 and r2fs2 are equivalent cusps modulo Г, we can choose
В е Г so that
Then the conditions A3.10) follow, which shows that they are necessary.
Using A3.10), we easily check that no two of d,..., C12 are equivalent cusps
modulo Г. Then an application of Lemma 13.1 shows that
2 ЛГ(Г; С,) = б(р + 1)(<? + 1) = (ГA):Г),
which, by Rankin's book [2, Eq. B.4.10)], shows that d,..., A2 is a complete
set of inequivalent cusps for Г.
Now suppose that the conditions A3.10) hold. Choose i and j so that
ri/si ~ d and r2/s2 ~ Cj modulo Г. But then by A3.10) and the definitions of
i = j.Thus,rl/sl ~ r2fs2, and the proof is complete.
Recall that for a cusp С and a modular form F, the order of F with respect
to Г at C, Ordr(F; ?), and the invariant order of F at ?, ord(/; ?), are related
by the equality
Ordr(F^C) = iV(r; C) ord(F; C), A3.11)
where, as above, N(T; ?) is the width of Г at (. We also recall from Lemma
0.1 that if M = (" Jj) is an integral matrix with determinant m = ad — bo 0,
and if (r,s) = 1, then
ord(F|M;r/s) = —ord(F;M-), A3.12)
m \ sj
where g = (ar + bs, cr + ds).
Now let d be one of the factors p, q, or pq of и = pq and choose В =
(« S) e ГB) such that
S s 0 (mod d) and у = 0 (mod e),
where e = n/d. We call
a Fric/ce involution of level d for Г = ГB) п Г0(и). For completeness, we also
define Мг = Ц \).

20. Modular Equations of Higher and Composite Degrees 405
The following lemma is very important for later computations.
Lemma 13.3. Let r/s e Q и {oo} and let F be any modular form on Г. Let
M = Md, as above. Then
OrdrmM; - J = Ordr(F; M-Y
In our proof of Lemma 13.3, we use the following simple, general lemma.
Lemma 13.4. Let M and Abe 2 x 2 and 2x1 integral matrices, respectively,
and let G = MA. Let m, a, and g denote the greatest common divisors of the
entries of M, A, and G, respectively, and let d = det(M). Then
ma\g and mg\da.
Proof. The first conclusion is clear. The second claim follows from the first
and the equality
dM~xG = dA.
Proof of Lemma 13.3. We employ the elementary properties ((a, b), c) =
(a, b, c) and (a + b, b) = {a, b).
Let (r, s) = 1, let M = ($ g), and set
With g = (r1, s'), the width of Г at r'/s' is, by Lemma 13.1,
2 2 2
By Lemma 13.4, g\d, since (r, s) = 1. Thus, since d\5 and (d, ft) = 1,
g = (d, g) = (d, r', s') = {d, dar + j8s) = (d, j8s) = (d, s). A3.14)
We now apply Lemma 13.4 to
-СИ K)-
We see that m, a, g, and d of Lemma 13.4 are replaced by d, 1, g':— (nr', s'),
and nd, respectively. We thus conclude that g'\n. Hence,
g' = (n, g') = (n, ndxr + nj8s, dyr + Ss)
= (n, dyr + Ss)
S
dl
= d{e, s\ A3.15)
J
= dl e, yr + -s

406 20. Modular Equations of Higher and Composite Degrees
since e\y and (e, S/d) = 1. Finally, from A3.13)-A3Л5),
2^.
d{e, s)
Thus, by Lemma 13.1, A3.11), A3.12), A3.14), and A3.16),
(d,
= Ordr(F;M-l,
which completes the proof.
A3.16)
Lemma 13.5. The Fricke involution Md interchanges the cusps (i. Сг> •••> Си
of Г = ГB) п Г0(и) pairwise. More precisely, we can group the indices into pairs
{i, j} such that Mdd ~ Cj and MdCj ~ Ci modulo Г, where j is determined by
i + 9, for i = 1, 2, 3, when d — pq,
i + 3, for i = 4, 5, 6, when d = pq,
i + 6, for i = 1, 2,..., 6, when d = q,
i + 3, for i = 1, 2, 3, 7, 8, 9, when d = p.
J =
Proof. The result follows from Lemma 13.2 and the definition of Md.
We conclude that if we know the orders of F at ?b B> and Сз» then we can
use Lemmas 13.3 and 13.5 to determine the orders of F at A; B,..., ?i2-
In order to uniformly examine F(t), F(pz), F(qx), and F(pqx), where F = r\,
/j, <jj, or hj, 0 < j < 2, we consider the effect of Md on a modular form F(/mz),
where /|d and m|e. Then also t\5 and m\y, and so
y/m
md
A3.17)

20. Modular Equations of Higher and Composite Degrees 407
In order to simply determine the multiplier system vF above, we impose
further restrictions on the matrix B. We require that B = (l °) (mod 243e), in
addition to our previous requirements that у = 0 (mod e) and 6 = 0 (mod d).
An examination of the multiplier systems @.14), @.16), and @.18)-@.23) for r\,
fi, gJt and hj, 0 < j < 2, now shows that
Ф(/а, mp, y/m, b/t) = 0 (mod 48),
in each case. Furthermore, since 6 = 1 (mod 8) and ? is odd, b(( = ? (mod 8).
Hence, by A3.7),
m e/?J \ 5/? j\
where
_{y, if F = ti,fi,g0,gi,g2,hl,
Уо~{2у, if F = ho,h2.
Employing A3.18) in A3.17), we conclude that
F(?rrn:)\Md
if F = t],f1,g0,gug2,h1,
Now apply Md to the proposed identities A3.4)—A3.6). First, M3 and M7
transform A3.4) into the reciprocal identity, that is, the identity relating
modular forms that corresponds to the reciprocal modular equation (ii). On
the other hand, M21 (and Mt) preserves A3.4). Second, M7 and M21 transform
A3.5) into its reciprocal, and Mt and M3 preserve A3.5). Formula A3.6) is
preserved by M3 and M7 but is transformed into its reciprocal by M21.
If we clear denominators in A3.4)-A3.6) and collect terms on one side, we
obtain, in each case, an equation F = 0, where F is a modular form of weight
r = jv on ГB) о Г0(п), and where v is the number of factors in each term of
F. Since r\, g^ and hp 0 < j < 2, have no poles, neither does F.
Let F* denote the modular form corresponding to F that arises from
the reciprocal relation, and suppose we now calculate the coefficients of
q°, q1,..., q" in both F and F* and find that these coefficients are equal to 0.
Then by Lemmas 13.3 and 13.5,
Ordr(F; Q > ц + 1,
for i = 1, 4, 7, 10. By the valence formula @.30), @.6), and @.24), we may
conclude that F = 0, provided that
4(ц + 1) > ^Г(ГA): Г) = ^(р + № + 1),

408
20. Modular Equations of Higher and Composite Degrees
ОГ
v(p
16
A3.19)
In the present applications, p = 3 and q = 7; that is, (p + i)(q + 1)/16 = 2.
The following table records the minimum value of ц necessary to effect a proof
of the identity, at least according to the reasoning we have used.
V
A3.4)
10
20
A3.5)
8
16
A3.6)
12
24
Using the computer algebra system MACSYMA, we have calculated the
coefficients off and F* through q" and have verified that all, indeed, are equal
to zero. This then completes the proof of A3.4)—A3.6) and therefore of Entry
13 as well.
Entry 14. Let ft, y, and S be of the third, eleventh, and thirty-third degrees,
respectively. Then
(i)
and
(ii)
ay
A -
- a)(l - y)J
/0A -
«УГ + ,A-«)A-
A - № -
ay(l-«)(l-y)\vi2
ay(l - a)(l - y)
- № ~ &)
wtere m and m' are the multipliers associated with the pairs a, fi and у, ё,
respectively.
Proof of (i). Applying C6.10) with ц = 6 and v = 5 and utilizing Entry 18(iv),
both in Chapter 16, we find that
П) = 2qf(q55, q71)Aq, q") + 2qi3f(q33,
9, q21)
= 2qf(q, qll)f{q55, q11) + 2q*f(q\ q9)f(q33, q")
i2) -f(-Q, -92)}{/(вп,в22)+/(-в11, -в22)}

20. Modular Equations of Higher and Composite Degrees 409
,«22)-/(-«n, -q22)}
, q22)~f(-q, ~q2)f{-ql\ ~q22)
W*\ A4.1)
where we have used Entries 30(ii), (iii) in Chapter 16. Now by Entry 19, B2.3),
and Entry 22, all in Chapter 16, we find that
-• A4.2)
Using A4.2) in A4.1), we deduce that
(p( — q3)(p( — q33)
A4.3)
Translating A4.3) by Entries 10(ii), ll(i),and 12(ii),(vi) in Chapter 17, we arrive
at
W33/ I V <*У /
Next, replace q by — q in A4.3) and transcribe the resulting identity via
Entries 10(i), ll(ii), and 12(i), (v) in Chapter 17 to find that
Z3Z33/
(-l + 2-1'3{ay(l - «)A - y)}). A4.5)
Now we apply C6.3) with ц = 6 and v = 5 and Enry 18(iv) seven times,
both in Chapter 16, to deduce that
= f(qi32, q132)uq12, q12) + q12f(q116, q*8)f(q32, q~8)
+ q**f(q220, q")f(q52, q~28) + q1O8f(q26\ D/(«72, <Г48)
+ q192f(q308, q-")f(q92, <T68) + q3OOf(q352, q'88)Rq112, q~88)
= cp(q12)cp(q132) + 2qW, q16)f(q8\ q116)
+ 2q16f(q\ q20)M\ q220) + 4q36^(q2^(q26*). A4.6)

410 20. Modular Equations of Higher and Composite Degrees
But by Entries 25(i), (ii) in Chapter 16,
) + vi-
). A4.7)
Substitute A4.7) in A4.6) and replace q* by — q2 to deduce that
<p(-q2)<p(-q22)-4q6<l,(q*)il,(q")
= <p(-qe)<p(-qee) - 2q2f(q\ q*)f(q", q**)
+ 2q*f{-q2, -q">)f{-<?\ -q"°) - 4д
Now replace q by q* in A4.3) to obtain the equality
Combining the last two equalities, using A4.2), and invoking Example (v) of
Section 31 in Chapter 16, we deduce that
V(-q2M-q22) = cp(-q6M-q66) + 2q*f(-q2, -q">)f(-q22, -q"°)
-2q2f(-q*)f(-q")
= <p(-q*)<p(-q'*) + 2qs^(q6)x(-q2mq66)x(-q22)
-2q2f(-q*)f(-q").
Translating this via Entries 10(iii), 11 (ill), and 12(iv), (vii) in Chapter 17, we
find that
y
- a)(l - y)) + 21'3{ay(l - a)(l - y)}1'12
1/2Г / 2 2 \1/12
Finally, we add A4.4) and A4.8) and subtract A4.5) to deduce that
A -
P
j
+ 2~1/3{ay(l -a)(l -у)
( zizii j /3 + 2'3 {ay(l - a)(l - ¦
\Z3Z33/
+ (ayI'4 - 1)}. A4.9)

20. Modular Equations of Higher and Composite Degrees 411
By Entry 7(i), the last expression in parentheses on the left side is equal to
-24/3{Д<5A - p){\ - <5)}1/12, while the last expression in parentheses on the
right side is equal to -24/3{<xy(l - <x)(l - y)}1'12. Hence, A4.9) reduces to
Entry 14(i), and the proof is complete.
Proof of (ii). Formula (ii) is simply the reciprocal of (i).
Entry 15. // P is of the twenty-third degree, then
(i) (a/?I'8 + {A - a)(l - P)}m + 22l3{<xP(l - «)A - Д)}1'24 = 1,
(ii) 1 + (ф1'* + {A - a)(l - Р)}Щ + 24/3{<^d - «)A - W1/12
= {2A + (a/JI/2 + {A - a)(l - P)}1'2)}1'2,
and
(iii) m
m
- a)(l - P)}1'12 -
P)}1
Ramanujan's formulation of Entry 15(iii) (p. 249) appears to be incorrect,
and we have corrected it here.
Entry 15(i) is due to Schroter [3], [4] who does not give his proof. A more
complicated modular equation of degree 23 is proved in his thesis [1] and is
also found in his paper [2]. Since the proof of (i) is decidedly nontrivial and
since there seems to be no proof in print, we provide one here.
Formula (ii) follows from (i) by elementary algebra, and so we begin with
this deduction.
Proof of (ii). Put
(a/3I/8 + {A - a)(l - /5)}1/8 = 1 - t and {<xj3(l - a)(l - P)}m = u.
Then (i) is equivalent to the equality t3 = 4м. Now,
= {2A + A - tL - 4«A - iJ + 2u2)}112
= 2{1 -2t + 3t2 - ft3 + §f4 - it5 + ^t6}1'2
= 2-2t + 2t2- |t3
= 1 + A - tJ - 2u + DuJ/3
= 1 + (ajSI'4 + {A - a)(l - Р)}щ + 24'3{аДA - «)A - P)Ytn,
which completes the proof.

412 20. Modular Equations of Higher and Composite Degrees
Proof of (i). We derive two modular equations of degree 23 and then combine
them to deduce (i).
In C6.11) of Chapter 16, set ц = 12, w = 1, and Q = q23 to deduce that
Hf(AQ,Q/A)f(Bq,q/B)+f(-AQ, -Q/A)f(-Bq, -q/B)}
= У q*-2B-2jl ^4^, ^^ / ABq2*-*\ * . A5.1)
Apply A5.1) with A = В = 1; A = -B = i; A = Q, В = q; and A = iQ,
В = — iq, in turn, to deduce that
¦ <p(-QM-q)}
ii
л=0
ч,'«) + /(-«б, *
z
п=0
.1) +л-е2, -i)/(-«2, -1)}
л=0
and
iiQ2, ~i)f(-iq2, 0 +A-iQ\ i)f(iq2, -0}
= Z (- l)V2T(e24+4", е24~4в)Л«48и, «4")- A5-5)
We now use Entries 18(ii), (iii) of Chapter 16 in A5.4). Also, easy exercises
show that
f(iq,-iq) = 9(-q4) and f(iq\ -i) = A - Ш-q2).
Multiply both A5.4) and A5.5) by q6 to get A5.4)' and A5.5)', say, respectively.
Then add A5.2) and A5.5)' and subtract A5.3) and A5.4)' from them. Using
the aforementioned simplifications, we deduce that
<p(-QM-q)} - <p(-Q*M-q*) - 2
= 2 X ^<2»-1>У(е20 + 8И) 028-8^28-8^ ^20+8»)
n=l
6
_ 2 V д4Bл-1J-2Bи-1) + 6/-//}20+8л л28-8п\/-/_52-8и
л=1

20. Modular Equations of Higher and Composite Degrees 413
6
л V „M2n—lJfif\20+8n л28-8п\ f г/„28-8п „20+8п\
=l L, ч J^si >vj )\j\q ,q )
n=l
= 2 t <Z4<2"-1>2/(e20+8", e28-8")/(-q4+4n, -q*-*n), A5.6)
л=1
where we have applied G.1) with a = — q8" and b = — q4+4n. Combining
the terms with indices n and 7 — n, 1 < n < 3, and employing Entries 18 (Hi),
(iv) in Chapter 16 and G.1) once again, we find that, from A5.6),
з
8, e20) - е4ле44. е4)}/(-«4, -
Employing A4.7) with q11 replaced by g, we find that the foregoing equality
becomes
Next, replace q2 by q above, and, consequently, Q2 by Q. Translate the
resulting equality by means of Entries 10(iii), (iv), ll(i), (ii), (iv), and 12(iii) in
Chapter 17 to obtain the modular equation
- {A - ofl(l - P
= 2*/3{аДA-а)A
that is,
(а/JI'8 + {A -
+ (аДI'2 + {A - a)(l - P)}112)}1'2. A5.7)
This is the first of the two modular equations that we have sought.
To obtain the second, we let ц = 16, со = 3, and Q = q23 in C6.14) of
Chapter 16. Hence,
<p{QMq) + cp(-QM-q) + 4q6^(Q2)il,(q2)
= 2 x «4и2ле32+4",е32-4|1)л«8л.«8+611)
n=O

414 20. Modular Equations of Higher and Composite Degrees
= 2cp(Q32Mqs) + 2q256f(Q6\ 1)/(<T4°, q56)
+ 4 X <24/(e32+4", Q32~*n)f(qs~6n, qs+6n),
n=l
where we have combined the terms with indices n and 16 — n, 1 < n < 7.
Combining the terms with indices n and 8 — n, 1 < n < 3, and utilizing Entries
18(ii), (iv) in Chapter 16, we find that
4>{-QM-q) +
8, e36) + е6яе4, е
24, S40) + Q*f(Qs, Q56)}f(q\ qi2)
4q26{f{Q20, Q") + Q2f(Q12, Q52)}f(q6, q10)
= Ш8) + 2
+ {№*) - 2Q2№16)}{<p(qs) - 2q2il/(q16)}
+ 2q3{f(Q6, e10) + Qf{Q2, e14)} {f(qe, q10) + qf(q\ q14)}
- 2q3{f(Qe, e10) - QfiQ2, e14)} {f(qe, q10) - qf(q\ q1*)}
+ W(G*№(e*), A5.8)
where in the penultimate equality we employed G.1) three times as well as the
equality
<p(q) = <P(q*) + 2#(<A A5.9)
which is a ready consequence of Entries 25(i), (ii) in Chapter 16. Now use A5.9)
four times and G.1) also four times to reduce A5.8) to the identity
<p(QMq) + <p(-Q)<p(-q) + 4q4iQ2mq2)
= <P(Q2Mq2) + <P(-Q2M-q2) + 2q4
Lastly, we translate this identity with the aid of Entries 10(i)-(iv) and
11 (i)—(iv) in Chapter 17 to deduce that
{A - a)(l - W
= i(l + уГ^I/2A + УГ^I'2 + {A - a)(l -
which can be recast in the form

20. Modular Equations of Higher and Composite Degrees 415
I'4 + {A ~ «)A - P)}m ~ W)vs ~ {A - «)A - P)}m
= {|A + WI'2 + {A - a)(l - P)}112)}1'2, A5.10)
which is the second of the modular equations that we sought.
Equating the left sides of A5.7) and A5.10), we deduce that
For small a, the expression within the outside curly brackets on the left side
is approximately equal to ?a > 0. Thus, in taking the square root of both sides,
we arrive at
1 - (ф1'* - {A - a)(l - P)}1* = 22'3{<*PA - a)(l - P)}ll2\
which is formula (i).
Observe that, by adding A5.7) and A5.10), we obtain (ii).
Proof of (iii). We employ the theory of modular forms. We first translate (i)
by means of Entries 10(i), (ii), 11 (i), (ii), and 12(i), (iii) in Chapter 17 to find that
cp\q) - 23tp V3)
= 2(VW«№(«23) - cp(-q2)cp(-q*6))(llcp(q)<p(q23) - 26qf(q)f(q23)
+ 36q2f(-q2)f(-q*6) - Щ3ф(-9)ф(-q23)
q
<p{q)<p(q23)
Using @.13), we write this proposed identity as an identity involving modular
forms,
g\{x) - ?
= DЛ0(т)М23т) - 2й2(т)М23т))
+ 36f,(T)f,B3t) - 28^(t)^B3t) + ^f^X A5-11)
Using @.14), @.16), @.19), and @.21)-@.23), we may easily verify that each
expression in A5.11) has a multiplier system identically equal to 1.
Letting Г = ГB) n ГОB3), we see from @.6) and @.24) that ft- = 12. If we
clear fractions in A5.11) and rewrite A5.11) in the form F := Ft + F2 H +
F12 = 0, we observe that the weight r of each form equals 3. Thus, by @.30),
in order to proveA5.11), we need to showthat the coefficients of qJ,0 < j < 36,
in F are equal to 0. However, if M is as in @.31)-@.37), we readily find that
F\M = —F. Thus, by @.38), we need only show that the coefficients of qj,

416 20. Modular Equations of Higher and Composite Degrees
0 < j < 18, in F are equal to 0. Using MACSYMA, we have, indeed, verified
this, and so the proof of A5.11), and hence of (iii), is complete.
Entry 16. // /i is of degree 19, then
1 - а У \a) V a(l - a)
and
Observe that (ii) is the reciprocal of (i), and so it suffices to establish (i).
However, we use the theory of modular forms and apply @.38). Thus, in fact,
(i) and (ii) will be established simultaneously.
Previously proven modular equations of degree 19 by Sohncke [2], Schlafli
[1], Fiedler [1], Weber [1], Russell [1], and Fricke [1] do not have the
simplicity of Ramanujan's modular equations in Entry 16.
Proof of (i). Utilizing Entries 10(i), (iii), (iv) and ll(i), (ii), (iv) in Chapter 17,
we rewrite (i) in the form
<p(q2Mq38) + 4q
2 7№) 2 7
9(-q2)
_J2 ,ФЧ-д)\9(-д) vHg) 7
\ Ф(Ф 1 <PA2) 9(q) q
Written in terms of modular forms, by @.13), the foregoing equality becomes
+ 4<?2BтH2C8т)
Л2(т) Л0(т) h
:A9т) з3A9т)
Л2(т) в1(т) ~ Л0(
ЛоA9т)")
A6.1)

20. Modular Equations of Higher and Composite Degrees 417
We now must examine the multiplier systems for each term in A6.1). By
identically the same proof, with 11 replaced by 19, we deduce as in Lemma
7.1 that дх{2х)д1(Ъ^х) + 4^2Bт)д2C8т) is a modular form on Г = ГB) п ГоA9)
with the multiplier system (^). If F = gu h0, h1, or h2, then by the same
reasoning that gave G.16), we find that F3A9t)/F(t) has a multiplier system
equal to (^). Thus, upon multiplying out in A6.1), we find that each term has
a multiplier system identically equal to 1.
Multiplying out and clearing denominators in A6.1), we may rewrite A6.1)
in the form F := Fx + F2 + ••• + F19 = 0. The weight r of each modular form
Fp 1 <j < 19, is equal to Ц-. Also, by @.6) and @.24), pr = 10. By @.38), if we
verify that the coefficients of qJ, 0 <j < 27, in both F and F\M are equal to
0, then the proof of A6.1), and hence of both (i) and (ii), is complete. Using the
computer algebra system MACSYMA, we have, indeed, found that all of the
required coefficients are equal to 0.
Entry 17. We have
(i) <p(q)<p(q35) = <p(-q)cp(-q35) + Aqf{-ql0)f{-q^) + Ц9ф(Ч2)^0),
(ii) cptfMq1) = H-q'M-q1) + 4Ч3ф^°)ф(Ч1*) - 4q3f(-q2)f(-q10),
(iii) cp{qMqi35) = cp{-qy0)cp{-q5A) + 2qf{q9)f{q15) + 2q^(q5mq21),
and
(iv) <p(q5)cp(q21) = cp(-q2)(p{-q210) + 2qllHq)il>{q^) + 2q2f(q3)f(q^).
Proof of (ii). First, putting A = В = 1, ц — 6, and v = 1 in C6.2) of Chapter
16, we find that
5
_ у _12«2+14n+7/Y 35A4+4в) _35A0-4л)чд 26+4л „-2-4я\
n=0
2
_ 2 у
7г/ 35Ц4+4Л) 35A0-4n)WY 26+4л^ -2-4и\
where we have employed Entry 18(iv) of Chapter 16 to show that the terms
with index и and 5 — n, 0 < и < 2, are equal. Second, setting ц = 6 and v = 1
and replacing q by q2 in C6.10) of Chapter 16, we arrive at
2
_ 2 у д12л2+12л+3/Уд35A4+4и) _35A0-4л)\ /Y_14+4n _10-4л\
л=0
Subtracting the latter equality from the former, employing Entry 18(iv) in
Chapter 16, and noting the cancellation of the terms with index 1, we see
that

418 20. Modular Equations of Higher and Composite Degrees
?7) — (p( — qs)(p( — q1)} — 2q3i//(q10)il/(q14)
0,q350){q5f(q2,q22)- <?3/(<Л 414)}
q14)-q15f(q2,q22)}
= -2q3{f(q™, q") - q2f(q2, q22)} {f(q3™, q*90) - qlof(q10, q110)}
= -2q3f(-q\ -q*)f(-q10, -q140)
= -2q3f(-q2)f(-q™),
where we have applied G.1) twice. This concludes the proof of (ii).
Proof of (i). In (ii), replace q by qil5 to obtain the equality
<p(q)<p(q715) = <p(-q)cp(-q115) + 4Ч3'5ф(Ч2)ф(Ч^5) - 4q3'sf(-q2's)f(-q14).
Employing Entries 10(i)-(iii) in Chapter 19, we deduce that
(p(q){<p{q35) + 2q7l5f(q2\ q*9) + 2q2SI5f(q7, q63)}
5f( — Q21 —fl49) + 2a28l5f( — a1 — q63)}
°) +f(q2s, q*2) + «14/5/(«14. q56)}
\-qlo)+f2(-q*,-q')
If we now equate rational parts on both sides, we obtain (i).
Proof of (iii). The proof of (iii) is very troublesome, indeed. First, replace q
by — q in (iii) to find that
= <p(-q)<p(-q135).
We now translate this into a modular equation via Entries 10(ii), (iii), 11 (ii),
and 12(ii) in Chapter 17. With the subscript on each parameter below corre-
corresponding to its degree, we find that
(z5z27I/2{a5a27(l - as)(l -
Note that we have obtained a modular equation involving six moduli! Now
replace each modulus an by its complementary modulus 1 — a135/n to obtain
the reciprocal formula
(z27z5I'2(a5a27I/8 - (z15z9I/222/3(a9a15I'6{(l - a9)(l - a^)}1'24
+ (z27z5I/2{a5a27(l - a5)(l - a,,)}1'8 = (z135z1I/2(a1a135I/4.

20. Modular Equations of Higher and Composite Degrees 419
Now this formula can be translated into the theta-function identity
= V WWta270), A7.1)
via Entries ll(i)-(iii) and 12(iv) in Chapter 17. Since A7.1) and Entry 17(iii)
have been shown to be equivalent, it suffices to prove A7.1).
In the remainder of the proof, all references, unless otherwise stated, will
be to results found in Chapter 16.
In C6.8), set ц = 16, v = 11, and Q = q135 to deduce that
+ ? qi^-lln+^Ql6+2nQl6-2n^q22nq32-22ny
l
n=l
Replacing q by — q and adding the result to the foregoing equality, we find that
6, <T144)
2 ? q6**-22.*l64 ^^
n=l
2, Q20)f(q12, q20) + 2q136f(Q8, Q2*)f(q8, q2*)
+ 2q3O6f(Q\ Q2S)f(q\ q2S),
where we have made several applications of Entry 18(iv). Rewriting the right
side with the aid of Entries 25(i), (ii) and 30(ii), (iii), we find that
к34{ле2,е6)+л-е2. -e6)}{/(«2.«6)+/(-«2. -в6»
к34{ле2, е6) -л-е2. -е6)
A7.2)

420 20. Modular Equations of Higher and Composite Degrees
Next, we employ C6.1) with q replaced by q18 and ц = 4, v = 1, A = q30,
В = q18, and Q = qi35. Accordingly,
H/(«3W2)/(«60,«120)+/(-«36, -q12)f(~q60, ~q120)}
n=0
16)f(q96
= <P(Q16)f(q96, q192) + q
, q2W) + q1212f(Q~8, S4<W2°,
where we have utilized Entry 18(iv) several times and employed Entries 25(i),
(ii) and 30(ii), (iii) as well. Employing once again these four entries as well as
G.1) in this chapter, we deduce that
,q120) +f(~q36, -q12)f(-q60, ~q120)}
= 2q*<p(Q*)f(q12, q60) + 2q*cp(-Q*)f(-q12, -q60)
+ 2q\i-<?)f{-q12, ~q60)
+ fi-q3, -q15)}
= q<P(Q)f(q\ q15) ~ q<p(-Q)f{-q3, ~q1S)
+ 2q*<p(-Q*)f(-q12,-q60)
= Ы<1)Ш - <p(q9)} + Ы-6) W-e) - <p(-q9)}
-<p(-Q*)W(-q*)-<p(-q36)}, (П.З)
where we have used Corollary (i) in Section 31 three times.
Third, we invoke C6.2) with q replaced by q18, ц = 4, v = 1, A = q30,
В = q~18, and Q = q135. Then in the second equality below, we apply Entry
18(iv) several times. In the third, Entries 30(ii), (iii) are employed. In the fourth,
we make four applications of G.1) of the present chapter with a = ±q6,
b= ±qee and a = ±q30, b = ±q*2. Two further applications of G.1), with
a = ±q6, b = q12, are made in the penultimate equality. Lastly, Corollary (ii)
in Section 31 is utilized. Thus,
М/(взб.в72)/(вб0.?120)-/(-в36. -q12)fi-q60, -q120)}
з
_ у _144n2+192n+120r//j20 + 8n /Jl2-8n\r/fl372 + 72d _-84-72в\
n=0

20. Modular Equations of Higher and Composite Degrees 421
20)f(q8\ q2M) + q3OOf(Q4, Q28)f(q132, q156)
Q28)f(q60, Q228) + q6Of(Q12, Q20)f(q12, q216)
, Q6){q36f(q8\ q204) + q3Of(qi32, q156)
+ Ы-Q2, -Q6){q36f(q84, q204) ~ q3of(q132, q156)
-q42f(q6°,q228) + q60f(q12,q216)}
mQ2){q36f(q6, q66) + q3Of(q30, q42)}
+ ^(-Q2){q36f(~q6, ~q66) ~ q30A-q3°, ~q42)}
k34(Q2)f(qe, q12) - h34(-Q2)f(-qe, q12)
- k30*H-Q2)№-q2) + q2H-q18)}-
Rearranging slightly, in summary, we have shown that
2q4{f(q36,q12)f(q6O,q12°)-f(-q36, -q12)f(-q60, -qi20)}
+ <И-418Ж-е2)}- A7.4)
Subtracting A7.4) from A7.3), we deduce that
q4f(~q36, -q12)f(-q60, ~q120)
= HvteMQ) + <p(-qM-Q) - 2(p{-q4M-Q4)
<P(~q9M-Q) ~ 2<p(-q36)<p(-Q*)
2 ~ 4q36*(-q18m-Q2)}. A7.5)
Quite remarkably, the latter expression within curly brackets vanishes, as we
demonstrate in the next paragraph.
Let ц = 8 and v = 7 in C6.3). Combine the terms with indices n and 8 - n,
1 < n < 3, with the aid of Entry 18(iv). Then apply Entries 25(i), (ii), 18(iv)
again, and 30(ii), (iii). Accordingly,
. у „16n2/Y_240+60n _240-60п\/Уд16+28л „16-28п\
и=0
2 ? q16n2f(q2*°-6On,
и=1

422 20. Modular Equations of Higher and Composite Degrees
) - 9(-q60
2q*f(q180, q3°°)f(q12,
0, q90) +f(-q30, -q9°)}{f(q2, <Z6) +f(-q\ ~q6)}
O,q9O)-n-q30, -q9°)}{f(q2,q6)-A-q2, -q6)}
If we replace q by q9 in the identity above, we see that we have, indeed, shown
that the latter expression in curly brackets on the right side of A7.5) is equal
to 0. Thus, A7.5) reduces to the equality
= HvtoMQ) + 9(-qM-Q)} - <p(-q*M-Q*)
+ 2q3^(-q^(-Q2). A7.6)
If we now multiply A7.2) by 2 and then subtract A7.6) from the resulting
equality, we obtain A7.1), which is what we sought to prove.
Proof of (iv). Replacing q by q115 in (iii), we find that
<p(qll5)9(q27) = <p(-q2M-q5AI5) + 2qll5f(q9'5)f(q3)
Applying Entries 10(i)-(iii) of Chapter 19, we find that the previous formula
becomes
+ 2q^f{q3, tf) + 2q^f(q, q9)}
5) +f2(-q18, q21) - ql8'5f2(q9, -
135) + f(qs\ q81) + q2*5f(q2\ qi0*
Equating rational parts on both sides, we complete the proof.

20. Modular Equations of Higher and Composite Degrees 423
Entry 18. Let /?, y, and 5 have degrees 5, 7, and 35, respectively. Let m and m'
denote the multipliers connecting a, /? and y, 8, respectively. Then
(i) ИI'4 + {A - a)(l -
+ (Py)m + {(i - № - y)V'A + 2^{/fy(i - № - y)Y112
= 1 + {1 + 24/>jSy<5(l - a)(l - /0A - y)(l - <5)}1/24}2,
(ii) {(a^I/4 + {A - a)(l - ^)}1/4 + 24/3{a^(l - a)(l - ^)}l/12}
- /*m - y)}m + 24/3{iSy(i - № - y)Yli2}
- a)(l - /0A - 1/24
x «a^jW8 + {A -
(iii) (a*I* + {A - a)(l - 3)}1'4 +
(iv) (ДуI'4 + {A - />)A - у)}1'4 -
- у)}1'24 -
- y)}1/24 + {ЩуA -
- a)(l - <5)}1/8 + {16a5(l - a)(l - <5)}1/24
Ту) Л<Г
+ 2
-№-y)
-y)Y12
Proofs of (iii), (iv). If we transcribe (iii) and (iv) via Entries 10(i), (ii), 11 (iii),
and 12(iii) in Chapter 17, we obtain Entries 17(i) and 17(ii), respectively.
Hence, the proofs are complete.
Proofs of the remaining modular equations are accomplished by means of
the theory of modular forms.
Proofs of (i), (ii), (v)-(vii). Using Entries 10(i)-(iii), ll(i)-(iii), and 12(i), (iii) in

424 20. Modular Equations of Higher and Composite Degrees
Chapter 17, we translate the six modular equations to be yet proved into the
proposed theta-function identities
д9Ф(д2Жд70) <р(-дМ-д35) .g3R-g2)f(-g70)
(M35) ()C5)
,дЧ(д10Жды) <p{-gsM-g7) ,<
cptfMq1) cptfMq1) <p(qsMq7)
<p(qsMq7)
<p(-q)<p(-q35) + 4д3Я-д2)Л-д70))
x Dд3ф{д1О)ф(д14) + q>{—gs)<p{—g1) + 4gf(—g10)f(—g14))
, oq2fW(q5)f(q7)f(q35)
x {4д6ф(д)ф^ъ)ф№)
<р(д5)<р(д7)(Ы(-д5Ж-д7)
= <р(д)<р(д35)Лд5)Яд7),
<p(qMq35)W(-q5m-q7)-
= 9(g5)<p(g7)fW(g35),
д3ф(.д)ф{дзъ) (р(—д2)(р(—д
Ф(Я
+ 2
¦2q
70)
5) + <p(-g2)<p(-910)(p(—q14)(
'4*Ф(-яЖ-ч35) +f(q5)Aq7))
3ф(-д)ф(-д35)-ГШй35))
д3Ф(-дЖ-д35)
p(-q70)),
A8.2)
A8.3)
A8.4)
Ф(д5Жд7) <p(-q10M-g14) Ф(-q5Ж-q7)
, 2д2А-д2Ж-д70)_.
f(-qlo)fi-q1A) '
and
) <P(-q2M-q10) eV(
+ 2^(~^ 2)/irg7o)) = -1- A8-6)
Next, we employ @.13) to rewrite A8.1)—A8.6), respectively, as
до(т)доC5т) + ^ i?(t)i?C5t)
4.»/Et)i/Gt)

20. Modular Equations of Higher and Composite Degrees
425
х DМт)ЛоEт)М7т)йоC5т) + Л2(т)М5т)М7т)Л2C5т)), A8.8)
A8.9)
A8.10)
= 1, A8.11)
Mt)/ioC5t) Mt)M35t)
йоEт)ЛоGт) Л2Eт)й2Gт)
and
= -1. A8.12)
hoEx)hoGx) h2Er)h2(lx) i
Л0(т)Л0C5т) + Л2(т)й2C5т) '
We now apply the theory developed in Section 13. By employing the
multiplier systems @.14) and @.18)-@.23) in A3.7) and A3.8), we may easily
verify that each expression in A8.7), A8.8), A8.11), and A8.12) has a multiplier
system identically equal to 1, while each expression in A8.9) and A8.10) has a
multiplier system equal to exp {ni(b — c)d/2}. In conclusion, for each proposed
identity, the terms have identical multiplier systems.
We now apply the operator Md to the proposed identities A8.7)—A8.12). In
all cases, A8.7) and A8.8) are invariant under the Fricke involutions. Each of
A8.9)—A8.12) is transformed into its reciprocal by M5 and M7 but is left
invariant under M35.
Proceed now as in Section 13. Here p = 5 and q = 7. Thus, (p + l)(q + 1)/16
= 3. Using A3.19), we obtain the following table of values for ц and v.
V
A8.7)
8
24
A8.8)
8
24
A8.9)
4
12
A8.10)
4
12
A8.11)
8
24
A8.12)
8
24
Using the computer algebra system MACSYMA, we have calculated the
coefficients of F and F* (in the notation of Section 13) through q» and have
verified that, in each of the six cases, all, indeed, are equal to 0. This completes
the proofs of A8.7)-A8.12) and hence also of Entry 18.
If we multiply (vi) and (vii) together, we obtain a modular equation first
discovered by Weber [1].

426 20. Modular Equations of Higher and Composite Degrees
Entry 19.
(i) <p(q)<p(qe3) - vtfMq9) = 2qf(q3)f(q21).
(ii) Hq'mq9) - ч'ФШ(ч63) =f(-qe)f(-q*2)-
(iii) Let /?, y, and S have degrees 3, 13, and 39; or 5, 11, and 55; or 7, 9, and 63,
respectively, in each case. Then
{A - a)(l - 5)}1'8 - (adI'8
{A - P)(l - y)yi* + (fly)V* ~ {A -
_ (adI'8 + {ad(l - a)(l - ё)}1'8
к
where m is the multiplier associated with a and /?, and m' is that associated
with у and 3. In the third expression, the plus sign is to be taken in the first
two cases, and the minus sign is chosen in the third instance.
(iv) // P, y, and 3 are of degrees 3, 13, and 39 or of degrees 5, 7, and 35,
respectively, in each case, then
a<5\1/8 /A - a)(l - <5)\1/8 /a<5(l - a)(l - <5)V/8
) + ) ^
Jy) + \{\ - P)(l - y)) ^7A - № ~ V)
aS(l - a)(l -
+ 2
and
ад) + \A - <x)(l - 3)) ~ \aS(l - a)(l - ,
J)V/12=+ /m
where the plus sign is taken in the first case, and the minus sign is assumed
in the second. The multipliers m and m' have the same meaning as in (iii).
In fact, in the case of degrees 5, 7, and 35, the two formulas of (iv) are
identical to Entries 18(vi), (vii), respectively.
The first and third equalities in (iii) appeared in Ramanujan's [10, pp. xxix,
353] second letter to Hardy.
When p, y, and 3 have degrees 3, 13, and 39, respectively, multiplying the
two modular equations of (iv) yields a modular equation first found by Weber
[1].
It will be simplest to first prove (ii) and then deduce (i) from it.
Proof of (ii). Unless otherwise stated, all references in this proof are to results
in Chapter 16.

20. Modular Equations of Higher and Composite Degrees 427
First, we employ Corollary (ii) of Section 31 and Entry 18(ii). With the use
of Entry 18(iii), we next apply C6.1) with ц = 4, v = 3, q replaced by q9'2, and,
in turn, A = q21'2, q6*2 and В = q'9'2, q~b>2. Hence,
= {f(q21, q42) +
= i/(?", q*2)f(l q9) - i/(i, q")f(q\ q6)
3
_ у 36п2 + 15у/ 42G + Зп) 42E-Зп)чд 42+54п 30-54п\
3
_ у „36л2 + 33п+6л„42G+3л) „42E-Зя)\у-/„66 + 54я _6-54в\
п=О
3
_ у -36в2+15яу-/ 42G + 3n) _42E-3n)W у-/_42 + 54п _30-54п\
0
и=0
If we now apply G.1) of this chapter with a= —q6 and b = -q12,v/e easily
deduce that
f(-q6, ~q12) =f(q30, qA1) ~ q6f(q6, q66)- A9.2)
Furthermore, applying Entry 18(iv) twice with n = 2, we find that
f(~q6, ~q12) = q132Aq~102, <?174) - q9Of(q-78, q150)- A9.3)
Using again Entry 18(iv) with n = 2, we see that
f{q~2\ q96) - ЧгШ*\ q120) = 0, A9.4)
while employing Entry 18(iv) with n — 5, we arrive at
/(<T132, q204) ~ q6Of(q-156, q22S) = 0. A9.5)
Using equalities A9.2)—A9.5) to simplify the terms with indices 0, 2, 1, and 3,
respectively, we find that A9.1) reduces to the equality
= f(q210, q29*)f(-q6, ~q12) ~ qS4f(q~42, q546)f(~q6, ~q12)
= f(-q6){f(q210, q29A) ~ q*2f(q42, <?462)}
where we have applied Entry 18(iv) and then utilized A9.2) above with q
replaced by q1. This concludes the proof of (ii).
Proof of (i). Let <Xj have degree j, and let z, = q>2(qJ), as usual. Translating
Entry 19(ii) via Entries ll(i) and 12(iii) in Chapter 17, we find that

428 20. Modular Equations of Higher and Composite Degrees
Replacing a, by its "complement" 1 — a63/j, we deduce that
(z9z7)^{(l - a9)(l - a,)}1/8 - (z63z1I/2{(l - «бзШ - «i)}1/8
= 21'3(z21z3I/2{(l - a21)(l - a3)a21a3}1'12.
Translating this formula with the aid of Entries 10(iii) and 12(iii) in Chapter
17, we deduce that
cp(-ql*M-q18) - <p(-q2M-q126) = 2q2f(-q6)f(-q*2).
If we replace q2 by — q, we obtain (i) at once.
Proof of (iii). All references in this proof are to Chapter 16, unless otherwise
stated.
Apply C6.6) with ц = 8 and put Q = q64~v\ Then
S:= MPGX?) /(ZXV)} qWq)Hq
t 2v", q*~2vn). A9.6)
t
я=О
By two applications of Entry 18(iv), with n = 1, 2v, we find that
_16(8-и)у/д16+4(8-я) Ql6-4(8-n)\^4+2v(8-n) 4-2v(8-n)\
_ „16(8-nJQ-16+4«_D-2v(8-n))vBv+l)+D + 2v(8-n))vBv-l)
~4n, al6+4n)/(<z4~2vn, ?4+2vn)
Using this equality in A9.6) for 1 < n < 3, we find that
S = <p(Q16)<p(q4) + 2q256iP(Q32)nq*+8\ q4'8")
+ 2 E ?16b2/(G16+4b. Q16-4")/^4-2, ?4+2"-). A9.7)
n=l
If v is an odd integer, we apply Entry 18(iv) with n = (v — l)/2 and twice
with n = v to deduce that, respectively,
and
Employing these results in A9.7), we find that
2q16{f(Q12, Q20) + Q2f{Q\ Q28)}f(q*+2\ q*~2v)
+ 2q16f(Q2, Q6)f(q4+2\ q*~2v). A9.8)

20. Modular Equations of Higher and Composite Degrees 429
In the last equality, we applied Entries 25(i), (ii) and G.1) in this chapter with
a = Q2 and b = Q6. Now in the three cases of part (iii), v = 5, 3, and 1,
respectively. By using Entry 18(iv) and Entries 25(i), (ii) once again, we see
that we can write A9.8), in all cases, in the form
S = HV(Q)V(9) + <p(-QM-q)} + 2(Qq)ll4iHQ2)iHq2).
In summary, for v = 5, 3, and 1,
= <p{QMq) + <p(-QM-q)
Translating this formula by means of Entries 10(i), (ii) and 1 l(iii) in Chapter
17, we deduce that
(z8+vz8_,)«2(l + {A -
= (z^z^a + {A - a)(l -
This establishes the first part of (iii).
Next, the three parts of the corollary in Section 37 may be collectively
written in the form
where v = 5, 3, and 1, and where the plus sign is chosen in the first two cases
and the minus sign is assumed in the last case. Translating this formula via
Entries 11 (i), (ii) in Chapter 17, we find that
- /00 - V)}1/8)
= (z64_v2ZlI/2((a<5I/8 ± {a<5(l - a)(l - S)}1'*), A9.9)
which establishes the third equality of (iii).
The reciprocal of the preceding modular equation is obtained by replacing
a, /?, y, and 8 by 1 — S, 1 — y, 1 — /?, and 1 - a, respectively. Accordingly,
/?)}!/<> _ {A -
= (г1г64_У01/2({A - 6)A - a)}1'8 + {A - 8){1 - a^a}1'8). A9.10)
For brevity, set A = (a<5I/8, A' = {A - a)(l - ^)}1/8, В = (PyI/B, and
B' = {A - 0)A - y)}1'8. Then combining A9.9) and A9.10), we deduce that
A + AA' A' + AA'
В - BB' B! - BB'
Consulting the statement of (iii), we see that it suffices to prove that
A ± AA' _ A' - A
B-BB' ~ B' -B'

430 20. Modular Equations of Higher and Composite Degrees
By cross-multiplication, it is easily seen that the last two equalities are equi-
equivalent. Thus, the proof of (iii) is complete.
Proof of (iv). As we have already seen, we need only prove the case when /?,
y, and 8 are of degrees 3, 13, and 39, respectively. As in the proofs of Entries
18(vi), (vii), we employ the theory of modular forms. Since the modular equa-
equations to be established are exactly of the same shapes as those of Entries 18(vi),
(vii), the proofs are almost identical. Thus, we forego almost all of the details.
In particular, we do not record the relevant theta-function and modular form
identities.
In the instance at hand, p = 3 and q = 13, and so (p + l)(q + 1)/16 = \.
Also, in each case, v = 8 as before. Thus, ц = 28 in each case. With the same
notation as in Sections 13 and 18, we need to show that the coefficients of q°,
q1, ...,q2S are equal to 0 for both F and F*. Using MACSYMA, we have,
indeed, done this, and so the proof of (iv) is complete.
Entry 20.
(i) Let /?, y, and 8 have one of the following sequences of degrees:
3,21, 63;
5,19, 95;
11, 13, 143;
7,17,119;
9, 15, 135;
respectively. Then
+ {A - a)(l - <5)}1/2)}1/2
+ {A - a)(l - <5)}1/8 + {aS(i -
where the plus sign is taken in the first three cases and the minus sign is
chosen in the latter two cases. Here m is the multiplier associated with a and
P, and m' is that attached to у and 8.
(ii) Let P, y, and 8 have one of the following sequences of degrees:
5,19, 95;
7,17,119;
11,13,143;
respectively. Then
i2 + {(i-P)(i-y)Vi2)V12
- y)V18 - {№ - № - y)V18
m

20. Modular Equations of Higher and Composite Degrees 431
where the minus sign is chosen in the first two cases and the plus sign is
assumed in the last case. The multipliers m and m' are as in part (i).
We first prove the three formulas of (ii). Then the second, fourth, and third
formulas in (i) can be deduced immediately. Lastly, the first and fifth formulas
of (i) are established. Because 3 and 21 as well as 9 and 15 have a factor in
common, a somewhat different argument is needed to establish these two
formulas.
Proof of (ii). Unless otherwise stated, all references in this proof are to
Chapter 16.
Let v denote one of the integers 7, 5, and 1 and put Q = g144~v2. First, in
C6.8), set fi = 12 to deduce that
122 2 322v, <?24~12v)
Replace q by — q and subtract the result from the equality above to arrive at
n=O
Second, in C6.2), let ц = 12, set A = В = 1, and replace q by q2. Accord-
Accordingly,
ii
_ V „<4n + 2)A2+v)+48n2/YQ52 + 8n Q44-Sn\f/ 96+4v+8vn _-4v-8vi>\ B0 2)
«=0
Third, in C6.10), let ju = 12 and replace q by q* to deduce that
5
_ V „48n(n+iyv052 + 8n л44-8лч y-/-48+4v+8vn _48-4v-8vn\
n=0
= 1 E ^8«(«+1)дд52 + 8П) g44-8n^48+4v+8vB) ^8-4v-8vB)
n=O
We have extended the sum to 0 < n < 11 by using the identity
-48(ll-B)A2-n)r(Q52 + 8(U-n) ?L4-8(l l-n)\ yv 48+4v + 8v(ll-n) 48-4v-8v(l l-n)\
_ -48n(n+l)/Y/j52 + 8n /j44-8n\y-/_48+4v+8vn _48-4v-8vn\
which is established by two applications of Entry 18(iv), with n = 1, v
there.
Combining B0.2) and B0.3), we find that

432 20. Modular Equations of Higher and Composite Degrees
+2X<?24v) - <p(-<?24+2X-<?24-2v)} + 241 W8+4v)<A(<?48-4v)
11
_ у „48n(n + l) + 12/YQ52 + 8n л44-8п\ Г „12 + 2v+4viyv 96+4v+8vn „-4v-8vn\
в=0
_ у ~48n(n+l) + 12/Y/}52 + 8n /}44-8пч/Y_12 + 2v+4v« „12-2v-4vn\
n=0
_ V „48л(и+1)+12Г/ул52 + 8л л44-8п\
, л14+4илд100 + 8я Q-4-Sn\\ уу„12 + 2у+4ул „12-2v-4vb\
5
_ у „48п(л+1) + 12/у/}14+4л /}10-4л\ «„12 + 2v+4vn „12-2v-4vn\
n=0
2
_2 у д48п(п+1)+12/У014+4л л10-4п\/-/„12 + 2v+4vn „12-2v-4vn\ B04)
n=0
where we have applied G.1) of this chapter with a = ^12+2v+4vn and b =
^i2-2v-4v^ utiiizeci Entry 18(iv) with n = v to combine the terms with indices
n and n + 6, 0 < n < 5, invoked G.1) once more but with a = Q14+4n and
b = Q10", and lastly utilized Entry 18(iv) again with n = v to show that the
terms with indices n and 5 — n, 0 < n < 2, are equal.
Combining B0.1) and B0.4), applying G.1) of this chapter with
a = -g3-'-2 and b = -q3+*+2™, invoking Entries 18(iv), (iii), and employ-
employing G.1) once again but with a = — Q2 and b = —Q*, we find that
2
2 У д48л(и+1)+12уу/}14+4п /}10-4л\ f r/ 12 + 2v+4vn „12-2v-4vn\
n=O
™3-v-2vn/'/™4vn + 2v 24-4vn-2v\\
2
2У „48п(п+1)+12у-/л14+4в /}10-4п\/у_ 3-v~2vb _дЗ+у+2ул\
и=О
= 2q12{f(Q10, G14) - Q2f(Q2, Q
= 2q12f(-Q2, -Q*)f{-q3-\ -q3+l-
When v = 7, 5, and 1, the last expression has the values
-2q*f(-Q2)f{-q2), -2q">f(-Q2)f(-q2), and 2q12f(-Q2)f(-q2),
respectively, by Entry 18(iv). In summary, we have shown that

20. Modular Equations of Higher and Composite Degrees 433
'1 + </>(-<724+2>(-<z24-2v)
- 2qM-q12+v)H-q12'v) ± 4qQ)vl2f(-Q2)f(-q2), B0.5)
where the minus sign is taken in the first two cases and the plus sign is chosen
in the last case.
We now translate this equality via Entries 10(iii), (iv), ll(i), (ii), (iv), and
12(iii) in Chapter 17. If we furthermore use the identity
= {i(i + s/Py + У0 - /*)(i - ?))}1/2, B0.6)
which is easily established by squaring both sides, we complete the proof.
Proof of (i). In B0.5), replace qhyqm2~v) to deduce that
We now equate the rational parts on both sides. In order to do this, we must
employ Entries 10(i)-(iii) in Chapter 19 for the case v = 7, Entries 17(iii)-(v)
in Chapter 19 for the case v = 5, and Entry 6 in Chapter 20 in the case v = 1.
The details are somewhat tedious as each case must be examined separately.
However, the details are straightforward, as in the similar proof of Entry
17(iv), and we eventually find that
<p(Q2Mq2)
= 2Ш1/8<А«2Ж<7) + <p(-Q2M-q2) ±
+ V/(-424+2v)/(-<z24~2v),
where the plus sign is correct when v is equal to 7 or 1, and the minus sign is
chosen when v = 5. Employing Entries 10(iii), (iv), ll(i), (ii), (iv), and 12(iii) in
Chapter 17, we see that the foregoing equality transcribes into the modular
equation
= (<x<5I/8 + {(l-a)(l-<5)}1/8±
I*
т
If we use B0.6), we find that the equality above yields the required identity.
We now establish part (i) in case 1. In C6.8) of Chapter 16, put ц — 8 and
v = 1. Then apply Entries 25 (i), (ii) and Corollary (ii) in Section 31 of Chapter

434 20. Modular Equations of Higher and Composite Degrees
16. Accordingly,
4 ?
в=1
V/(<?630, q378)f(q2, 414) + WW82, q126)f(q6,
2)>A(94)
~ cp(-q126M-q2) + 4Ч
- <p{-q126M-q2)
q6i) - 2Ч*ф{
Hence, by Entry 19(ii) of this chapter,
V>A(«)>A(963) + 4q2f(-q6)f(-
= q>(q126Mq2) - ф(-9«
Transcribing this equality via Entries 10(iii), (iv), ll(i), (ii), (iv), and 12(iii) in
Chapter 17, we complete the proof of (i) in case 1.
Lastly, we establish (i) in the fifth case. Rewriting A7.6) and using Entries
25(i), (ii) in Chapter 16, we find that
2<Z3W)<A(<7270) + <p{-<fM-qS4°) - Wf-^W-?270)
+ 4q4f(-q36)f(-q60)
(-q)} {<p(qi35) + v(-q135)}
Replacing q by y/q, we find that
<p(q2Mq210) + W(94)>A(9540)
= 2q,W(q135) + q>(-q2M-q270) -
+ 4q2f(-q18)f(-q30).

20. Modular Equations of Higher and Composite Degrees 435
Transcribing this equality by the same entries in Chapter 17 as in the previous
four cases, we complete the proof.
Entry 21.
(i) Let a. and /? has degrees 1, 7; 3, 5; or 1, 15, respectively. Then
(a/?I'8 + {A - a)(l - P)}1'8 ± {a/?(l - a)(l - 0)}1'8
where the minus sign is chosen in the first two cases and the plus sign is
selected in the last case.
(ii) Let P, y, and 8 have one of the following sequences of degrees:
3, 13, 39;
5,11,55;
7, 9,63.
Let m and m' denote the multipliers associated with the pairs a, /? and y, 5,
respectively. Then
{A - a)(l - б)}1* + ((PyI*8 + {Py(l - /)( у)})
and
{A -
m
where the minus sign is correct in the first two cases and the plus sign is
correct in the last case.
Russell [2, p. 388] has derived a modular equation of degree 15 similar to,
but more complicated than, Ramanujan's modular equation in Entry 21(i).
All references in the proofs of (i) and (ii) are to Chapter 16 unless otherwise
stated.
Proof of (i). We first apply C6.12) and C6.13) when ц = 8 and ш = 3 and
subtract the results. Second, we employ G.1) of this chapter with a = -q*~6n
and b = -q4+6n. Third, we apply Entry 18(iv) to the term of index 4 to find
that
f(-q20, -q2S)=-q-36<p(-q4).
By using Entry 18(iv), we next show that the terms with indices n and 8 — n,
1 < n < 3, are equal. Then we use Entries 25(i), (ii) and further simplify the
terms by using Entries 18(iii), (iv). Lastly, we invoke G.1) again but with

436 20. Modular Equations of Higher and Composite Degrees
a = —q1* and b = — q42. Accordingly,
= ? q4n2f{q112+2Sn, q112~2Sn)
n=0
= Z <z4/(<z112+28n, q112~28n)f{-q4~6n, -q4+6n)
n=0
= <p(q112)<p(-q4) - 2<?2 W2X-44)
2 t
n=l
, q**)f{-q2, -q6) + 2qi6f(q196, q28)f(-q2, -q6)
= <p(-q2S)<p(-q4) - 2Ч2ф(-Ч^)ф(-Ч2).
Replacing q by y/q, we find that
Transcribing this identity by means of Entries 10(iii), (vi), (vii) and ll(i), (ii) in
Chapter 17, we arrive at
+ {A -
Using an obvious analogue of B0.6) above, we conclude the proof of (i) in the
first case.
To prove (i) in the third case, we again first employ C6.12) and C6.13), but
now with ц = 8 and со = 1. Subtract the results and employ G.1) in this
chapter with a = -qA~2n and b = -q4+2n. We then apply Entry 18(iv) to
simplify the term with index 4 and to show that the terms with index n and
8 — n, 1 < n < 3, are equal. Next apply Entries 25(i), (ii) and 18(iii), (iv) to
further simplify the terms. Lastly, we appeal to G.1) again but with a = — q30
= -g90.Thus,
<p(-q15M-q)} - 2<Л
л=0
7
= ? <z4/(<z240+60n, q2*°~6On)f(-q*~2n, -<?4+2в)
п=0
- ,л^/,240ч
= <p(q

20. Modular Equations of Higher and Composite Degrees 437
+ 2 ? q4ny(q240+60n,q2*°-60n)f(-q4-2n, -q4+2n)
n=l
0, q18°W-q2, ~q6)
-2q3*f{q*2O,q6o)f{-q2,-q6)
= <p(-q60M-q4) + 2q^(-q30)iH-q2). B1.1)
Replacing q by y/q, we find that
<p(-q30)q>(-q2) + 2«V(-«ls№(-«).
Employing Entries 10(iii), (vi), (vii) and ll(i), (ii) in Chapter 17, we easily
transcribe the equality above into the desired modular equation in case 3. The
details are completely analogous to those in the first case.
In order to establish the desired modular equation in the second case, we
first replace q by q113 in B1.1) to arrive at
Using Corollaries (i), (ii) in Section 31, we equate rational parts on both sides
above and deduce that
When q is replaced by y/q, the foregoing equality becomes
H<p(q5l2)<p(q312) + <P(-q5/2M-q312)}
Translating this equality by the same set of results from Chapter 17 that we
used above, we complete the proof of (i) in the second case.
Proof of (ii). Apply C6.3) and C6.4) with ц = 8 and v = 5, 3, or 1. Set
Q = qM~. The theorems that we use below are precisely the same that we
used in the proof of part (i), and so we proceed without further comment.
Hence,
л=0
- <72v"+4/(<Z32+4vn,
7
n=0

438 20. Modular Equations of Higher and Composite Degrees
+ 2 t q16»2f(Q16+*\Q16-*")f(-q*+2™,
n=l
M-44) + 2q16f(Q2°,
- 2q16Q2f(Q28, Q4)f(-q4+2\ -<Z4~2v)
= v(-Q*M-q*) ± 2№)v4{-Q2m-i2\ B1.2)
where the plus sign is taken when v = 1 and the minus sign is chosen when
v = 3 or 5. (We emphasize that we used Entry 18(iv) several times above.)
Replacing q by y/q, we deduce that
= <p(-Q2M~q2) ±
Employing Entries 10(iii), (vi), (vii) and ll(i), (ii) in Chapter 17 and an obvious
analogue of B0.6) above, we readily find that
Ш1 + Jfl
= ({A - a)(l -
\Z8-vZ
8+v
Replacing each modulus by its complementary modulus, that is, taking the
reciprocal of this modular equation, we obtain the second part of (ii).
To prove the first part of (ii), return to B1.2) and replace q by ^1'<8"v) to
find that
p(q)p(-<Z4/(8~v)) ± 2(^8)^(^)^(^),
where the plus sign is correct when v = 1 and the minus sign is correct when
v = 3 or 5. We now equate rational parts on both sides. In the case v = 5, we
use Corollaries (i), (ii) in Section 31. For v = 3, we appeal to Entries 10(i), (ii)
of Chapter 19, and for v = 1, we employ Entries 17(iii), (iv) of Chapter 19.
Omitting the straightforward details, we conclude that, in all three cases,
+ v(-QM-q)} - 2Ш1/4>А(е2Ж<72)
Replacing q by v/q, we deduce that
H<p(Qll2Mq112) + <p(-QV2)<p(-q112)} -

20. Modular Equations of Higher and Composite Degrees 439
By Entries 10(iii), (vi), (vii) and ll(i), (ii) in Chapter 17, the translation of this
is the modular equation
+ УаI^ + y^I'2 + |(i -
Z1Z64_V2
Using an analogue of B0.6) and replacing each modulus by its complementary
modulus, we obtain the first part of (ii).
Entry 22. Each of the following modular equations is of degree 31.
(i) Let
o, p) =
+ {(I + yfiKl + у/Р)}112}112).
Then
O(a, p) + 0A - ft 1 - a) = 81/4.
(ii) i+(«/j)V* + {(i-«)(i _/?)}»/¦
- 2((а/?Г + {A - a)(l - P)}1* + {ф1
= 2{a/?(l - a)(l - ^)}1'16A + (a/?I'8 + {A - a)(l -
(Hi) 1 + (a/?I'4 + {A - a)(l - P)}114 - (i{l +^Гр + ,/A -
= (a^I'8 + {A - a)(l - Р)У* + mi - a)(l - ^}1/8-
The statement of (i) in the second notebook (p. 252) is somewhat obscure.
The first and third of these modular equations of the thirty-first degree are
new. Entry 22(ii) is due to Russell [1]. See also Greenhill's book [1, p. 327].
The only other modular equation of degree 31 which is comparable to Rama-
nujan's in simplicity is due to Schroter [1], [2], [3], who showed that
This can be proved with the aid of C6.8) in Chapter 16, but we do not give
any details. Schroter [2] further remarks that "une autre forme de cette

440 20. Modular Equations of Higher and Composite Degrees
equation modulaire, plus analogue aux formes precedents, mais plus compli-
quees." Of course, it is pure speculation to conjecture that Schroter had in
mind one of the three modular equations given above.
Considerably more complicated modular equations of degree 31 were
found by Weber [1], Berry [1], and Hanna [1].
It will be convenient to prove Ramanujan's modular equations in the
reverse order in which they are stated.
All references in the proof of Entry 22 are to Chapter 16, unless stated
otherwise.
Proof of (iii). Put Q = q31. We first apply C6.14) with ц = 16 and со = 1. The
resulting term of index 8 is
i, a64) = 2<z2
by Entries 18(ii), (iv). For 1 < n < 7, we apply Entry 18(iv) to show that the
terms of index n and 16 — n are equal. After combining terms with the aid of
Entry 18(iv), we employ Entries 25(i), (ii) and G.1) of this chapter three times.
Lastly, we invoke Corollary (ii) in Section 31. Accordingly,
E ч^Яч8'2"' q8+2n)AQ32'^ Q32+4n)
и=О
+ 2 E q*n2f(q8-2\ q8+2n)AQ32-*n, а32+4и)
n=l
i<P(q8){<p(Q8) + <p(-Q8) + <p(Q8) - v(-Q8)}
+ 2q*f(q6, qlo){f(Q2s, Q36) + Q6f(Q\ Q60)}
+ 2q16f(q\ q12){f(Q2\ QA0) + QAf{Q\ Q56)}
+ 2q36f(q\ q^){f(Q20, Q44) + Q2f(Q12, Q52)}
<p(q8MQ8) + VW6)<A«216) + 2qARq6, qlo)f(Q6, Q10)
+ 2q'6nq\ ql2)f(Q\ Q12) + 2q36f(q2,
U<p(,q2) + <p(-q2)} Ш2) + <p(-Q2)}
~ 9(-q2) 2
= H<p(q2MQ2) + v(-q2M-Q2)} + 2*?
+ q*{mm) + H-q)H-Q)}.
Using Entries 10(i)-(iv) and 11 (i)—(iv) in Chapter 17 to translate the fore-

20. Modular Equations of Higher and Composite Degrees 441
going formula, we find that
1 + {A - a)(l - /?)}1/4 + W)m
+ 1A - УГ^I/2A - УП^I'2 + {A - a)(l -
Using B0.6) to simplify, we complete the proof of (iii).
Proof of (ii). Let
(«fli/8 + {A _ a)(i _ p)}W = t and {aK(l - a)(l - 0)}1* = u.
Then (iii) assumes the form
l-t + t2-3u = {i(l + t4 - 4ut2 + 2u2)}112.
Squaring both sides, we arrive at
A - tf - 4иC - 3t + It2) + 16и2 = О,
or
{A - tJ - 4и}2 = 4мA + t). B2.1)
Since P = O(a31) as a tends to 0, we find that
1 - t - lju ~ 1 - A - |a) + O(a2) ~ |a,
as a tends to 0. Thus, when a is small and positive, A — tJ > 4и. Hence, taking
the square root in B2.1), we find that
A - tJ - 4и = 2y/u(l + t).
Rephrasing this equality in terms of a and /?, we deduce that
1 - 2((«/?)W + {A - «)A - 0)}1'8) + ((a^I'8 - {A - a)(l - /?)}1/8J
= 2{a0(l - a)(l - /?)}1/16A + (Ф118 + {A - a)(l - ^)}1/8I/2,
which readily is seen to be equivalent to (ii).
Proof of (i). The proof of (i) is a bit more difficult than those for (ii) and (iii).
As above, we set Q = q31. First, we apply C6.3) with ц = 16 and v = 15.
For the term of index 8, we apply Entry 18(iv), with n = 1 there, and Entry
18(ii) to deduce that
We then show, with the aid of Entry 18(iv), that the terms with index n and
16 — n, 1 <: n < 7, are equal. Employ next Entries 25(i), (ii). We now simplify
the terms somewhat by using Entry 18(iv) to show that
д32и2/D32 + б0В, ^32-60») = q4ny^q32+4n^ q32-4n)
Next, we combine the terms of index n with those of index 8 — n, 1 < n <, 3,
and then make several applications of Entries 30(ii), (iii). Accordingly,

442 20. Modular Equations of Higher and Composite Degrees
= I
и=0
= v(q32MQ32
= v(q32MQ32) + 4«2
+ 2 ? q32f(q32+60n, <f2-60n)/(Q32+4", б32"
2 ? q*n
n=l
2 i «4и2{/(«32+4п, q32-4-)/(G32+4", е32и)
"n), -G2D+n))}
+ {/D2D~"W2D+n)) -f(-4m'n\ -q2D+n))}
x {/(G2D-"),G2D+"))-/(-G2D-'I), -а2D+и))})
H<p(q*MQs) + <p(-q*)<p(-Q8)} + 2<Z6W
*\ q10)f(Q6, Q10) +f(~q6, -qlo)f(-Q6, -Q10)}
q\ q14)f(Q2, QU) +f(~q2, -q14)f(-Q2, -Q14
v(-q8M-Q8)} + 2<z6W6)<A(G16)
*) + <А(-<74Ж-<24)}
(-Q, -a3)}
+f№, ~i
+ i«*{/(e, q3) -A-q, -q3)} {/(Q, a3) -Л-Q, -Q3)}
= Hv(q8MQ8) + v(-q8M-Q8)}
4{Aq, q3)AQ, Q3) +A-q, -q3)A-Q, -Q3)
, -iq*)A-iQ, iQ3) +/(-*«, iq3)AtQ, -iQ3)}

20. Modular Equations of Higher and Composite Degrees 443
Replacing q by q11*, we find that
- <p(q2MQ2) - <p(-<i2M-Q2)
1>(~q)*(-Q)}- B2.2)
By Entries 25(iv), (i), (ii),
Using this in B2.2), we find that the left side of B2.2) equals
x {A + ia1/4I'2(l - i^1'4I'2 + A - ia1'4I/2(l +
2 2 /
x {2 + 2(a,
x {2 + 2(a)8I/4 ^ У
= 21'4(z1z31I'2Q(a,/?), B2.3)
where we have employed Entries 1 l(iii), (vii)-(ix) in Chapter 17 and analogues
of B0.6) above.
Using B2.3) and employing Entries 10(iii), (iv), (viii), (ix) and ll(i), (ii), (iv)
of Chapter 17 in B2.2), we deduce the modular equation

444 20. Modular Equations of Higher and Composite Degrees
21'Ап{<х, /?) = A + a1/4)(l + p1'*) + A - a1/4)(l - p1»)
- {A - a)(l - P)}m
- (a/?I'8
- {a/?(l - a)(l - 0)}1/8
= 2 + 2(aKI/4 - (a/?I/8 - {A - a)(l - /?)}1/8
Next, take the reciprocal of the modular equation above and add it to the
original equation to deduce that
21/4П(а, P) + 21/4QA - p, 1 - a)
= 4 + 2
by (iii). Thus, (i) follows immediately.
Entry 23.
(i) // P is of degree 47, then
-(а/О^ + Ш-аШ-/?)}1'8).
(ii) // P is of degree 71, then
1 + (a/?I'4 + {A - a)(l -
' + {(!-
?(l - a)(l - K)}1/24A - (a^I'8 - {A - a)(l -
These two modular equations are the climax of Ramanujan's modular
equations involving two moduli only.
The first modular equation of degree 47 was offered without proof and with
two sign errors by Hurwitz in a paper by Klein [2]. Russell [1] corrected and
proved the result shortly thereafter. More complicated modular equations of
degree 47 were established by Fiedler [1] and Hanna [1]. Fiedler [1] also
constructed a modular equation of degree 71. Simpler forms of Fiedler's
equation were obtained soon thereafter by Weber [1] and Russell [2].
Before embarking on a proof of Entry 23, we show that Russell's modular
equations can easily be derived from those of Ramanujan.

20. Modular Equations of Higher and Composite Degrees 445
Set
P=l + (aj?I/8 + {(l-a)(l-0)}1/8 and R = {a/?(l - a)(l - /?)}1/8-
Then Russell's modular equation of degree 47 assumes the form
(P - 2J - PDRI/3 - 2DКJ/3 - AR = 0, B3.1)
while Ramanujan's equation takes the shape
2(|{(P - IL + 1 - AR{P - IJ + 2K2}I/2 = (P - IJ + 1 + PDRI/3 - 2R.
Squaring Ramanujan's equation and rearranging the terms, we derive the
equality
P{P - (ARI13} {(P - 2J - PDRI/3 - 2DКJ/3 - AR} = 0.
By examining each of the first two factors above as a tends to 0, we see that
they cannot vanish identically. Thus, the third factor must vanish; that is,
Russell's equation B3.1) holds.
In the case of degree 71, set
p = (a^I'8 + {A - a)(l - 0)}1/8 - 1 and R = -
Then Russell's equation takes the form
P3 - 4DRI/3(P2 + P + 1) + 2PDRJ/3 - AR - (AR)*3 = 0, B3.2)
while Ramanujan's equation of degree 71 assumes the shape
P2 + P + 1 - PDRI/3 + R
= (H(p + IL + 1 + 4R(P + IJ + 2R2}I/2.
Squaring and rearranging Ramanujan's equation, we arrive at
lp{P3 _ 4DK)i/3(p2 + P+l) + 2PDRJ'3 - AR - (AR)*^} = 0. B3.3)
Now as a tends to 0,
P~0(ix9)+ l-|a + ----l.
Thus, P cannot identically vanish. Hence, the second factor in B3.3) is identi-
identically equal to 0; that is, B3.2) is valid.
It is interesting to note that, by Entry 19(i) in Chapter 19, P does vanish
identically in the case that /? is of degree 7.
All references in the proofs of (i) and (ii) are from Chapter 16, unless
indicated otherwise.
Proof of (i). Our proof rests on two representations for 47, namely,
47 = 3 ¦ 24 - I2 = 3 • 25 - 72.
In C6.14), let ц = 48 and со = 7 and set Q = q41. After combining the terms
with index n and n + 24, 0 <, n < 23, by an application of Entry 18(iv), and
using G.1) in this chapter, we deduce that

446 20. Modular Equations of Higher and Composite Degrees
= E ^/(G96+4n,G96")/(^4+14n>«244")
n=0
= E
n=0
х/(9
= E q4n2f(Q2*+2n, G24~2n)/(924+14n^244")- B3.4)
q4n2f(Q2*+2n G24~2n)/(924+14n^244")
Second, in C6.12), set /i = 24, со = 1, and Q = <?47. Replacing <? by ^fq, we
find that
= E <?2я2ла24+2в,
я=0
Subtracting B3.4) from the last equality, we deduce that
S := kWV2112 ll2112
= E
и=1
= 2 ё 92в2ле24+2л,а24-2в){л^24+2в,«24-2в)
п=1
244в)}- B3.5)
In the analysis above, we used the fact that the term with index 12 vanishes
and that the terms of index n and 24 — n, 1 < n < 11, are equal. These
deductions are easily made with the help of Entry 18(iv). Next, by repeated
applications of Entry 18(iv), the terms with indices n = 3, 4, 6, 8, 9 likewise
vanish. Further transforming via Entry 18(iv), we find that
S = 2q2f{Q22, Q26){f(q22, q26) - q2f(q10, <?38)}
+ 2<?8Ле20, a28)№°, q28) ~ qW, <Z44)}
+ 2<Z5O/«214, <234)№4, <Z34) - <74Л<72, <Z46)}
+ 2q9*f(Q10, Q38){f(q10, <?38) - Ч~2Ячг\ q26)}
+ 2q2OOf(Q\ Q"){f{q\ q") - q^fiq20, q28)}
+ 2q2*2f{Q\ Q*6){f(q2, q*6) - q^fiq1*, <?34)}
= 2q2{f(q22, q26) - q2f(q10, q38)} {f(Q22, Q26) - Q2f(Ql°, Q38)}
+ 2q™{f{q'\ q3*) - cff(q2, q*6)} {/(G14, Q3*) - Q*f(Q2, Q*6)}
+ 2q»{f(q20, q28) - q*f(q*, q")} {f(Q20, Q2*) - Q*f(Q\ Q")}.

20. Modular Equations of Higher and Composite Degrees 447
By Entries 30(ii), (iii) and G.1) in this chapter,
2{f{q22,q26)-q2f{q10,q38)}
= Л<?5, q1) +f(~q\ -q1) ~ qf(q, q11) + qf(-q, -<?U)
= f(-q) +f(q) B3.6)
and
11) + qf(-q, -<?U) ~f(q5, q1) +f(~q5, -q1)
= f(q)-f(-q)- B3.7)
Consequently,
s = Wifi-q) +f(q)}{f(-Q) +f(Q)}
+ W{f(q) ~f(-q)}{f(Q) -f(-Q)} + 2q8f(-q4)f(-Q4)
= q2f(Q)f(q) + q2f(-Q)f(-q) + 2q8f{-Q4)f(-q*).
Referring back to B3.5) for the definition of S and transcribing the equality
above via Entries 10(i), (ii), (vi), (vii), 11 (iii), and 12(i), (ii), (iv) in Chapter 17,
we find that
{A - «)A - /?)}1/4) - |(a/?I/4
P)}1'2* + 2-1/3{(l - «)A - P)}1I6(«PI124
Simplifying by an obvious analogue of B0.6) above and rearranging terms, we
complete the proof.
Proof of (ii). Our proof depends on two representations for 71, namely,
71 = 3-25 — 52 = 3-26 — II2.
In C6.14), let ц = 48, со = 5, and Q = q71. Combine the terms with indices
n and n + 24,0 < n < 23, with the aid of Entry 18(iv). Then using G.1) in this
chapter, we deduce that
? <?4/(G96+4l>, Q96-*n)f(q24+i0n, q2*'10")
я=0
E 94п2лб24+2и' е24пш«24+10п. q240")- B3.8)
я=0

448 20. Modular Equations of Higher and Composite Degrees
Apply C6.14) once again, but now with ц = 96, со = 11, and q replaced by
q2. Proceeding as above, we find that
H<p(Q2M<i2) + <p(-Q2M-q2)} + 2q36HQ*№(q*)
95
_ у _8я2/-/0384+8я /-K84-8п\/у_96+44л _96-44я\
n=0
47
_ у д8я2/ /Ул384+8л /}384-8я\ ¦
л=0
х f(q96+"", д964")
= I q8n2f(Q.96+4n, Q96-4n)f(q96+*4n, q96-AAn). B3.9)
п=0
Third, let ц = 96 and со = 11 in C6.13) to infer that
2 4
I
я=0
Replace ^2 by q and — ^, in turn. Add the resulting two equalities to find that
= ? q«»2-22n+24f(Q96+4nQ96-4n}f(q192-44nq44ny B3.10)
n=O
Adding B3.9) and B3.10), employing G.1) of this chapter, and combining
the terms with indices n and n + 24, 0 < n < 23, we deduce that
+ <P(~Q2M-q2)} + 2q
я=0
2
я=О
= Z 98
и=О
x f(q24+22n, q24'22")
= z 98п2ле24+2",е24-2"Ш924+22"^24-22и),
я=О
by a calculation made in B3.4) above.
By subtracting the last result from B3.8), we see that
s := HviQMi) + v(-QM-q)} +
- Hv(Q2M<i2) + <p(-Q2M-q2)} -

20. Modular Equations of Higher and Composite Degrees 449
= e Л(е24+2и,е24-2я){я<?24+10",<?24-10п)
и=1
- q4l>1f(q24+22n, q24'22")}
= 2 ё 94п2/(е24+2л,е24в){/(«24+10л,9240'1)
я=1
q24'22")},
where we have used Entry 18(iv) to show that the term with n = 12 vanishes
and that the terms with indices n and 24 — n, l<n<ll, are equal. With
several applications of Entry 18(iv), it is easy to show that the terms with
indices n = 3, 4, 6, 8, 9 vanish. Thus, by further applications of Entry 18(iv)
and the same calculations that we made in the proof of (i), we deduce that
S = 2q*f(Q22, Q26){f(q14, q34) ~ qAf{q\ q46)}
°, Q28){f(q\ q") ~ «"W0, q28)}
*, Q34){f(q22, q26) ~ q2f(ql°, q3*)}
10, Q38){q4f(q2, q46) ~f(q14, q34)}
\ Q"){f(q20, q28) - q*f(q\ q44)}
+ 2q3"f{Q2, Q46){q2f{qi0, q38) -f(q22, q26)}
= V{/F" C26) - Q2f(Q10, Q38)} {Пч14, Q34) - q4fD2, q46)}
+ 2q14{f(Q14, Q34) - Q4f(Q2, Q46)} {f(q22, q26) - q2f(qi0, q38)}
- 2q12{f(Q20, Q28) - Q4f(Q4, Q44)} {f(q20, q28) - q4f(q4, q44)}
= k3{f(-Q) +f(Q)}{f(q) -/(-«)}
+ k3{/@ -f(-Q)} if(-q) +/(«)} - V2/(-<?4)/(-e4)
= q3f(Q)f(q) - q3f(-Q)f(-q) ~ 2q12f(-qA)f(-Q4),
by B3.6) and B3.7).
Finally, we employ Entries 10(i)-(iv), ll(i)-(iv),and 12(i), (ii), (iv) in Chapter
17 to transcribe the equality above. Using B0.6), we immediately obtain (ii)
to complete the proof.
Entry 24. Let P, y, and S have one of the following sequences of degrees:
3,29, 87;
5, 27, 135;
11,21,231;
13,19,247;

450 20. Modular Equations of Higher and Composite Degrees
7, 25, 175;
9, 23, 207;
15,17,255.
Let m and m! denote the multipliers associated with the pairs a, ft and y, 5,
respectively. Then
(i) (Ml + Vfy + V(l - /0A - y)}I'2
' + {A-.
(ii) v
+ (a<5I/8 + {A -
where the minus sign is chosen in the first four cases and the plus sign is assumed
in the last three cases.
A phrase about the appropriate signs is absent in the notebooks (p. 252).
Entry 24(i) can be found in Ramanujan's [10, p. 353] second letter to Hardy.
All references in the proofs of (i) and (ii) are to Chapter 16, unless otherwise
stated.
Proof of (i). First, we invoke C6.6) with ц = 16, v taking the values 13,11, 5,
3, 9, 7, and 1, respectively, q replaced by q2, and Q = <j256~v\ Thus,
+ 2v4,./_32-2v\ , „./•_„32+2у\,„/_ „32-2v\
2v) + <p(-q32+2vM-q32~~2v) + 4q
= 2 X q64nlf{Q64+8n, Q64-8n)f{q16+4vn, g16™).
»=o
Second, let ц = 16 in C6.4) with q replaced by ^/q. This yields
2<zV(<Z16+v)i/'(<Z16~v) = X q16n2+vn+*f(Q16+2n, Q16-2n)f(q32+2vn, q~2vn).
n=O
Replace q by — q and add the result to the preceding two equalities. Then
combine the terms with indices n and 8 + n, 0 < n < 7, by making use of Entry
18(iv). Lastly, we apply G.1) of this chapter twice. Accordingly, we find that
:= <p(q )(p(q ) + <P( — q )<p\ — q )

20. Modular Equations of Higher and Composite Degrees 451
7
= 2;
n
+
+
+
= 2 ;
n
? <764
=0
Q16+4y
7
2 ? «f
7
=0
{/(G64+8">
(б128+8я, С
^32+4vb. ^
дб1б+4я, с
g64-8B)
!-8")}/(916+4vB, 916v")
^16+411 /il6~
Ли»
-4B)/(^32+4v,
In the next step, we combine the terms with indices n and n + 4,0 ^ n ^ 3,
with the aid of Entry 18(iv). We then apply G.1) again. The terms of indices 1
and 3 are now found to be equal, by Entry 18(iv). Thus,
s = 2 i q64nl{f(Qi6+*n, e16") + е4+2вле32+4в, eB)}/(«4+2vB. «4~2VB)
з
Now we apply Entry 18(iv) with n = (v + l)/4, according as v = T1 (mod 4),
to discover that
We also apply Entry 18(iv) with n = (v — l)/2 to deduce that
/(^4+4v^4-4v) = 2ql-vV(q8)
Employing lastly Entries 25(i), (ii), we find that
s =
Finally, employing Entries 10(i)-(iv) and 11 (i)-(iv) in Chapter 17, we trans-
translate the equality above into the sought modular equation and so complete the
proof of (i).
Proof of (ii). In the extremal parts of B4.1), we replace q by qm 6~v) to find that

452 20. Modular Equations of Higher and Composite Degrees
We now equate rational parts on both sides to deduce, except when v = 7, that
<p(Q2)<p(q2) + <P(-Q2M-q2
^ ±
B4.2)
where the plus sign is chosen when v = 9 or 1 and the minus sign is taken
when v = 13,11, 5, or 3. The details in demonstrating the validity of B4.2) are
rather tedious, and we shall be content with merely indicating the requisite
steps. Each of the six cases must be examined separately. Corollaries (i), (ii) of
Section 31 in Chapter 16 are used when v = 13. Entries 10(i), (ii) of Chapter
19 are employed when v = 11. For v = 9, the rational parts are obtained by
using Entries 17(iii), (iv) in Chapter 19. When v = 5, utilize Entries 6(i), (ii) of
the present chapter. For v = 1,3, Ramanujan has not explicitly recorded the
appropriate formulas, but they are very easily obtained from Entry 31 of
Chapter 16 in the same manner as the aforementioned results were derived.
We emphasize that when v = 7, B4.2) is not obtained, because when rational
parts are equated, additional terms arise. Note that 16 — 7 = 32.
The translation of B4.2) into Entry 24(ii) uses precisely the same formulas
from Chapter 17 that were employed in the proof of part (i).
There remains the proof of (ii) in the case v = 7. In this instance, the
requisite formula to be established is
<p(Q2)<p(q2)
+ <p(-Q2M-q2)
= <p(q23)<p(q9) + 44 Vta46)^18) + <p(-q23)<p(-q9), B4.3)
where Q = q201. The translation of B4.3) into (ii) is exactly the same as above,
and so the proof will be completed on establishing B4.3).
We apply C6.6) with \i = 16 and v = 7. Using Entry 18(iv), we first combine
the terms with indices n and 8 + n, 0 < n < 7. Simplify the resulting sum with
the aid of G.1) in this chapter. After separating the terms with n = 0, 4 and
simplifying with the aid of Entries 18(ii), (iv), we use Entry 18(iv) to show that
the terms with indices n and 8 — n, 1 < n < 3, are equal. Next, we employ
Entries 25(i), (ii), and lastly we invoke Corollary (ii) of Section 31. Accordingly,
we deduce that
H<p(i23M<i9) + <p(-q23M-q9)}
= z 9з
л=0

20. Modular Equations of Higher and Composite Degrees 453
л=0
: ? 932„у(д8 + 2В; g8-2B)/(q8+14« q8-14B)
n=0
= /лСЛ)8^л^8)
<p(Q8Mq8)
+ 2 ? q32n2f(Q8+2n, Q8-2n)f(q8
л=1
iW(Q2) + <p(-Q2)} Ш) + <p(-q2)}
+ U<p(Q2) - <p(-Q2)} W(q2) - <p(-i2)} + 2q26f(Q6, Q10)f(q6,
+ 2ql0*mAmqA) + 2q23*f(Q2, Q14)f(q2, q1*)
2Ыч2) + l2<p(-Q2M-q2) + 2910V(e4)^(94)
q26({f(Q6, Q10) + Qf(Q\ G14)} {f(q6, Я10) + qf(q2, qlA)}
{f(Q6, Q10) - Qf(Q2, G14)} {f(q6, qw) ~ qf(q2, <?14)})
Thus, the proof of B4.3) and, consequently, the proof of (ii) in the sixth case
are complete.
This concludes a truly fascinating chapter!

CHAPTER 21
Eisenstein Series
Chapter 21 concludes the organized portion of Ramanujan's second note-
notebook; after Chapter 21, there are 100 pages of unorganized material. Chapter
21 constitutes only four pages and thus is the shortest chapter in the second
notebook. Almost all of the previous chapters are twelve pages in length.
The focus of this chapter is similar to those of the immediately preceding
chapters. However, whereas in Chapters 19 and 20, the goal was to establish
identities involving theta-functions, here our task is to prove equalities relating
a certain linear combination of Eisenstein series with theta-functions. From
the viewpoint of modular forms, just as in Chapter 20, both the Eisenstein
series and theta-functions are forms on ГB) п Г0(п) for some odd integer
n>3.
The key to establishing Ramanujan's formulas is apparently B.3) below.
This formula is not explicitly stated by Ramanujan, but we conjecture that it
is this formula to which Ramanujan makes allusion in Entry 2(v). Unfortu-
Unfortunately, we have not always been successful in applying this formula or the
related formula E.3). Thus, for seven of the results in this chapter, we have
had to rely on the theory of modular forms that was developed in Chapter 20.
As in the last chapter, the theory of modular forms provides the best means of
explaining why these identities exist. However, again as before, it is necessary
to know the identity in advance, and so the proofs are more properly called
verifications.
As in previous chapters, we employ the notation introduced in Chapter 17,
especially in Section 6.
We shall precisely quote Ramanujan (p. 253) for Entry 1.
Entry 1.

21. Eisenstein Series 455
is a complete series which when divided by z2 can be expressed by radicals
precisely in the same manner as the series
and the series
1-504I^t (L3)
when divided by z* and z6, respectively.
(iii) 1 - 240
(iv) 1 + 504 ? *~^Г" = г6A - 2х){1 + 32x(l - x)}.
л=1 С I V
The Eisenstein series A.1)—A.3) were introduced by Ramanujan in Section
9 of Chapter 15 and were denoted by L — 3/y, M, and N, respectively. In
Ramanujan's celebrated paper [6], [10, pp. 136-162], the series L, M, and N
are designated by P, Q, and R, respectively.
The definition of "complete" is given rather vaguely by Ramanujan in
Section 10 of Chapter 15 (Part II [9, p. 320]).
In fact, (i) is not quite accurately stated by Ramanujan, since the condition
that y2/n2 be rational should be added. With this additional stipulation,
(i) was established by Ramanujan in his paper [2], [10, pp. 32, 33]. The
reader should consult [2] and the Borweins' book [2, Chap. 5] to learn how
Ramanujan used such results to derive excellent approximations to л.
It might be noted that, in general,
where, as usual, К and K' denote the complete elliptic integrals of the first kind
attached to the moduli к and k', respectively, while E and E' are the complete
elliptic integrals of the second kind associated with к and k', respectively.
To prove A.4), recall from Entry 2 of Chapter 18 that
s = \iFA-l i; i; *) - 2 + x - -^.
Since x = к2, у = лК'/К, and z = 2K/n, we rewrite this last equality in the
form
2A 4AA
From C.7) in Chapter 18, 2F1(-i, i; 1; к2) = 2?/я. Using also Legendre's

456 21. Eisenstein Series
relation (Whittaker and Watson [1, p. 520]), we find that
3? „ ,, Ъ(Е Е
82 + *\ +
which yields A.4) at once. (A recent, somewhat simpler proof of Legendre's
relation has been given by Almkvist and Berndt [1].)
With regard to A.2) and A.3), we recall from Entries 13(i), (ii) of Chapter 17
that
and
respectively.
Proof of (ii). By Entries 13(viii), (ix) in Chapter 17,
= 2z2(l - |x) - z2(l + x)
= z2(l - 2x).
Proof of (iii). We use the procedure of "obtaining a formula by change of
sign," described in Section 13 of Chapter 17. Thus, in Entry 13 (iii) of Chapter
17, replace x by —x/(l — x), which induces the replacements of e~y by — e~y
and z by z^/l — x. Hence,
which upon simplification yields the proposed formula.
Proof of (iv). We employ the "change of sign" process to Entry 13 (iv) of
Chapter 17 to find that
After simplification, we obtain the desired result.
Entry 2.
V B" +

21. Eisenstein Series 457
я=1
d , ,,
(v) By differentiating the equation for m once or the equation for a, /? twice we
can calculate the value of the first series.
Parts (i)-(iv) are quite easy to prove. However, the meaning of (v), for which
we have quoted Ramanujan (p. 253) exactly, is rather opaque. Perhaps Rama-
Ramanujan is referring to a more precise version of Entry l(i), or to a certain
formula, B.3) below, which will be needed to prove many of the formulas in
the remainder of the chapter.
Proof of (iii). By Entry 23 (Hi) in Chapter 16,
Differentiating both sides with respect to q, expanding the summands, and
inverting the order of summation, we find that
(-q)
= 1 - 24 X qk X Щ
fc=l П=1
k(n~l)
СЮ 00
= 1 - 24 X n X qnk
which completes the proof.
Proof of (iv). From Entry 23 (iv) in Chapter 16,
^ = & Log a + J; щ

458 21. Eisenstein Series
Proceeding in exactly the same manner as in the proof above, we deduce that
1 + 24 ? (-l)V + q3k) t
fc = l П=1
00 00
1 + 24 X « Z (-ljV2"' + qkBn+i))
n=l *=1
72" +
,_„„№ +IX™. ч
„=о \ 1 + q2n+1 l + <z2B+1
Bn + l)q2n+1
Proof of (i). From Entry 22(i) and B2.2) in Chapter 16, we can easily see that
Log <p(q) = Logiq^fi-q2)} - 2 Log{ql<2*/x(q)}.
Differentiating and employing parts (iii) and (iv) above, we find that
Proof of (ii). Differentiating Entry 23 (ii) of Chapter 16 and proceeding as in
the proofs of (iii) and (iv), we find that
2Ш?
- 3 I 21 У {2П + 1)g 24 У 2Ш
-3 + 24,u i-«2B+1 A 1^
. Dn + 2)g^
i4B+2 У
- 3 I 21 У Bn + l)q2"+1 21
" Л 1+2B+1
from which (ii) follows.

21. Eisenstein Series 459
Proofs of Two Interpretations of (v). We first prove a formula on which
much of the remainder of this chapter is based.
Let n denote any positive integer, and put и = q2 = ey. By using Entry
2(iii) above, we find that
dy
funl2fl\-un)
-24м —Log
du
l
Now let)? have degree n over a, and let m denote the multiplier associated
with a and p. By Entries 12(iii) and 9(i) in Chapter 17,
Thus, combining B.1) and B.2), we derive the important formula
To derive our second possible interpretation of (v), we utilize Entry 27(iii)
of Chapter 16 in the form
ll4f(-e-2"'), B.4)
where y, y' > 0 with yy' = n2. Set q = e~y and q' = e~y'. Logarithmically
differentiating B.4) with respect to у and using Entry 2(iii) above, we see that
or

460 21. Eisenstein Series
- НО-
Now set _y = n/y/n, where n is a natural number. Thus, у' = я^/и. Combin-
Combining B.3) and B.5), we conclude that
= „f 1 - 24 ? -4—) - «A - ф? I Log
V h 2«kS» \) ' d<x
or
n( 1-24 У.
Entry 3.
_ ty\q) + 3#4(g3)|
1 </'(9)«/'(93) J
= f/12(-9) + 279/12(-q3)l2/3
1 f4-q)P(-q3) J
4(p(q)(p(q3)
(iii) Let)? haue decree 3 with respect to a. Then
oo fcfl2* oo t/76*

21. Eisenstein Series 461
Ramanujan actually expresses (iii) in the variable y, where q = e~y. It will
be convenient to prove (ii) and (iii) before (i).
Proof of (ii). Putting n = 3 in B.3), we find that
= |«(l-«)zjf Log *T " . C.1)
da \m6a(l — a)/
Now from E.2) and E.5) of Chapter 19,
.. ч (m2 - 1)(9 - m2K P(l - P) m\m2 - IJ
' W^) = (9-m2J
and
dm 16m4
da. {9-m2J'
Using these equalities and the chain rule in C.1), we arrive at
m2) , d
l dm L°8
v
7' ( }
which establishes the first equality in (ii).
From the second equality in C.3) and C.2),
(9 - m2)(m2 - 1)
by Entry 11 (ii) of Chapter 17. Thus, the second equality of (ii) has been proved.
Proof of (iii). From C.3) and from E.8) of Chapter 19,
ZiZn .
+ 3J = |Zlz3{l + Уа0 + V(l
from which the truth of (iii) is manifest.
Proof of (i). Once again from C.3), and from E.1) of Chapter 19,

462
21. Eisenstein Series
C + mf + 3(m - 1JV
J
4\fi
C.4)
by Entry 1 l(iii) of Chapter 17. If we now replace q2byq, we obtain the equality
between the first and fourth expressions in (i).
By Entry 1 l(i) of Chapter 17, E.2) and E.3) of Chapter 19, and Entry 11 (Hi)
of Chapter 17,
3 3(w- 1)
4m
6m _ 3)
. C.5)
We next invoke Entries 3(i), (ii) of Chapter 19. For each summand of even
index, we use one of the two equalities,
qn
In
l-q2" l-qn \-q2n
We thus deduce that
<p(q)<P(q3)
and
1 + q2n 1 - q2n 1 -
- q l-q2 l-q4 l-q5
C.6)
Combining C.5) and C.6), we establish the equality between the second and
fourth expressions of part (i).
Next, by Entry 13(i) in Chapter 17 and C.2) above,
1^
*=i 1 — i
- a + a2) + 9z$(l - ,

2
21. Eisenstein Series 463
- m2a(l - a) - -^A -
9 (m2 - 1)(9 - m2K 9(m2 - 1K(9 - m2)
+-2 256^ 256^
4 + 12m2 + 54 + 108m~2 + 81»T4)
Ш?0*
by the same calculation as in C.4). Replacing q2 by q, we establish the equality
between the third and fourth expressions in (i).
By Entries 24(i), (iii) in Chapter 16,
Я-q) = M*^q) =/(-V)*(-<?) = ФШЧ-Ч). C.7)
Using Entries l(i), (iv) of Chapter 20, the definition of v given in Entry 1 of
Chapter 20, and C.7), we find that
''V) = #V)D + ^
#V)/27 +
f3i-q)f4-q3)
Thus, we have shown that the fourth and fifth expressions in (i) are equal. In
conclusion, the equality of all five expressions of (i) has been established.
Entry
(i)
4.
14-
6f k4k ¦
fc=i 1 — qk
{f12(-q) +
ф\д) + 2дф
22дГ
2№2
kqSk
1 — д5к
4-q)f4~q
f(~q)f(-
'(д5) + 5д2ф-
5L
q5)
4qS
- П5дТ2(-д*)У>2
)
х {ф\д) - 2дф2(д)ф2(д5) + 5д2ф*(д5)У'2.

464 21. Eisenstein Series
(iii) Let /? have degree 5 ouer a. Then
- «)(i - P)}
Ramanujan has stated (iii) in terms of y, where q = e~y.
Proof of (i). Setting n = 5 in B.3), we find that
D.1)
Now take A4.2) of Chapter 19 and differentiate both sides with respect to m.
After simplification, we find that
da. <x(l-<x)/25-20m-m2
D.2)
dm 1 — 2a \m(m — 1)E — m)
Furthermore, from A4.2) and A4.4) of Chapter 19,
j8(l - P) = (ib - lLm4
«A-я) E-mL '
Lastly, in Entry 14(ii) of Chapter 19, make the substitution p = (m — l)/2,from
A4.1), simplify, and use the definition of p given in A3.3) of the same chapter.
Accordingly, we find that
B5 - 20m - m2)p
1 — 2a = 5 .
8m3
Employing these last three equalities in D.1), we deduce that
(m - IL \
m2E - mf)
p(m - 1)E - m)z\ d ( {m-\f
Log f
32m2 dm 6 \m\5 - mf

21. Eisenstein Series 465
_ p(m2 + 2m + S)z\
16m3
-2
= т^Ц^7 + 2m6 + llm5 + 12m4 + 55m3 + 50m2 + 125mI/2. D.3)
16m
By Entry 12(iii) in Chapter 17 and Entry 13(iv) in Chapter 19,
2
-1).
/(9) K/»(!-/»)/ 16 V»
Also, by Entry 12(iii) in Chapter 17 and D.2),
- /?)Y/2 = (m - IJ
-<x)J mE-mf
=
^ /6(-<?2) rn3\<x(l
Employing these last two results in D.3), we find that
f5(-q2)(m6 + 2m5 + llm4 + 12m3 + 55m2 + 50m + 125\1/2
7FV )
m2E - mL
- mL + 22(m - lJmE - mJ + 125(m - IL
f(-qw)\ m2E - mL
11/2
П-q2) ' '-* f12{~q2)J ' D)
from which, upon the replacement of q2 by q, the first equality of (i) is apparent.
We now prove the second equality of (i). First, from D.3) and the equality
m = <p2(q)/$2(q5),
„ p(m2 + 2m + 5)z2
<p*(q) + 2<p2(q)<p2{qs) + 5<p*(q5) 2
= 5 {(p (q) — 2<p (q)<p (q ) + 5<p (q )} ' .
Observe that D.4) is invariant under a change of sign of q. Thus, by D.4), the
first part of (i), and the foregoing equality,
{P*(-qi) + 22q2f6(-q2)f4-q10)-
<P2(q)<p2(q5)A-q2)f(-q10)
_ <p\-q) + 2<p2(-q)<p2(-q5) + 5<p\-q5)
l6<p2(q)<p2(q5M-q)<p(-q5)
x {<p\-q) - 2<p2(-q)<p2(-q5) + 5<p\-q5)Y12. D.5)
Converting D.5) into a modular equation via Entries 10(i), (ii) and 12(iii) in
Chapter 17, we find that
{m3a(l - a) + 22ya/?(l - «)A - /Q

466 21. Eisenstein Series
- a) + 2/A - a)(l -
- Р)}щ
x {m(l - a) - 2^/A - a)(l - P) + 5m A - P)}1'2. D.6)
The left side of D.6) is self-reciprocal and so is equal to the reciprocal of the
right side of D.6). Hence, by D.5) and D.6),
{f12(-g2) + 22q2f6(-q2)f6(-q1°) + l25qT2(-q10)V12
5m-lp + 2ja.p + ma /—
mi/4 Wm P — 2va^ + ma}1'2
ф*(д2) + 2д2ф2(д2)ф2(д10) + 5дАф\д10)
х {ф\д2) - 2д2ф2(д2)ф2(д")
by Entry ll(iii) in Chapter 17. Replacing g2 by g, we complete the task of
showing that the second and third expressions in (i) are equal.
Proof of (ii). By Entry 12(iii) of Chapter 17 and A3.8) of Chapter 19,
(m — 1)E — m)
D.7)
8m
m2 + 2m + 5
8m '
Similarly, by Entry 12(v) in Chapter 17 and A3.8) in Chapter 19,
'-1
1 (iti - 1)E - m)
4m
m3 — 2m2 + 5m p2
4m2 4m2'
by A3.3) of Chapter 19. Hence, combining D.7) and D.8), we conclude that
Л _ 2 f2(-q)f2(-q10)\fl _ 4q Y'2 _ p(m2 + 2m + 5)
V q <p2(q)<p2(q5) A x\q)x*(q5)) ~ ^
D.8)
foV 16m2
Appealing to D.3), we finish the proof of (ii).
Proof of (iii). By A3.7) in Chapter 19,

21. Eisenstein Series
467
Thus,
з + ^ + jo^hT7!) = 2 + {i + y^ +
D.10)
Hence, by D.9) and D.10),
_ p(m2 + 2m + 5)
16m3 '
By D.3), we complete the proof of the first equality of (iii).
By A3.7), A3.8), and A3.3) of Chapter 19,
к.2 \2
2m:
-1 +
256m3
64m2
256m
(m6 + 2w5 + llm4 + 12m3 + 55m2 + 50m + 125)
М2 + 2m + 5J.
(It is quite clear that we have omitted a heavy dosage of tedious algebra.)
Taking the square root of both sides and using D.3) again, we complete the
proof of the second equality of (iii).
Entry 5.
¦*
J-q^ l-q2 l-q3 ' l-q* l-q5 l-q6
r,8 \") 2
/n the ntfle expression, the cycle of coefficients has length 1.

468 21. Eisenstein Series
1
*=i l —
(iii) Let /? /iare degree 1 over a. 77ien
oo t^zt oo ka14k
28 L
1 i и V
1 + 4 L
k=i i — q k=i l — q
In our proof of Entry 5, all references of the form A9.-) arise from Chapter
19.
Proof of (i). We begin by replacing q by — q on the left side of (i). Thus, we
first derive an analogue of B.3) wherein q2 has been replaced by — q.
With q = e~y and и = q", where n is a positive integer,
dy
by the same argument that we used in B.1).
On the other hand, by Entries 12(i) and 9(i) in Chapter 17,
d
= 2 jr
Thus, combining E.1) and E.2), we conclude that
Now, setting и = 7 in E.3), we find that

21. Eisenstein Series 469
= 1A - 2)8J?, - Ml - 2ф2 - 2a(l - ф2 -r- Log m, E.4)
da
where A9.17) has been utilized.
Let t > 0 be defined by A9.2). Observe that, for any positive integer n,
+ lYi(m -l) = (m + -Y (l + —)— = -— E5)
m) dt\ mj \ mj \ m1) dt m dt'
Also, in general, if /? has degree n and F is any differentiable function, then,
from A9.17), it is easily shown that
E.6)
Thus, from E.4)-E.6),
A dt d
S = 6zi^71 —A — 2)8) — m{\ ~ 2a) — 12ma(l — a)-7- -r Log m
\m da. dt
By A9.23),
and so
m--= -6 + 16t-12t2 + 8t3, E.8)
(m__)= 16-24t + 24t2. E.9)
Л\ ml v
From A9.20) and A9.22), respectively,
m= -3 + 8t-
and
1 = 3 _ 8t + 6t2 - 4t3 + 2R,
m
where R is defined by A9.6). Thus,
m + I = 4R. E.10)
m

470 21. Eisenstein Series
Hence, from E.7), E.9), and E.10),
S = \z,zn{im + Л (« - P) + (m - ^j(« + P - 1)
Our next goal is to obtain a suitable expression for (dt/da) (dt/dfl). From
A9.2),
dP -,dt , „ da. „ , dt
B + a— = St — and a + a— = ы —.
da. da dp dp
Therefore, from these equalities, A9.17), and A9.18),
dt dt 1 / dp\( da>
Щ1-Р)\( т2а(\ - a)
7A-/»
«*«/?(! - aJ
448а2/92A - «)A - P)m
64aP(l-a)(l-P)(t-a)(t-P)
t2{(t - P)(l - a) - (t - a)(l - P)}2
64a/9(l - a)(l - P)(t - a)(t - P)
t2(a - PJ{1 - tJ
- a)(l - P)(a - t)(t - PY
As we observed after A9.19), a > t and p < t. Thus, taking the square root on
each side above, we deduce that
dt dt\112 t(a - P)(\ - t)
dp) 8W?(l - a)(l - P)(a - t)(t -
Hence, from E.8), E.10), and E.11),
S = \z^znUR(a - P) + (-6 + Ш - Ylt2 + St3)(a + p - 1)
EЛ2)
From A9.18) and A9.19),
(a - t)(t -P) = ^(t - PJ = 7t2(l - tJ(l - t + t2J.

21. Eisenstein Series 471
Hence,
1/2 1
- t)(t - p)) t(i - 0A -1 +12)'
By A9.5), a — /9 = 2BR and a + /? = 2A Substituting these expressions and
then the values of A, B, and R into E.12), we discern that
S = \z,Zl (sBR2 + (-6 + 16t - 12t2 + ШЪ)BА - 1) -
l-t + t2
- 3t + 2t2)B - t + B)A - t + 2t2)
+ (-6 + Ш - 12t2 + 8t3){(l + t8) - A - O8 - 1}
- 12r(l - 0B - 3t + 3t2)
= 2X27A -2t + 2t2J
= 2127{(l-tJ + t2}2, E.13)
where the algebra was effected by a computer algebra package. Utilizing A9.2),
A9.3), and Entries 10(ii) and 11 (Hi) in Chapter 17, we deduce that
S = 2127(
= Wi-qM-q1) + ^WWfa14)}2- E-14)
Lastly, we use Entries 17(i), (ii) in Chapter 19 to find that
<p(q)9(q7) + 4
E.15)
where the cycle of coefficients is of length 7. Changing the sign of q in E.14)
and combining the result with E.15), we complete the proof of the first equality
in part (i).
In order to establish the second equality of (i), we first observe, by Entry
12(i) in Chapter 17, A9.2), and A9.3), that
2/3
It follows that
Z!Z,A -2t + 2t2J
-6t
E.16)

472 21. Eisenstein Series
On the other hand, by Entry 12(i) in Chapter 17, A9.2), A9.3), A9.14),
A9.15), A9.6), E.8), and E.10),
«) M
)
f*(q)
/ 49
+ A - 2t)K m2 - -j
\ m
m - -VB - 7t + lit2 - 8t3 + 4t4)L - -
+ A - 2()я(т + - j j + 14B - It + lit2 - 8t3 + 4t4)
LJ2(-3 + 8t - 6t2 + 4t3)U2 - 7t + lit2 - 8t3 + 4f4)
2A - 2t)B - 3t + 2t2)B - t + t2)(l - t + 2t2)
7B-7t+ Ilt2-8t3
-3 + 8t - 6t2 + 4t3)B - 3t - 3t2 + 2t3)
?A - t)
+ 7B - It
1
B + t + 23t2 - 64t3 + 72t4 - 48t5 + 16f6). E.17)
Combining E.16) and E.17), we deduce that
f(q)f(q1)

21. Eisenstein Series
473
Using the equality above in E.13) and changing the sign of q, we complete the
proof of the second part of (i).
Proofs of (ii), (Hi). Replacing q by — q2, we showed in E.14) that
+ 4<?W)«M<?28)}2.
E.18)
By Entries 10(iv) and ll(iv) in Chapter 17, B0.6) in Chapter 20, A9.2), and
A9.3),
E.19)
where we have invoked Entries 10(i) and 11 (ii) of Chapter 17. Substituting the
far right side of E.19) into E.18), we complete the proof of (ii). Furthermore,
substituting the second expression on the right side of E.19) into E.18), we
deduce (iii).
Entry 6.
(i) // /? has degree 3, then
(ii) // P has degree 5, then
(iii) // /9 fcas degree 7,
{A -

474 21. Eisenstein Series
Proof of (i). By Entry 3(i), the series on the left side of Entry 6(i) is equal to
Using Entry 11 (ii) of Chapter 17 and E.2) and E.5) from Chapter 19, we deduce
that
z2(9 - m2) - 3z|(m2 -
(9 - m2) - 3(m2 - 1)
8m л
(m + 1)C - m) (m - 1)C + m)\2
4m
4m
F.1)
Hence, the truth of (i) is made manifest.
Proof of (ii). By Entry 4(i), the series on the left side of Entry 6(ii) is equal to
By Entry 11 (ii) in Chapter 17 and A4.2) and A4.4) in Chapter 19,
-a)}1/2 4m {«A - a)}1/4 10
x (iz2{a(l - a)}1/2 + izlZs{a/9(l - a)(l -
, 5 ум /5
m2 ( — — 1 J 2m(--l
m
m
зм
(m -
(m -
5 V/4
5/2
3/2
(m - IK'2
5(m-lM'2 -
a-от

21. Eisenstein Series 475
- т)г'2E ~ m)(m""+5(n -
x (E - mJ + 2E - m){m - 1) + 5(m - 1JI/2
= _l!iZi!MLD0 - 32m + 8m2)B0 - 8w + 4m2I'2. F.2)
iH-4)(m- IK
From Entry ll(ii) in Chapter 17 and A3.3)-A3.5) in Chapter 19,
A - jgM\1/8
)
- IK ~ z\l\m - IK V «A -a)
4z}/2(m-lK 4zJ/2 4m3/2'
Using the calculation above in F.2) and then employing A3.6) and A3.7) in
Chapter 19, we discern that
2m14m
which completes the proof of (ii).
Proof of (iii). Observe that E.14) is precisely Entry 6(iii), and so the proof has
already been accomplished.
Entry 7.
f2(-q)f4-q9
„25*
25k
-q
q25) + 5q2f2(~q25)}112-

476 21. Eisenstein Series
Throughout the proofs of (i) and (ii), we use the notation of Section 3 in
Chapter 20.
Proof of (i). It will again be judicious to make a change of sign. By F.1),
Vi 12
j i Vn. г ^д^ — m) (m — 1)C + m)
' + l)C-m') (w'-l)C + m')>2
2V
Our next task is to express these last expressions in terms of t. By C.10)
and C.11) of Chapter 20 and G.1),
3-m2V
)
_2329 /{3A + 8t3) - A + 2tL}2 3{3-
16 \ A + 2tJ(l + 8t3) 1 + 8t3
- It + 4t2) - A + 2tK}2 + 12A - 4f3J)
16A + 8Г3)
_({1 _ 6t _ 4?3}2 + 3A -
-^A -3t + 9t2 - 8t3 + 12t4 + 16t6)
8t
t + t2)(l -2t + At2J
l + 8t3V
Z3*9 ,,
G.2)
Next, we attempt to identify this last expression with the middle expression
of (i). By Entry 12(i) of Chapter 17 and C.7)-C.11) of Chapter 20,
fW(q9) B,29)^2^ {«A - фA - у)}

21. Eisenstein Series 477
Z3Z9m'3/2t1/3(l - {3J/3(l + 2tI/3
m1/2(l - tI/3(l + 8t3J/3
z3z9(l + 8t3)t1/3(l - t3J'3(l + 2tI/3
A + 2t)(l - гI/3A + 8t3J'3
z3z9f1/3(l - 2t + 4t2I'3(l - tI/3(l + t + ?2J/3
A + 2tI/3
Hence, we may rewrite G.2) in the form
_ qf(q)((l + t + t2)(l - 2t + 4f2JV/3
fW(q9) \ t(i + 2tJ(i -
2tJ(l-t)/ '
where the last equality is verified by straightforward algebra.
It now suffices to prove that
P(q) _ A + 2tJ(l - t)
for upon substituting G.4) into G.3) and changing the sign of q, we then obtain
the first equality of part (i).
Employing Entry 12(i) in Chapter 17 and C.7), C.9), C.10), and C.11) in
Chapter 20, we find that
f\q) fa
as desired.
To establish the second part of (ii), we first observe that by Entry 11 (ii) in
Chapter 17 and C.7)-C.11) in Chapter 20,
A - t)(l - t3)^2
Thus, by G.2), we see that
s = iVV^-^-i9)^ "У J
J G.5)

478 21. Eisenstein Series
By Entry 11 (ii) in Chapter 17 and C.7)-C.11) of Chapter 20,
iA4(-<?3) m'{0(l - P)}112 l + t + t2 Inc.
— —_—.— — = 1 G.6)
Substituting G.6) into G.5) and changing the sign of q, we establish the second
part of (i).
Proof of (ii). By part (i), the series on the left side of (ii) is equal to
Utilizing Entry 12(iii) in Chapter 17 and C.7)-C.11) in Chapter 20, we deduce
that
x < —
z3t2>3(l-
(l - tJ/3(l
2t)(l - tf
3t2 + At3 + 9t4 + 6t5
m A + 8t3)(l ~2t + 4t2I/3
1 + 8t3 A + t + t2J
A + 2tJ 1 + 8t3
2t
3
from which (ii) now readily follows.
Proof of (iii). We employ the notation
4

21. Eisenstein Series 479
which is a modification of A1.7) in Chapter 19. Returning to Entry 4(i) and
utilizing A1.8) of Chapter 19, we deduce that
_/5(-g)A + 22/6(-45) + i2Sq^-q5)Y
22 f6(-q5) 125 /12(-«?5)
(
Я-q5)\ ™6 q5f6(~q25) w12 <210/12(-<Z25),
125 V/2
+ -n(wS + 5w* + 15w3 + 25w2 + 25wJ
J
n( + + 15w + 25w + )
w J
22w9 + 235w8 + 1580w7 + 7425w6 + 25550w5
~/(-«5)w5V"
+ 65625w4 + 125000w3 + 171875w2 + 156250w + 78125I/2
q5f5(-q25)
= —j. jr—(w + 10w3 + 45w + lOOw + 125)(w2 + 2w
Again using Entry 4(i) and proceeding in the same manner as above, we
discover that
q5f4-q25)( f12(-q5)
A-q5)
a5f5(-a25)
, „ /(-g) , „
q5f4~q25)
15w3 + 25w2 + 25wJ
_ I ((w
+ 22(w5 + 5w4 + 15w3 + 25w2 + 25w) + 125I/2
}~\ V + 55w8 + 200w7 + 525w6 + 1022w5
+ 1485w4 + 1580w3 + 1175w2 + 550w + 125I/2
= 4J y ; ;(w4 + 4w3 + 9w2 + lOw + 5)(w2 + 2w + 5I/2.
jy~q )
Multiplying G.9) by g and G.10) by f, adding the two resulting equalities,
and using A1.8) of Chapter 19 and G.8), we conclude that

480 21. Eisenstein Series
10w3 + 45w2 + lOOw + 125)
f(-q5)
qf4-qS)
+ §(w4 + 4w3 + 9w2 + 10w + 5))(w2 + 2w + 5I/2
-(w4 + 5w3 + 15w2 + 25w + 25)(w2 + 2w + 5I/2
A-q) \q2f4-q2S) qA-q25)
from which the truth of Entry 7(iii) is evident.
Entry 8.
(ii) Let /J feare degree 11 orer a.
5 + 12 У q - 132 У feg
+ Й1-«а A I-
2"
+ {A -
(iii) If fi has degree 19, then
1^ 1^
t=i 1 — q *=i 1 — q
= (p2(q)cp2(qi9)(l + (a/?I'2 + {A - a)(l -
+ {A -
We are unable to prove either (i) or (iii) using B.3) or E.3). Thus, the proofs
of (i) and (iii) will be deferred until Section 11 where the theory of modular
forms will be invoked to establish several results that we cannot otherwise
prove. Part (ii) follows from (i), and so this proof will be the only one given in
this section.
Proof of (ii). By Entries 11 (ii) and 12(i), (iii) in Chapter 17 and Entry 7(i) in
Chapter 20,

21. Eisenstein Series 481
5<p2(q)<p2(q11)-20qf2W2(qli)
+ 32q2f2(-q2)f2(-q22) -
P)Y'6 - 5{a(l
= 2l2llE - 5A - (a/?I'4 - {A - a)(l - jS)}1'4)
+ 2A - (ajSI/4 - {A - a)(l - JS)}1'4J
- 5{a(l - «HA - jS)}1'4)
+ {A - «)A - P)
Hence, (i) implies the truth of (ii).
Entry 9.
oo t/j2* oo
<0 " + 12 ? з^-д - 276 ?
*=i 1 — q k=\ 1 — q
- 8{16ajS(l - a)(l - jS)}1/12(l + (a)?I'4 + {A -
(ii) // fi is of degree 15, then
oo bn2k oo кпъок
7 + 12 V g - 180 Y q
{A - «)A -
(iii) // P is of degree 31, then
5 Ц f kg2' 121 f
- /?)}1/4}2
}
+ {1 + (a/9I'4 + {A - a)(l - /?)}
- 2{a)S(l - «)A - /J)}1/8{1 + (a/?I/8 + {A - a)(l - /9)}1/8}).
We defer the proofs for Entry 9 until Section 11, where we employ the
theory of modular forms.

482 21. Eisenstein Series
Entry 10.
(i) Let ft have degree 5 with respect to a. Then
= cp2{q)cp2{q5m\ + ajS + A -
(ii) // jS has degree 9, then
(iii) //jS is of degree 17,
- «)A - P)} {16a)S(l -
Proof of (i). Comparing Entry 10(i) with Entry 4(iii), we see that it suffices to
prove that
{1-yfiP- V(l - a)(l - /J)}2 = 4{16a^(l - a)(l - jS)}1'3.
But this equality follows immediately from Entry 13(i) of Chapter 19, and so
the proof is complete.
Proof of (ii). Employing the notation of Section 7, G.7), and C.10) and C.11)
of Chapter 20, we deduce that the left side of (ii) is equal to
'V'2 (\ + t + e\2 A + 8t3I/2
J =22
A - 2t + 4*2)УA + t + t2)
2J

21. Eisenstein Series 483
Now by C.17), C.7), and C.9) in Chapter 20,
A0J)
A
and
Employing A0.2) and A0.3) and performing a very laborious calculation, we
find that
A - 2t + 4t2)(l + t + t2L 1(
A + 2tM l{
_ A - 2t + 4t2)(l + t + t2L 8t(l - t3) 32t2(l - t3J
A + 2t)s + A + 2tJ A + 2tL
16t5(l - tL(l - t3)
+ A + 2tL(l + 8t3)
_ - 18t2 + 36t3 - 69t4 + 30t5 - 78t6 + 168t7 - 39t8 + 42t9 - 72t10
A + 2fK(l + 8t3) '
A0.4)
Next, from A0.2)-A0.4),
3t4(l - It - 2t2 + 8t3 - 7t4 + 2ts)
A + 2tK(l + 8t3)
3t4(l - if
2tJ(l + 8t3)
I+2t>/ 1 + 8t3A8t(l - t3).
3
2 1 - (a^I/2 - {A - a)(l - ^)}1/2*
By combining A0.1) with the result just obtained, we obtain (ii) to complete
the proof.

484 21. Eisenstein Series
The proof of (iii) will be deferred until Section 11.
Entry 11. If (i has degree 35 over a, then
^ ^
{A
In fact, Entry 11 is listed as ll(i) in the second notebook (p. 256), but no
further result is stated in this section.
There remain now seven formulas in Chapter 21 which we have not yet
proved but which we now establish via the theory of modular forms. Our first
task is to identify the series on the left sides of these formulas as modular forms.
Theorem. Let
where q = e"" and x e Ж. Define
Fn(x) = E2(x) - nE2(m).
Then Fn is a modular form on T0(n) of weight 2 and trivial multiplier system.
Proof. Set E%(x) = ?2(т) + Ъ1(жу), where у = Im(r). It is well known (e.g., see
the treatises of Rankin [2, pp. 194-195] or Schoeneberg [1, p. 68]) that if
V = (" J) e ГA), then ?*(T) satisfies the transformation formula
E*(Vx) = (ex + dJE$(x) - — (ex + d). A1.1)
n
Observe that Fn(x) = ?f (x) - nE$(nx). So, by A1.1) and @.4) of Chapter 20,
if Ve T0(n),
Fn(Vx) = (ex + dJE%(x) - —(ex + d)
n
с Л2„... . die (с
-nx + d
- n((-nx + JO) l
\\n J nn \n
= (ex + dJFn(x).
The desired conclusion now follows.

21. Eisenstein Series 485
Proofs of Entries 8(i), (iii), 9(i)-(iii), 10(iii), 11. We first translate those six
entries written in terms of a. and /? into proposed identities involving theta-
functions. Thus, using Entries 10(i)-(iii), ll(i)-(iii), and 12(iii) in Chapter 17,
we find that Entries 8(iii), 9(i)-(iii), 10(iii), and 11 can be written as the
respective theta-function relations
~38fc
2(q38) + <P2(-q)<P2(-q19)
>(q)9(q19)9(-q)v(-q19)
I, (И.2)
и +12 ? kq2\k- °° k"A6k
*=i 1 — q
7 112 f ^ ISO f kq
" 415) + <p(-q2)<p(-q30)L
-q15)), (П.4)
5 + 4 У Kg _ 124 У К<?
+ Й1-?3' ^il-96"
A1.5)
- <p2(-q)<P2(-q")) - 48«6/4(-q2)/4(-q34), A1.6)

486 21. Eisenstein Series
and
oo hn2k „о ЬпТ0к
17 + 12 У q - 420 Y q
- 9(-q)<p(-q35)
35)f
-4eV(-«)^2(-935). (И.7)
Next, we rewrite Entry 8(i) and A1.2)—A1.7) as proposed identities relating
modular forms. Thus, by @.13) in Chapter 20, we find that, respectively,
oo bfP-k oo kn22k
5 + 12 У q - 132 У q
+ kl-q2k U\kl-q22k
= 50?(tH?A1t) - 2ОГ?(т)Л2(Пт) + 32^(т)^2A1т) - 20^(т)й?(Пт),
A1.8)
3 И f kg2t 76 f kg38*
+ Al-92* A 1-е38*
= ef (r)ef A9t) + 16д|(т)в1A9т) + д20(т)д20A9т)
(т)воA9т)
A1.9)
16д|(тH!B3т)
т)@1(тH1B3т) + 43
- 40*,2(ф2B3т), (НЛО)
7 I 12 f kqU 180 f kq3°k
A5r)(gl(T)glA5T) + 2ЛоМЛоA5т) + Й2(т)й2A5т)L
A1.11)
(g1(x)g1CU) + Ад2(т)д2C1т) + go(*)9o№)J
01C1т) + 2Ло(т)ЛоC1т) + Л2(т)й2C1т)Х
A1.12)

21. Eisenstein Series
487
„34*
k~i l - q"
256д%{х)д\{\Ъ)
- 16д2(т)д2(Пт) - д20(т)д2(Пт))
A1.13)
and
*=1 1 —
1
- 4Л?(т)й?C5т).
A1.14)
We next demonstrate that the multiplier system for each term on each side
of A1.8)—A1.14) is trivial, for transformations belonging to Г = ГB) п Г0(п),
where n = 11,19, 23,15, 31,17, and 35, respectively.
First, by the theorem proved above, the multiplier system of each of the
seven left sides is trivial. Using @.14), @.16), and @.18)-@.23) in Chapter 20,
we easily check that each term on the right sides in A1.8)—A1.14) has a trivial
multiplier system. We note that @.27) in Chapter 20 facilitates the computa-
computations. Furthermore, for A1.11), we need to use the remarks made after A3.8)
in Chapter 20, because here 3\n.
After clearing denominators in A1.11) and A1.14), we write each of the
proposed modular form identities A1.8)—A1.14) in the form
The following table indicates the weight r of each modular form F, the value
pr calculated from @.6) and @.24) in Chapter 20, and the number /z determined
from @.30) in Chapter 20.
A1.8)
A1.9)
A1.10)
A1.11)
A1.12)
A1.13)
A1.14)
r
2
2
2
4
2
4
3
Pr
6
10
12
12
16
9
24
12
20
24
48
32
36
72
By the theory surrounding @.30) in Chapter 20, if we can show that the

488 21. Eisenstein Series
coefficients of q°, q\ ..., q" for F are equal to 0, then F = 0. With the help of
the computer algebra system MACSYMA, we have, indeed, verified that the
required coefficients are equal to 0. Hence, the truths of A1.8)—A1.14) have
been established, and so the proofs are complete.
After the statement of Entry 11, Ramanujan draws a short horizontal bar
and offers below it three equalities relating <p at certain arguments. Although
the material is unrelated to the subject matter of Chapter 21, we provide proofs
here because the results appear on the last page (p. 256) of Chapter 21.
Final Entry. // the principal branch of each root is taken, then
<p(q) ~ <?(-<?) __ /Ф V) ~ <р\~Ч
and
A1.16)
Proof. Squaring and employing Entries 10(i)-(iv) in Chapter 17, we find that
the first proposed identity of A1.15) is equivalent to the equality
'1 _ (l _ х)У*у
+ A - X) /
This equality is obvious.
From A0.1) and Entries 10(iii), (iv), all in Chapter 17, we find that
Taking the fourth power of the extremal sides of A1.15), using Entries 10(i),
(ii), (v) in Chapter 17, and utilizing A1.17) above, we deduce the equivalent
identity
- xI/4
+ (i _ x)m
A modest calculation verifies the truth of the latter identity.
Squaring both sides of A1.16), we find that it is equivalent to the formula
<p(q) + M-9) = 9(9) + W\q) - 2(p2{q2)}w.
By Entries 10(i), (ii), (iv) in Chapter 17, the last identity is equivalent to the
equality
1 + i(l - xI/4 = 1 + {1 - A + ,/1 - x)}112,
which is obvious.
Some of the results in this chapter were independently derived by S. Ghosh
in her doctoral dissertation [1].

References
Abramowitz, M. and Stegun, I. A., editors
[1] Handbook of Mathematical Functions, Dover, New York, 1965.
Adiga, C.
[1] A Study of Some Identities Stated by Srinivasa Ramanujan in His "Lost" Note-
Notebook and Earlier Works, Doctoral Thesis, University of Mysore, 1983.
Adiga, C, Berndt, B. C, Bhargava, S., and Watson, G. N.
[1] Chapter 16 of Ramanujan's second notebook: Theta-functions and q-series,
Mem. Am. Math. Soc, No. 315, 53A985), American Mathematical Society,
Providence, 1985.
Agarwal, A. K. and Andrews, G. E.
[1] Rogers-Ramanujan identities for partitions with "N copies of N," J. Comb.
Theory (Ser. A) 45A987), 40-49.
Alder, H. L.
[1] Generalizations of the Rogers-Ramanujan identities, Pacific J. Math. 4A954),
161-168.
Almkvist, G.
[1] Aritmetisk-geometriska medelvardet och ellipsens baglangd, Nordisk Mat.
Tidskr. 25-26A978), 121-130.
Almkvist, G. and Berndt, В. С
[1] Gauss, Landen, Ramanujan, the arithmetic-geometric mean, ellipses, л, and the
Ladies Diary, Am. Math. Monthly 95A988), 585-608.
Al-Salam, W. A. and Ismail, M. E. H.
[1] Orthogonal polynomials associated with the Rogers-Ramanujan continued
fraction, Pacific J. Math. 104A983), 269-283.
Andrews, G. E.
[1] On q-difference equations for certain well-poised basic hypergeometric series,
Q. J. Math. (Oxford) 19A968), 433-447.
[2] On Ramanujan's summation of гфх (a; b; z), Proc. Am. Math. Soc. 22A969),
552-553.
[3] On a transformation of bilateral series with applications, Proc. Am. Math. Soc.
25A970), 554-558.

490 References
[4] Partition identities, Adv. Math. 9A972), 10-51.
[5] On the general Rogers-Ramanujan theorem, Mem. Am. Math. Soc, No. 152,
American Mathematical Society, Providence, 1974.
[6] A general theory of identities of the Rogers-Ramanujan type, Bull. Amer. Math.
Soc. 80A974), 1033-1052.
[7] Applications of basic hypergeometric series, SI AM Rev. 16A974), 441-484.
[8] Problems and prospects for basic hypergeometric series, in Theory and Applica-
Application of Special Functions, R. A. Askey, ed., Academic Press, New York, 1975,
pp. 191-224.
[9] The Theory of Partitions, Addison-Wesley, Reading, MA, 1976.
[10] An introduction to Ramanujan's "lost" notebook, Amer. Math. Monthly 86A979),
89-108.
[11] The hard-hexagon model and Rogers-Ramanujan type identities, Proc. Natl.
Acad. Sci. USA 78A981), 5290-5292.
[12] Ramanujan's "lost" notebook. I. Partial ^-functions, Adv. Math. 41A981),
137-172.
[13] Ramanujan' s "lost" notebook. III. The Rogers-Ramanujan continued fraction,
Adv. Math. 41A981), 186-208.
[14] L. J. Rogers and the Rogers-Ramanujan identities, Math. Chronicle 11A982),
1-15.
[15] Multiple series Rogers-Ramanujan type identities, Pacific J. Math. 114A984),
267-283.
[16] Hecke modular forms and the Kac-Peterson identities, Trans. Amer. Math. Soc.
283A984), 451-458.
[17] The fifth and seventh order mock theta functions, Trans. Amer. Math. Soc.
293A986), 113-134.
[18] q-Series: Their Development and Applicationin Analysis, Number Theory, Combi-
Combinatorics, Physics, and Computer Algebra, CBMS Regional Conference Series in
Mathematics No. 66, American Mathematical Society, Providence, 1986.
[19] The Rogers-Ramanujan identities without Jacobi's triple product, Rocky
Mountain J. Math. 17A987), 659-672.
[20] Physics, Ramanujan, and computer algebra, in Computer Algebra, D. V. Chud-
novsky and R. D. Jenks, eds., Marcel Dekker, New York, 1989, pp. 97-109.
[21] On the proofs of the Rogers-Ramanujan identities, in q-Series and Partitions,
D. Stanton, ed., Springer-Verlag, New York, 1989, pp. 1-14.
Andrews, G. E. and Askey, R.
[1] Enumeration of partitions: the role of Eulerian series and q-orthogonal poly-
polynomials, in Higher Combinatorics, M. Aigner, ed., D. Reidel, Dordrecht, 1977,
pp. 3-26.
[2] A simple proof of Ramanujan's summation of the 1i^1, Aequa. Math. 18A978),
333-337.
[3] Another q-extension of the beta function, Proc. Amer. Math. Soc. 81A981),
97-100.
Andrews, G. E. and Baxter, R. J.
[1] A motivated proof of the Rogers-Ramanujan identities, Amer. Math. Monthly
96A989), 401-409.
Andrews, G. E., Baxter, R. J., and Forrester, P. J.
[1] Eight-vertex SOS model and generalized Rogers-Ramanujan-type identities, J.
Stat. Phys. 35A984), 193-266.
Andrews, G. E., Berndt, B. G, Jacobsen, L., and Lamphere, R. L.
[1] Variations on the Rogers-Ramanujan continued fraction in Ramanujan's note-
notebooks, in Number Theory, Madras, 1987, K. Alladi, ed., Lecture Notes in Math.
No. 1395, Springer-Verlag, Berlin, 1989, pp. 73-83.

References 491
Askey, R.
[1] The q-gamma and q-beta functions, Appl. Anal. 8A978), 125-141.
[2] Ramanujan's extensions of the gamma and beta functions, Amer. Math. Monthly
87A980), 346-359.
[3] An elementary evaluation of a beta type integral, Indian J. Pure Appl. Math.
14A983), 892-895.
[4] Ramanujan's ,^t and formal Laurent series, Indian J. Math. 29A987), 101-105.
[5] Beta integrals in Ramanujan's papers, his unpublished work and further exam-
examples, in Ramanujan Revisited, Academic Press, Boston, 1988, pp 561-590.
[6] Beta integrals and ^-extensions, in Proceedings of the Ramanujan Centennial
International Conference, 15-18 December 1987, Annamalainagar, The Rama-
Ramanujan Mathematical Society, Madras, 1988, pp. 85-102.
[7] Orthogonal polynomials and theta functions, in Theta Functions Bowdoin 1987,
Part 2, Proceedings of Symposia in Pure Mathematics, vol. 49, American Mathe-
Mathematical Society, Providence, 1989, pp. 299-321.
[8] Ramanujan and hypergeometric series, in Ramanujan International Symposium
on Analysis, N. K. Thakare, ed., Macmillan India, Madras, 1989, pp . 1-83.
Askey, R. and Roy, R.
[1] More «-beta integrals, Rocky Mountain J. Math. 16A986), 365-372.
Askey, R. and Wilson, J.
[1] Some basic hypergeometric orthogonal polynomials that generalize Jacobi
polynomials, Mem. Amer. Math. Soc, No. 319, 54A985), American Mathe-
Mathematical Society, Providence, 1985.
Atkin, A. O. L. and Swinnerton-Dyer, P.
[1] Some properties of partitions, Proc. London Math. Soc. C) 4A954), 84-106.
Bailey, W. N.
[1] On the simplification of some identities of the Rogers-Ramanujan type, Proc.
London Math. Soc. C) 1A951), 217-221.
[2] A note on two of Ramanujan's formulae, Q. J. Math. (Oxford) 3A952), 29-31.
[3] A further note on two of Ramanujan's formulae, Q. J. Math. (Oxford) 3A952),
158-160.
[4] Generalized Hypergeometric Series, Stechert-Hafner, New York, 1964.
Baxter, R. J.
[1] Exactly Solved Models in Statistical Mechanics, Academic Press, London, 1982.
[2] Ramanujan's identities in statistical mechanics, in Ramanujan Revisited, Aca-
Academic Press, Boston, 1988, pp. 69-84.
Berndt, В. С
[1] Generalized Dedekind eta-functions and generalized Dedekind sums, Trans.
Amer. Math. Soc. 178A973), 495-508.
[2] Generalized Eisenstein series and modified Dedekind sums, J. Reine Angew.
Math. 272A975), 182-193.
[3] Modular transformations and generalizations of several formulae of Ramanu-
Ramanujan, Rocky Mountain J. Math. 7A977), 147-189.
[4] Analytic Eisenstein series, theta-functions, and series relations in the spirit of
Ramanujan, J. Reine Angew. Math. 303/304 A978), 332-365.
[5] Ramanujan's Notebooks, Part I, Springer-Verlag, New York, 1985.
[6] Remarks on some of Ramanujan's number theoretical discoveries found in his
second notebook, in Number Theory, K. Alladi, ed., Lecture Notes in Math. No.
1122, Springer-Verlag, Berlin, 1985, pp. 47-55.
[7] Ramanujan's modular equations, in Ramanujan Revisited, Academic Press,
Boston, 1988, pp. 313-333.
[8] Introduction to Ramanujan's modular equations, in Proceedings of the Rama-

492 References
nujan Centennial International Conference, 15-18 December 1987, Annama-
lainagar, The Ramanujan Mathematical Society, Madras, 1988, pp. 15-20.
[9] Ramanujan's Notebooks, Part II, Springer-Verlag, New York, 1989
[10] Ramanujan's formulas for Eisenstein series, in Number Theory and Related
Topics, Oxford University Press, Bombay, 1989, pp. 23-29.
[11] Ramanujan's Notebooks, Part IV, Springer-Verlag, New York, to appear.
Berndt, B. C, Biagioli, A. J., and Purtilo, J. M.
[1] Ramanujan's "mixed" modular equations, J. Ramanujan Math. Soc. 1A986),
46-70.
[2] Ramanujan's modular equations of "large" prime degree, J. Indian Math. Soc.
51A987), 75-110.
[3] Ramanujan's modular equations of degrees 7 and 11, Indian J. Math. 29A987),
215-228.
Berry, A.
[1] On elliptic modular equations for transformations of orders 29, 31, 37, Amer.
J. Math. 30A908), 156-169.
Bhagirathi, N. A.
[1] On certain continued fractions and ^-series, Math. Student 56A988), pp. 97-104.
Bhargava, S.
[1] On Rogers-Ramanujan identities, in Srinivasa Ramanujan A887-1920), Mac-
millan India Ltd., Madras, 1988, pp. 75-81.
Bhargava, S. and Adiga, С
[1] On some continued fraction identities of Srinivasa Ramanujan, Proc. Amer.
Math. Soc. 92A984), 13-18.
[2] Two generalizations of Ramanujan's continued fraction identities, in Number
Theory, K. Alladi, ed., Lecture Notes in Math. No. 1122, Springer-Verlag,
Berlin, 1985, pp. 56-62.
[3] A basic hypergeometric transformation of Ramanujan and a generalization,
Indian J. Pure Appl. Math. 17A986), 338-342.
[4] On two elliptic 0-function identities stated by Srinivasa Ramanujan, Int. J.
Math. Ed. Sci. Tech. 18A987), 318-320.
[5] Simple proofs of Jacobi's two and four square theorems, Int. J. Math. Ed.
Sci. Tech. 19A988), 779-782.
Bhargava, S., Adiga, C, and Somashekara, D. D.
[1] On some generalizations of Ramanujan's continued fraction identities, Proc.
Indian Acad. Sci. (Math. Sci.) 97A987), 31-43.
[2] On certain continued fractions related to 3q>2 basic hypergeometric functions,
J. Math. Phys. Sci. 21A987), 613-629.
[3] Parity results deducible from certain theta function identities found in Chapter
16 of Ramanujan's second notebook, Math. Student 57A989), 88-100.
[4] Ramanujan's ^ summation and an interesting generalization of an identity of
Jacobi, to appear.
Biagioli, A. J.
[1] A proof of some identities of Ramanujan using modular forms, Glasgow Math.
J. 31A989), 271-295.
[2] A proof of two identities of Ramanujan, preprint.
Blecksmith, R., Brillhart, J., and Gerst, I.
[1] Parity results for certain partition functions and identities similar to theta
function identities, Math. Сотр. 48A987), 29-38.
[2] Some infinite product identities, Math. Сотр. 51A988), 301-314.
[3] New proofs for two infinite product identities, Rocky Mountain J. Math., to
appear.

References 493
Borwein, J. M.
[1] Some modular identities of Ramanujan useful in approximating к, Proc. Amer.
Math. Soc. 95A985), 365-371.
Borwein, J. M. and P. B.
[1] Explicit algebraic nth order approximations to pi, in Approximation Theory and
Spline Functions, D. Reidel, Dordrecht, 1984, pp. 247-256.
[2] Pi and the AGM, Wiley, New York, 1987.
[3] Explicit Ramanujan-type approximations to pi of high order, Proc. Indian Acad.
Sci. (Math. Sci.) 97A987), 53-59.
[4] More Ramanujan-type series for l/я, in Ramanujan Revisited, Academic Press,
Boston, 1988, pp. 359-374.
[5] Ramanujan and pi, Sci. Am. 256A988), 112-117.
[6] Approximating л with Ramanujan's modular equations, Rocky Mountain J.
Math. 19A989), 93-102.
Borwein, J. M., Borwein, P. В., and Bailey, D. H.
[1] Ramanujan, modular equations, and approximations to pi or how to compute
one billion digits of pi, Amer. Math. Monthly 96A989), 201-219.
Bressoud, D. M.
[1] Analytic and combinatorial generalizations of the Rogers-Ramanujan identi-
identities, Mem. Amer. Math. Soc, No. 227,24A980), American Mathematical Society,
Providence, 1980.
[2] An easy proof of the Rogers-Ramanujan identities, J. Number Theory 16A983),
235-241.
[3] Hecke modular forms and g-Hermite polynomials, Illinois J. Math. 30A986),
185-196.
Bressoud, D. M. and Zeilberger, D.
[1] A short Rogers-Ramanujan bijection, Discrete Math. 38A982), 313-315.
Burnside, W.
[1] The elliptic functions оЦК, & с, Mess. Math. 12A883), 154-157.
Byrd, P. F. and Friedman, M. D.
[1] Handbook of Elliptic Integrals for Engineers and Scientists, 2nd ed., Springer-
Verlag, New York, 1971.
Carlitz, L.
[1] Note on some continued fractions of the Rogers-Ramanujan type, Duke Math.
J. 32A965), 713-720.
[2] Advanced problem 5196, solution by the proposer, Amer. Math. Monthly
72A965), 917-918.
[3] Some identities in combinatorial analysis, Duke Math. J. 39A972), 31-37.
Carlitz, L. and Scoville, R.
[1] A note on weighted sequences, Fibonacci Q. 13A975), 303-306.
Carlitz, L. and Subbarao, M. V.
[1] On the quintuple product identity, Proc. Amer. Math. Soc. 32A972), 42-44.
Cauchy, A.
[1] Memoire sur les fonctions dont plusieurs valeurs sont liees entre elles par une
equation lineaire, et sur diverses transformations de produits composes d'un
nombre indefini de facteurs, C. R. Acad. Sci. (Paris) 17A843), 523-531.
[2] Oeuvres, Ser. I, t. 8, Gauthier-Villars, Paris, 1893.
[3] Oeuvres, Ser. II, t. 7, Gauthier-Villars, Paris, 1889.
Cayley, A.
[1] An Elementary Treatise on Elliptic Functions, 2nd ed., Dover, New York, 1961.
Chudnovsky, D. V. and G. V.
[1] Computational problems in arithmetic of linear differential equations. Some
Diophantine applications, in Number Theory, New York 1985-88, D. V. Chud-

494 References
novsky, G. V. Chudnovsky, H. Cohn, and M. B. Nathanson, eds., Lecture Notes
in Math. No, 1383^Springer-Verlag, Berlin, 1989, pp. 12-49.
Churchhouse, R. F.
[1] Continued fractions of the Rogers-Ramanujan type, in Proceedings of the Third
Caribbean Conference in Combinatorics and Computing, University of The West
Indies, Cave Hill, Barbados, 1981, pp. 1-14.
Cohn, H.
[1] Iterated ring class fields and the icosahedron, Math. Ann. 255A981), 107-122.
[2] An explicit modular equation in two variables and Hilbert's twelfth problem,
Math. Сотр. 38A982), 227-236.
[3] Iterated ring class fields and the 168-tesselation, Math. Ann. 270A985), 69-77.
[4] Fricke's two-valued modular equations, Math. Сотр. 51A988), 787-807.
[5] Singular moduli and modular equations for Fricke's cases, in Theta Functions
Bowdoin 1987, L. Ehrenpreis and R. С Gunning, eds., Proceedings of Symposia
in Pure Mathematics, vol. 49, part 2, American Mathematical Society, Provi-
Providence, 1989, pp. 233-249.
[6] Iteration of two-valued modular equations, in Number Theory, New York
1985-88, D. V. Chudnovsky, G. V. Chudnovsky, H. Cohn, and M. B. Nathan-
son, eds., Lecture Notes in Math. No. 1383, Springer-Verlag, Berlin, 1989,
pp. 50-58.
[7] Computation of singular moduli by multi-valued modular equations, in Pro-
Proceedings of the Colloquium on Computational Number Theory, Debrechen, 1989,
to appear.
[8] A numerical survey of the reduction of modular curve genus by Fricke's involu-
involutions, to appear.
Cohn, H. and Deutsch, J.
[1] An explicit modular equation in two variables for Q(s/3), Math. Сотр. 50A988),
557-568.
Court, N. A.
[1] College Geometry, 2nd ed., Barnes & Noble, New York, 1952.
Coxeter, H. S. M.
[1] Introduction to Geometry, 2nd ed., Wiley, New York, 1969.
Darling, H. В. С
[1] Proofs of certain identities and congruences enunciated by S. Ramanujan, Proc.
London Math. Soc. B) 19A921), 350-372.
Denis, R. Y.
[1] On certain <j-series and continued fractions, Math. Student 44A983), 70-76.
[2] On basic hypergeometric functions and continued fractions, Math. Student
52A984), 129-136.
[3] On certain q-series and continued fraction identities, Math. Student 53A985),
243-248.
[4] On certain summation of ^-series and identities of Rogers-Ramanujan type,
J. Math. Phys. Set. 22A988), 87-99.
[5] On generalization of continued fraction of Gauss, Int. J. Math, and Math. Sci.
13A990), 725-730.
Dickson, L. E.
[1] History of the Theory of Numbers, vol. 2, Chelsea, New York, 1966.
Dyson, F. J.
[1] A walk through Ramanujan's garden, in Ramanujan Revisited, Academic Press,
Boston, 1988, pp. 7-28.
Ehrenpreis, L.
[1] The Rogers-Ramanujan theorem, in A Tribute to Emil Grosswald: Papers in

References 495
Number Theory and Related Analysis, Contemporary Mathematics, American
Mathematical Society, Providence, to appear.
Eisenstein, G.
[1] Transformations remarquables de quelques series, J. Reine Angew. Math. 27
A844), 193-197.
[2] Mathematische Werke, Band 1, Chelsea, New York, 1975.
Enneper, A.
[1] Elliptische Functionen, Louis Nebert, Halle, 1890.
Erdelyi A., editor
[1] Higher Transcendental Functions, vol. 1, McGraw-Hill, New York, 1953.
Euler, L.
[1] Introductio in Analysin Injinitorum, vol. 1, Marcum-Michaelem Bousquet,
Lausanne, 1748.
[2] Solutio generalis quorundam problematum diophanteorum quae vulgo nonnisi
solutiones speciales admittere videntur, Novi Comm. Acad. Sci. Petropolitanae
6A756/57), 1761,155-184.
[3] Nova series infmita maxime convergens perimetrum ellipsis exprimens, Novi
Comm. Acad. Sci. Petropolitanae 18A773), 71-84.
[4] Opera Omnia, Ser. I, vol. 2, B. G. Teubner, Lipsiae, 1915.
[5] Opera Omnia, Ser. I, vol. 8, B. G. Teubner, Lipsiae, 1922.
[6] Opera Omnia, Ser. I, vol. 20, B. G. Teubner, Lipsiae, 1912.
Evans, R. J.
[1] Theta function identities, J. Math. Anal. Appl. 147A990), 97-121.
Fermat, P.
[1] Oeuvres, t. 3, Gauthier-Villars, Paris, 1896.
Fiedler, E.
[1] Ueber eine besondere Classe irrationaler Modulargleichungen der elliptischen
Functionen, Vierteljahrsschr. Naturforsch. Gesell. (Zurich) 12A835), 129-229.
Fine, N. J.
[1] Basic Hypergeometric Series and Applications, American Mathematical Society,
Providence, 1988.
Flajolet, P.
[1] Combinatorial aspects of continued fractions, Discrete Math. 32A980), 125-161.
Flajolet, P. and Francon, J.
[1] Elliptic functions, continued fractions, and doubled permutations, Eur. J. Com-
Combinatorics 10A989), 235-241.
Forsyth, A. R.
[1] The elliptic functions of %K, Mess. Math. 12A883), 134-138.
Fricke, R.
[1] Neue Beitrage zur Transformationstheorie der elliptischen Functionen, Math.
Ann. 40A892), 469-502.
[2] Die Elliptischen Funktionen und Ihre Anwendungen, Bd. 1, B. G. Teubner,
Leipzig, 1916.
[3] Die Elliptischen Funktionen und Ihre Anwendungen, Bd. 2, B. G. Teubner,
Leipzig, 1922.
Garsia, A. M. and Milne, S. С
[1] A Rogers-Ramanujan bijection, J. Combin. Theory (A) 31A981), 289-339.
Gauss, С F.
[1] Mathematisches Tagebuch 1796-1814, Ostwalds Klassiker der Exakten Wis-
senschaften, Bd. 256, Akad. Verlagsgesell., Geest & Portig K.-G., Leipzig, 1976
(published also in Math. Ann. 57A903), 6-34).

496 References
[2] Werke, Bd. 3, Koniglichen Gesell. Wiss., Gottingen, 1876, pp. 413-432.
[3] Hundert Theoreme iiber die neuen Transscendenten, in Werke, Bd. 3,
Koniglichen Gesell. Wiss., Gottingen, 1876, pp. 461-469.
Ghosh, S.
[1] On the Evaluation of k2 and E in the Theory of Elliptic Functions, Doctoral
Dissertation, University of Bombay, 1985.
Glaisher, J. W. L.
[1] Proof of the addition equation for elliptic integrals of the second kind by means
of the q-series, Mess. Math. 12A883), 43-48.
[2] On the process of squaring the q-series for kp sn u, kp en и, р dn u, Mess. Math.
16A887), 145-149.
[3] On the q-series derived from the elliptic and zeta functions of %K and \K, Proc.
London Math. Soc. 22A891), 143-171.
Glasser, M. L.
[1] Definite integrals of the complete elliptic integral K, J. Res. Nat. Bur. Stand. (B)
Math. Sci. 80B A976), 313-323.
Glasser, M. L., Privman, V., and Svrakic, N. M.
[1] Temperley's triangular lattice compact cluster model: exact solution in terms of
the q series, J. Phys. Math. Gen. 20A987), L1275-L1280.
Gordon, B.
[1] Some identities in combinatorial analysis, Q. J. Math. (Oxford) B) 12A961),
285-290.
[2] Some continued fractions of the Rogers-Ramanujan type, Duke Math. J.
32A965), 741-748.
Gradshteyn, I. S. and Ryzhik, I. M.
[1] Table of Integrals, Series, and Products, 4th ed., Academic Press, New York,
1965.
Gray, J. J.
[1] A commentary on Gauss's mathematical diary, 1796-1814, with an English
translation, Expos. Math. 2A984), 97-130.
Greenhill, A. G..
[1] The Applications of Elliptic Functions, Macmillan, London, 1892.
[2] The seventeen-section of the elliptic function, Math. Ann. 68A910), 208-219.
Guetzlaff, С
[1] Aequatio modularis pro transformatione functionum ellipticarum septimi ordi-
nis, J. Reine Angew. Math. 12A834), 173-177.
Gustafson, R.
[1] The Macdonald identities for affine root systems of classical type and hypergeo-
metric series very-well poised on semisimple Lie algebras, in Ramanujan Inter-
International Symposium on Analysis, N. K. Thakare, ed., Macmillan India, Madras,
1989, pp. 185-224.
Hahn, W.
[1] Beitrage zur Theorie der Heineschen Reihen, Math. Nachr. 2A949), 340-
379.
Hall, H. S. and Knight, S. R.
[1] Higher Algebra, Macmillan, London, 1957.
Halphen, M.
[1] Sur une formule recurrents concernant les sommes des diviseurs des nombres
entiers, Bull. Soc. Math. France 5A877), 158-160.
Hancock, H.
[1] Elliptic Integrals, Wiley, New York, 1917.

References 497
Hanna, M.
[1] The modular equations, Proc. London Math. Soc. B) 28A928), 46-52.
Hansen, E. R.
[1] A Table of Series and Products, Prentice-Hall, Englewood Cliffs, NJ, 1975.
Hardy, G. H.
[1] Proof of a formula of Mr. Ramanujan, Mess. Math. 44A915), 18-21.
[2] Collected Papers, vol. 5, Clarendon Press, Oxford, 1972.
[3] Ramanujan, 3rd ed., Chelsea, New York, 1978.
Hardy, G. H. and Wright, E. M.
[1] An Introduction to the Theory of Numbers, 4th ed., Clarendon Press, Oxford,
1960.
Hecke, E.
[1] Ober einen neuen Zusammenhang zwischen elliptischen Modulfunktionen und
indefiniten quadratischen Formen, Nachr. K. Gesell. Wiss. Gottingen. Math.-
Phys. Kl. A925), 35-44.
[2] Mathematische Werke, Vandenhoeck & Ruprecht, Gottingen, 1959.
Heine, E.
[1] Untersuchungen iiber die Reihe
(l-g«)(l-g») (i-g')(i-g«+*)(i_g»)(i-gJ+i)
+
- q)(\ - q*) + (I - q)(l - q2)(l - q*)(l - q?+i) *+ '
J. Reine Angew. Math. 34A847), 285-328.
Hirschhorn, M. D.
[1] Partitions and Ramanujan's continued fraction, Duke Math. J. 39A972), 789-
791.
[2] A continued fraction, Duke Math. J. 41A974), 27-33.
[3] A continued fraction of Ramanujan, J. Austral. Math. Soc. (Ser. A) 29A980),
80-86.
[4] Additional comment on Problem 84-10* by D. H. Fowler, Math. Intell. 8, No. 3,
A986), 41-42.
[5] A generalisation of the quintuple product identity, J. Austral. Math. Soc. (Ser.
A.) 44A988), 42-45.
[6] Ramanujan's contributions to continued fractions, to appear.
Hoppe, R.
[1] Allgemeinste Auflosung der Gleichung x3 + y3 = z2 in relativen Primzahlen,
Z. Math. Phys. 4A859), 304-305.
Hovstad, R. M.
[1] A remark on continued fractions and special functions, preprint.
Ismail, M. E. H.
[1] A simple proof of Ramanujan's 1i/'1 sum, Proc. Amer. Math. Soc. 63A977),
185-186.
Ivory, J.
[1] A new series for the rectification of the ellipsis; together with some observations
on the evolution of the formula (a2 + b2 — lab cos ф)", Trans. R. Soc. Edinburgh
4A796), 177-190.
Jackson, F. H.
[1] Summation of q-hypergeometric series, Mess. Math. 50A921), 101-112.
Jackson, M.
[1] On Lerch's transcendent and the basic bilateral hypergeometric series 2X^2<
J. London Math. Soc. 25A950), 189-196.

498 References
Jacobi, С. G. J.
[1] Fundamenta Nova Theoriae Functionum EUipticarum, Sumptibus Fratrum
Borntrager, Regiomonti, 1829.
[2] Gesammelte Werke, Erster Band, G. Reimer, Berlin, 1881.
Jacobsen, L.
[1] Domains of validity for some of Ramanujan's continued fraction formulas,
J. Math. Anal. Appl. 143A989), 412-437.
Jacobsen, L. and Waadeland, H.
[1] Glimt fra analytisk teori for kjedebreker, Del II, Nordisk Mat. Tidskr. 33A985),
168-175.
[2] Computation of the circumference of an ellipse, to appear.
Joyce, G. S.
[1] On the hard-hexagon model and the theory of modular functions, Philos. Trans.
R. Soc. London 325A988), 643-706.
Kac, V. G. and Peterson, D. H.
[1] Affine Lie algebras and Hecke modular forms, Bull. Amer. Math. Soc. (N.S.)
3A980), 1057-1061.
Kepler, J.
[1] Opera Omnia, vol. 3, Heyder & Zimmer, Frankfurt, 1860.
Kiper, A.
[1] Fourier series coefficients for powers of the Jacobian elliptic functions, Math.
Сотр. 43A984), 1-13.
Kleiber, J.
[1] On a class of functions derivable from the complete elliptic integrals, and
connected with Legendre's functions, Mess. Math. 22A893), 1-44.
Klein, F.
[1] Ueber die Transformation der elliptischen Functionen und die Auflosung der
Gleichungen fiinften Grades, Math. Ann. 14A879), 111-172.
[2] Zur Theorie der elliptischen Modulfunctionen, Math. Ann. 17A880), 62-70.
[3] Vorlesungen uber die Theorie der elliptischen Modulfunctionen, zweiter Band,
B. G. Teubner, Leipzig, 1892.
Knopp, M. I.
[1] Modular Functions in Analytic Number Theory, Markham, Chicago, 1970.
Koblitz, N. I.
[1] q-Extension of the p-adic gamma function, Trans. Amer. Math. Soc. 260A980),
449-457.
Kohler, G.
[1] Theta series on the Hecke groups G(y/l) and G(y/b), Math. Z. 197A988), 69-96.
[2] Theta series on the theta group, Abh. Math. Sem. Univ. Hamburg 58A988),
15-45.
Kondo, T. and Tasaka, T.
[1] The theta functions of sublattices of the Leech lattice, Nagoya Math. J. 101A986),
151-179.
[2] The theta functions of sublattices of the Leech lattice II, J. Fac. Sci. Univ.
Tokyo Sec. IA 34A987), 545-572.
Koornwinder, Т. Н.
[1] Jacobi functions as limit cases of q-ultraspherical polynomials, J. Math. Anal.
Appl. 148A990), 44-54.
Lamphere, R. L.
[1] Elementary proof of a formula of Ramanujan, Proc. Amer. Math. Soc. 91A984),
416-420.

References 499
Landen, J.
[1] A disquisition concerning certain fluents, which are assignable by the arcs of
the conic sections; wherein are investigated some new and useful theorems for
computing such fluents, Philos. Trans. R. Soc. London 61A771), 298-309.
[2] An investigation of a general theorem for finding the length of any arc of any
conic hyperbola, by means of two elliptic arcs, with some other new and useful
theorems deduced therefrom, Philos. Trans. R. Soc. London 65A775), 283-289.
Langebartel, R. G.
[1] Fourier expansions of rational fractions of elliptic integrals and Jacobian elliptic
functions, SIAM J. Math. Anal. 11A980), 506-513.
Legendre, A. M.
[1] Memoire sur les integrations par d'arcs d'ellipse, Mem. Acad. Sci. Paris, 1786.
[2] Traite des Fonctions Elliptiques, t. 1, Huzard-Courcier, Paris, 1825.
Lepowsky, J. and Wilson, R. L.
[1] The Rogers-Ramanujan identities: Lie theoretic interpretation and proof, Proc.
Natl. Acad. Sci. U.S.A. 78A981), 701.
Ling, C.-B.
[1] On summation of series of hyperbolic functions, SI AM J. Math. Anal. 5A974),
551-562.
[2] On summation of series of hyperbolic functions. II, SIAM J. Math. Anal.
6A975), 129-139.
[3] Generalization of certain summations due to Ramanujan, SIAM J. Math. Anal.
9A978), 34-48.
Littlewood, D. E.
[1] A University Algebra, William Heinemann LTD, Melbourne, 1950.
Macdonald, I.
[1] Affine root systems and Dedekind's //-function, Invent. Math. 15A972), 91-143.
Maclaurin, С
[1] A Treatise of Fluxions in Two Books, vol. 2, T. W. and T. Ruddimans, Edinburgh,
1742.
MacMahon, P. A.
[1] Combinatory Analysis, vol. 2, University Press, Cambridge, 1915.
Mermin, N. D.
[1] Pi in the sky, Am. J. Phys. 55A987), 584-585.
Milne, S. C.
[1] An elementary proof of the Macdonald identities for А\1), Adv. Math. 57A985),
34-70.
[2] A U{n) generalization of Ramanujan's 1ф1 summation, J. Math. Anal. Appl.
118A986), 263-277.
[3] Multiple q-series and U(n) generalizations of Ramanujan's j ф1 sum, in Ramanu-
Ramanujan Revisited, Academic Press, Boston, 1988, pp. 473-524.
[4] The multidimensional 1ф1 sum and Macdonald identities for Al/\ in Theta
Functions Bowdoin 1987, Part 2, Proceedings of Symposia in Pure Mathe-
Mathematics, vol. 49, American Mathematical Society, Providence, 1989, pp. 323-359.
[5] Classical partition functions and the U(n + 1) Rogers-Selberg identity, Discrete
Math., to appear.
Mimachi, K.
[1] A proof of Ramanujan's identity by use of loop integral, SIAM J. Math. Anal.
19A988), 1490-1493.
Moak, D. S.
[1] The q-analogue of the Laguerre polynomials, J. Math. Anal. Appl. 81A981),
20-47.

500 References
[2] The g-analogue of Stirling's formula, Rocky Mountain J. Math. 14A984),
403-413.
Mordell, L. J.
[1] Note on certain modular relations considered by Messrs. Ramanujan, Darling
and Rogers, Proc. London Math. Soc. B) 20A922), 408-416.
[2] An identity in combinatorial analysis, Proc. Glasgow Math. Assoc. 5A962),
197-200.
Moreau, C.
[1] Plus petit nombre egal a la somme de deux cubes de deux facons, LIntermediate
Math. 5A898), 66.
Muir, T.
[1] On the perimeter of an ellipse, Mess. Math. 12A883), 149-151.
Nyvoll, M.
[1] Tilnaermelsesformler for ellipsebuer, Nordisk Mat. Tidskr. 25-26A978), 70-72.
Odlyzko, A. M. and Wilf, H. S.
[1] The editor's corner: n coins in a fountain, Amer. Math. Monthly 95A988),
840-843.
Paule, P.
[1] On identities of the Rogers-Ramanujan type, J. Math. Anal. Appl. 107A985),
255-284.
Peano, G.
[1] Sur une formule d'approximation pour la rectification de l'ellipse, C. R. Acad.
Sci. Paris 108A889), 960-961.
Perron, O.
[1] Die Lehre von den Kettenbriichen, Band 2, dritte Auflage, B. G. Teubner,
Stuttgart, 1957.
Petersson, H.
[1] Automorphic Forms and Discrete Groups, Unpublished lecture notes, Uni-
University of Wisconsin, Madison, 1974.
Preece, С. Т.
[1] Theorems stated by Ramanujan (VI): theorems on continued fractions, J. Lon-
London Math. Soc. 4A929), 34-39.
[2] Theorems stated by Ramanujan (XIII), J. London Math. Soc. 6A931), 95-99.
Privman, V. and Svrakic, N. M.
[1] Difference equations in statistical mechanics. I. Cluster statistics models, J. Stat.
Phys. 51A988), 1091-1110.
Rademacher, H.
[1] Topics in Analytic Number Theory, Springer-Verlag, New York, 1973.
Raghavan, S.
[1] On certain identities due to Ramanujan, Q. J. Math. (Oxford) B) 37A986),
221-229.
[2] Euler products, modular identities and elliptic integrals in Ramanujan's manu-
manuscripts, in Ramanujan Revisited, Academic Press, Boston, 1988, pp. 335-345.
Raghavan, S. and Rangachari, S. S.
[1] On Ramanujan's elliptic integrals and modular identities, in Number Theory
and Related Topics, Oxford University Press, Bombay, 1989, pp. 119-149.
Rahman, M.
[1] A simple evaluation of Askey and Wilson's q-beta integral, Proc. Amer. Math.
Soc. 92A984), 413-417.
Ramamani, V.
[1] Some Identities Conjectured by Srinivasa Ramanujan Found in His Lithographed

References 501
Notes Connected with Partition Theory and Elliptic Modular Functions—Their
Proofs—Inter Connection with Various Other Topics in the Theory of Numbers
and Some Generalizations, Doctoral Thesis, University of Mysore, 1970.
Ramamani, V. and Venkatachaliengar, K.
[1] On a partition theorem of Sylvester, Michigan Math. J. 19A972), 137-140.
Ramanathan, K. G.
[1] On Ramanujan's continued fraction, Acta Arith. 43A984), 209-226.
[2] On the Rogers-Ramanujan continued fraction, Proc. Indian Acad. Sci. (Math
Sci.) 93A984), 67-77.
[3] Remarks on some series considered by Ramanujan, J. Indian Math. Soc. 46A982),
107-136 (published in 1985).
[4] Ramanujan's continued fraction, Indian J. PureAppl. Math. 16A985), 695-724.
[5] Some applications of Kronecker's limit formula, J. Indian Math. Soc. 52A987),
71-89.
[6] Hypergeometric series and continued fractions, Proc. Indian Acad. Sci. (Math.
Sci.) 97A987), 277-296.
[7] Ramanujan's notebooks, J. Indian Inst. Sci., Special Issue A987), 25-32.
[8] Generalisations of some theorems of Ramanujan, J. Number Theory 29A988),
118-137.
[9] On some theorems stated by Ramanujan, in Number Theory and Related Topics,
Oxford University Press, Bombay, 1989, pp. 151-160.
[10] Ramanujan's modular equations, Acta Arith. 53A990), 403-420.
Ramanujan, S.
[1] Squaring the circle, J. Indian Math. Soc. 5A913), 132.
[2] Modular equations and approximations to n, Q. J. Math. (Oxford) 45A914),
350-372.
[3] Question 584, J. Indian Math. Soc. 6A914), 199.
[4] Some definite integrals, Mess. Math. 44A915), 75-85.
[5] Question 662, J. Indian Math. Soc. 7A915), 119-120.
[6] On certain arithmetical functions, Trans. Cambridge Philos. Soc. 22A916),
159-184.
[7] Question 755, J. Indian Math. Soc. 8A916), 80.
[8] Proof of certain identities in combinatory analysis, Proc. Cambridge Philos.
Soc. 19A919), 214-216.
[9] Notebooks B volumes), Tata Institute of Fundamental Research, Bombay, 1957.
[10] Collected Papers, Chelsea, New York, 1962.
[11] The Lost Notebook and Other Unpublished Papers, Narosa, New Delhi, 1988.
Ramanujan Centenary Prize Competition
[1] Mathematics Today 5A987), Nos. 2-3, 21-24; No. 4,45-48; No. 5,42-46,
Rankin, R. A.
[1] George Neville Watson, J. London Math. Soc. 41A966), 551-565.
[2] Modular Forms and Functions, Cambridge University Press, Cambridge, 1977.
Rao, M. B. and Ayyar, M. V.
[1] On some infinite products and series. II, J. Indian Math. Soc. 15A923-24),
233-247.
Riesel, H.
[1] Some series related to infinite series given by Ramanujan, BIT 13A973), 97-113.
Rogers, L. J.
[1] Second memoir on the expansion of certain infinite products, Proc. London
Math. Soc. 25A894), 318-343.
[2] Third memoir on the expansion of certain infinite products, Proc. London Math.
Soc. 26A895), 15-32.

502 References
[3] On the representation of certain asymptotic series as convergent continued
fractions, Proc. London Math. Soc. B) 4A907), 72-89.
[4] On two theorems of combinatory analysis and some allied identities, Proc.
London Math. Soc. B) 16A917), 315-336.
[5] Proof of certain identities in combinatory analysis, Proc. Cambridge Philos.
Soc. 19A919), 211-214.
[6] On a type of modular relation, Proc. London Math. Soc. B) 19A920), 387-397.
Rothe, H. A.
[1] Systematisches Lehrbuch der Arithmetik, Barth, Leipzig, 1811.
Russell, R.
[1] On кХ - к'Х modular equations, Proc. London Math. Soc. 19A887), 90-111.
[2] On modular equations, Proc. London Math. Soc. 21A890), 351-395.
Schlafli, L.
[1] Beweis der Hermiteschen Verwandlungstafeln fur die elliptischen Modularfunc-
tionen, J. Reine Angew. Math. 72A870), 360-369.
Schoeneberg, B.
[1] Elliptic Modular Functions, Springer-Verlag, New York, 1974.
Schoissengeier, J.
[1] Der numerische Wert gewisser Reihen, Manuscripta Math. 38A982), 257-263.
Schroter, H.
[1] De aequationibus modularibus, Dissertatio Inauguralis, Albertina Litterarum
Universitate, 1854, Regiomonti.
[2] Extrait d'une lettre adressee A. M. Liouville, J. Math. B) 3A858), 258-264.
[3] Ueber Modulargleichungen der elliptischen Functionen, Auszug aus einem
Schreiben an Herrn L. Kronecker, J. Reine Angew. Math. 58A861), 378-379.
[4] Beitrage zur Theorie der elliptischen Funktionen, Ada Math. 5A894), 205-208.
Schur, I. J.
[1] Ein Beitrag zur additiven Zahlentheorie und zur Theorie der Kettenbriiche,
S.-B. Preuss. Akad. Wiss. Phys.-Math. Kl. A917), 302-321.
[2] Zur additiven Zahlentheorie, S.-B. Preuss. Akad. Wiss. Phys.-Math. Kl. A926),
488-495.
[3] Gesammelte Abhandlungen C volumes), Springer-Verlag, Berlin, 1973.
Schwarz, H. A.
[1] Formeln und Lehrsatze zum Gebrauche der Elliptischen Funktionen. Nach Vorle-
sungen und Aufzeichnungen des Herrn Prof. K. Weierstrass, Zweite Ausgabe,
Erste Abteilung, Springer, Berlin, 1893.
Sears, D. B.
[1] Two identities of Bailey, J. London Math. Soc. 27A952), 510-511.
Selberg, A.
[1] Ober einige arithmetische Identitaten, Avh. Norske Vid.-Akad. Oslo I. Mat.-
Naturv. KL, No. 8, A936), 3-23.
[2] Collected Papers, vol. 1, Springer-Verlag, Berlin, 1989.
Selmer, E. S.
[1] Bemerkninger til en ellipse-beregning av en ellipses omkrets, Nordisk Mat.
Tidskr. 23A975), 55-58.
Singh, S. N.
[1] Basic hypergeometric series and continued fractions, Math. Student 56A988),
91-96.
Slater, L. J.
[1] Further identities of the Rogers-Ramanujan type, Proc. London Math. Soc. B)
54A952), 147-167.

References 503
Sohncke, L. A.
[1] Aequationes modulares pro transformatione functionum ellipticarum et un-
decimi et decimi tertii et decimi septimi ordinis, J. Reine Angew. Math. 12A834),
178.
[2] Aequationes modulares pro transformatione functionum ellipticarum, J. Reine
Angew. Math. 16A837), 97-130.
Srivastava, H. M.
[1] A note on a generalization of a q-series transformation of Ramanujan, Proc.
Japan Acad. Ser. A 63A987), 143-145.
Stanton, D.
[1] An elementary approach to the Macdonald identities, in q-Series and Parti-
Partitions, D. Stanton, ed., IMA volume in Mathematics and Its Applications, vol. 18,
Springer-Verlag, New York, 1989, pp. 139-149.
Stark, H. M.
[1] L-functions at s = 1. IV. First derivatives at s = 0, Adv. Math. 35A980), 197-
235.
Stieltjes, T. J.
[1] Sur la reduction en fraction continue d'une serie procedant suivant les puis-
puissances descendantes d'une variable, Ann. Fac. Sci. Toulouse 3A889), 1-17.
[2] Oeuvres Completes, t. 2, P. Noordhoff, Groningen, 1918.
Stubban, J. O.
[1] Fergestads formel for tilnaermet beregning av en ellipses omkrets, Nordisk Mat.
Tidskr. 23A975), 51-54.
Subbarao, M. V. and Vidyasagar, M.
[1] On Watson's quintuple product identity, Proc. Amer. Math. Soc. 26A970),
23-27.
Szekeres, G.
[1] A combinatorial interpretation of Ramanujan's continued fraction, Can. Math.
Bull. 11A968), 405-408.
Tannery, J. and Molk, J.
[1] Elements de la Theorie des Fonctions Elliptiques, t. 2, Chelsea, New York, 1972.
Thiruvenkatachar, V. R. and Venkatachaliengar, K.
[1] Ramanuian at Elementary Levels; Glimpses, to appear.
Venkatachaliengar, K.
[1] Development of Elliptic Functions According to Ramanujan, Tech. Rep. 2,
Madurai Kamaraj University, Madurai, 1988.
Verma, A.
[1] On identities of Rogers-Ramanujan type, Indian J. Pure Appl. Math. 11A980),
770-790.
[2] Basic hypergeometric series and identities of Rogers-Ramanujan type, in
Srinivasa Ramanujan A887-1920), Macmillan India Ltd., Madras, 1988,
pp. 66-74.
Verma, A., Denis, R. Y., and Rao, K. S.
[1] New continued fractions involving basic hypergeometric 3Ф2 functions, J. Math.
Phys. Sci. 21A987), 585-592.
Verma, A. and Jain, V. K.
[1] Transformations between basic hypergeometric series on different bases and
identities of Rogers-Ramanujan type, J. Math. Anal. Appl. 76A980), 230-269.
[2] Transformations of non-terminating basic hypergeometric series, their contour
integrals and applications to Rogers-Ramanujan identities, J. Math. Anal.
Appl. 87A982), 9-44.

504 References
Viete, F.
[1] Opera Mathematica, Georg Olms Verlag, Hildesheim, 1970.
Villarino, M. B.
[1] Ramanujan's approximation to the arc length of an ellipse, preprint.
Wall, H. S.
[1] Analytic Theory of Continued Fractions, van Nostrand, New York, 1948.
Watson, G. N.
[1] Theorems stated by Ramanujan (II): theorems on summation of series,
J. London Math. Soc. 3A928), 216-225.
[2] A new proof of the Rogers-Ramanujan identities, J. London Math. Soc. 4A929),
4-9.
[3] Theorems stated by Ramanujan (VII): theorems on continued fractions,
J. London Math. Soc. 4A929), 39-48.
[4] Theorems stated by Ramanujan (IX): two continued fractions, J. London Math.
Soc. 4A929), 231-237.
[5] Ramanujan's note books, J. London Math. Soc. 6A931), 137-153.
[6] Ramanujan's continued fraction, Proc. Cambridge Philos. Soc. 31A935), 7-17.
Weber, H.
[1] Zur Theorie der elliptischen Functionen, Acta Math. 11A887-88), 333-390.
[2] Elliptische Functionen und Algebraische Zahlen, Friedrich Vieweg und Sohn,
Braunschweig, 1891.
[3] Lehrbuch der Algebra, dritter Band, Friedrich Vieweg und Sohn, Braunschweig,
1908.
Whittaker, E. T. and Watson, G. N.
[1] A Course of Modern Analysis, 4 ed., University Press, Cambridge, 1966.
Wolfram, J.
[1] Tables of Natural Logarithms, included in G. Vega, Ten Place Logarithms,
reprint of 1794 edition, Hafner, New York, 1958.
Woyciechowsky, J.
[1] Sipos Pal egy kezirata es a kochleoid, Mat. Fiz. Lapok 41A934), 45-54.
Zhang, L.-C.
[1] «^-Difference equations and Ramanujan-Selberg continued fractions, Acta
Arith. 57A991), to appear.
Zucker, I. J.
[1] The summation of series of hyperbolic functions, SI AM J. Math. Anal. 10A979),
192-206.
[2] Some infinite series of exponential and hyperbolic functions, SI AM J. Math.
Anal. 15A984), 406-413.
[3] Further relations amongst infinite series and products. II. The evaluation of
3-dimensional lattice sums, J. Phys. A: Math. Gen. 23A990), 117-132.

Index
Adiga, С. 7,10,18, 24, 28, 32, 39, 49,
73,79,86,115,142
Agarwal, A. K. 78
Alder, H. L. 78
Almkvist, G. 10,147,149-150,456
Al-Salam, W. A. 79
Andrews, G. E. 10, 13-15,18, 28-29,
32, 36-37, 77-80, 83, 347, 398
Askey, R. 2,10,14, 29, 32, 77-79, 149
Atkin, A. O. L. 83
Ayyar, M. V. 140
Bailey, D. H. 6
Bailey, W. N. 15-17, 83, 89,111,120,
181, 262
base (of an elliptic function) 5,102
basic hypergeometric series 12
Baxter, R. J. 78
Berndt, В. С 2, 5, 7, 24, 29,44, 79,
111-112,140-141, 147,150,172,
197, 326,456
Bernoulli numbers 42, 61-64, 97
Berry, A. 440
Bhagirathi, N. A. 28
Bhargava, S. 7, 10,18, 24, 28, 32, 39,
49,73,79,115,142
Biagioli, A. J. 7, 10, 326, 346
Blecksmith, R. 73, 83
Borwein, J. M. 6, 78, 269, 305, 346,
355,456
Borwein, P. B. 6, 78, 269, 305, 346, 355,
456
Bressoud, D. 78, 398
Brillhart, J. 10, 73, 83
Brouncker, Lord 200
Burnside, W. 240
Byrd, P. F. 113
Carlitz, L. 18, 79, 83
Catalan's constant 154-155
Cauchy, A. 14, 140
Cayley, A. 2-3, 5,106-107,135, 138,
218, 220, 232, 241
change of sign 126,130,132,178
Chudnovsky, D. V. 168
Chudnovsky, G. V. 168
Churchhouse, R. F. 79
Clausen transformation 114
Cohn, H. 6
column-row method of
summation 114
complementary modulus 4,102
complete series 42,455
consistency condition 327
continued fraction, geometric and
arithmetic mean arguments 164

506
Index
continued fractions 19-29,92, 146,
151, 163-168,185-187, 206-208,
221-222, 345-347
cosine identity 345, 347-348
Court, N. A. 245
Coxeter, H. S. M. 245
cusp 328
cusp parameter 328
Darling, H. В. С 261
Dedekind eta-function 37,44, 330-
338
degree of a modular equation 4
degree of a modulus 229
degree of series 42
Denis, R. Y. 28, 78-79
Deutsch, J. 6
Dickson, L. E. 197, 200
Digby, K. 200
dimidiation 126,178
diophantine equations 197-200
divisor functions 62, 64-65
duplication 125,127-128,178
Dyson, F. J. 10, 83
eccentricity of an ellipse 145
Ehrenpreis, L. 78
Eisenstein, G. 28
Eisenstein series 7, 65,121-122,
126-139, 144-145,175-177,454-
488
Eisenstein series, values in terms of
elliptic function parameters 126-
129
ellipse, approximations to the perimeter
of 145-150,180-189
elliptic curve 6
elliptic functions 2-3
elliptic functions, notation 101-102
elliptic integral of the first kind 4,102
elliptic integral of the second kind
176-177,303-304
elliptic integrals 104-113, 238-243,
297-298
elliptic integrals, addition theorem for
elliptic integrals of the first kind
106-108
elliptic integrals, addition theorem for
elliptic integrals of the second kind
303
elliptic integrals, duplication formula
106
Enneper, A. 5, 72,220
Euler, L. 14, 37,147,150, 196-197,
199
Euler numbers 61, 63
Euler's diophantine equation 197-199
Euler's partition theorem 37
Euler's pentagonal number theorem
36-37
Evans, R. J. 7,10, 83, 274, 276, 337,
352, 373, 375
F(x) 91
Fergestad, J. B. 146
Fermat, P. 200
Fiedler, E. 5, 315, 364,416,444
Fine, N. J. 32
fixed point of a modular form 328
Flajolet, P. 80,168
Forrester, P. J. 78
Forsyth, A. R. 242
Francon, J. 168
Frenicle 200
Fricke involution 216,404
Fricke, R. 5, 83, 364,416
Friedman, M. D. 113
Frobenius, G. 1
fundamental set 328
Garsia, A. M. 78
Gauss, С F. 14,28, 36, 89,147,151,
181
Gauss' transformation 113
geometrical problems 190-196, 211-
213, 243-249, 298-302
Gerst, I. 73, 83
Ghosh, S. 488
Glaisher, J. W. L. 169, 242, 303
Glasser,M. L. 80,110,113
Gordon, B. 78-79, 83, 347
Gosper, R. W. 13
Gray,J.J. 28
Greenhill, A. G. 2,400,439

Index
507
Guetzlaff, С. 5, 315
Gustafson, R. 32
Hahn, W. 32
Halphen, M. 62
Hancock, H. 212
Hanna, M. 5, 440,444
hard hexagon model 78
Hardy, G. H. 2, 6, 9,11, 29-32, 36, 39,
45, 77, 79, 84, 86,126,162-164,197,
199, 262, 326, 346, 385,426,450
Hecke, E. 398
Hecke operator 373
Heine, E. 11,14-15,18,21
Heine's continued fraction 21
Hermite, С 135
Hirschhorn, M. 11, 28, 31, 79, 83, 347
Hoppe, R. 196
Hovstad,R. M. 79
Hurwitz, A. 5, 444
hyperbola, perimeter of 180
hyperbolic function series evaluations in
closed form 140-141,157-162
hyperbolic function series evaluations
in terms of elliptic function
parameters 132-139,153-157,
172-178
hyperbolic function series identity 162
hypergeometric differential equation
120-121
hypergeometric functions 3, 5, 88-104,
120-122,144-150,153-155,164,
185-186,188, 213, 238-239, 289-
290,455-456
invariant order of a modular form 328
inversion formula for base q 100
Ismail, M. E. H. 32,42, 79
Ivory, J. 146-147
Jackson, F. H. 14-15
Jackson, M. 32
Jacobi, С G. J. 3, 5, 11,14, 36, 39, 54,
87, 115-116,123,126, 135,143,
165-166,169,173,176-177, 207,
218, 220, 232, 234, 239-241
Jacobi triple product identity 11-12,
32, 35-36
Jacobian elliptic functions 3, 54, 87,
107-108,135-136,138-139,143,
162-163,165-180,207-208, 227,
242, 304
Jacobian elliptic functions, conversions
of old formulas into new formulas
173-174
Jacobi's identity 39
Jacobi's imaginary transformation 106,
154
Jacobsen, L. 10, 20, 22-24, 26-27, 79,
84,146
Jain,V. K. 78
Journal of the Indian Mathematical
Society 9,11,77,190,246
Joyce, G. S. 6
Kac, V. G. 398
Kepler, J. 147,150
Kiper, A. 169
Kleiber, J. 89
Klein, F. 5,315,377,444
Knopp, M. I. 42,44, 327, 330
Koblitz,N. I. 29
Kohler, G. 6
Kondo, T. 6, 72, 366
Koornwinder, Т. Н. 13
Kumbakonam 2
Lamphere, R. 10, 29, 79
Landen, J. 5, 146,181
Landen's transformation 113,126,
146-147, 213
Langebartel, R. 169
lattice gases 6
Legendre, A. M. 5,107,181,220,232,
234,244
Legendre functions 89
Legendre-Jacobi symbol 329
Legendre's relation 455-456
Lepowsky, J. 78
Lie algebras 78
Ling, C.-B. 140,142
Littlewood, D. E. 311
Littlewood, J. E. 2

508
Index
Macdonald, I. 32
Macdonald identities 32
Maclaurin, C. 146
MacMahon, P. A. 113
MACSYMA 10, 312, 369, 372, 377,
400, 408, 416-417,425,430,488
Mathematica 10
medial section 298
Mehler-Dirichlet integral 89
Mermin, N. D. 151
Metius, A. 194
Milne, S. 32,78
Mimachi, K. 32
Mittag-Leffler theorem 144
mixed modular equation, defintion 325
mixed modular equations, table of
degrees 325-326
Moak, D. S. 29
modular equation, definition 213
modular equations 3-8
modular equations of degree 2 214
modular equations of degree 3 230-
238, 352-353, 356
modular equations of degree 4 214-
215
modular equations of degree 5 280-
288
modular equations of degree 7 314-
324,435-437
modular equations of degree 8 216-
217
modular equations of degree 9 352—
358
modular equations of degree 11 363-
372
modular equations of degree 13 376-
377
modular equations of degree 15 383-
397,435-439
modular equations of degree 16 216
modular equations of degree 17 397-
400
modular equations of degree 19 416-
417
modular equations of degree 21 400-
408
modular equations of degree 23 411-
416
modular equations of degree 25 290-
297
modular equations of degree 27 360-
362
modular equations of degree 31 439-
444
modular equations of degree 33 408-
411
modular equations of degree 35 423-
426,430
modular equations of degree 39 426-
430,435-439
modular equations of degree 47 444-
449
modular equations of degree 55 426-
430,435-439
modular equations of degree 63 426-
435,435-439
modular equations of degree 71 444-
449
modular equations of degree 87 449-
453
modular equations of degree 95 430-
435
modular equations of degree 119 430-
435
modular equations of degree 135 430-
435,449-453
modular equations of degree 143 430-
435
modular equations of degree 175 449-
453
modular equations of degree 207 449-
453
modular equations of degree 231 449-
453
modular equations of degree 247 449-
453
modular equations of degree 255 449-
453
modular equations, table 8, 325-326
modular form, definition 328
modular forms 7, 326-345, 366-376,
399-408,415-417,423-425,430,
484-488
modular group 327
modulus 4-5,102
Molk, J. 6, 45, 72

Index
509
Mordell,L-J. 2,83,261
Moreau, С 200
Muir, T. 147,150
Miiller, R. 10
multiplier 5, 214, 230
multiplier system 7, 328-329
multiplier system of Dedekind
eta-function 330
multiplier systems for theta-functions
330-332
National Science Foundation 10
notation 10, 12, 88-89, 230-231, 326-
329
Nyvoll, M. 147
Odlyzko, A. M. 79
order of a modular form 328
orders of theta-functions at rational
cusps 333
partial fraction expansions 200-206
partition function 262
Paule, P. 78
Peano, G. 147, 150
pendulum 212-213, 243-244, 246,
299-301
perfect series 42
Perron, O. 166-167, 186, 208
Peterson, D. H. 398
Petersson, H. 326
Pfaff's transformation 17
pi, approximations to 151 -152,194-
196
Playfair, J. 146
Preece, С. Т. 163-164
Privman, V. 80
Proceedings of the London
Mathematical Society 77
psi function 88,90
pure series 42
Purtilo, J. M. 7, 10, 326
<j-analogue of DougalFs theorem 15
^-analogue of Gauss' theorem 14
<jr-beta integral 11,29
<j-binomial theorem 14, 32
<j-gamma function 13
g-series 11-12,14-19,21-34
quintic algorithm for calculating pi
269
quintuple product identity 11,32,
56-57, 59, 80-83, 338
Rademacher, H. 218, 273, 330
Raghavan, S. 7, 113, 262-263, 324
Rahman, M. 29
Rama Murthy, С 10
Ramamani, V. 18,31,54
Ramanathan, K. G. 4, 7,10, 20, 28, 79,
82, 84, 86, 221-222, 262, 265, 274,
276, 324, 347
Ramanujan Centenary Prize
Competition 246
Ramanujan's ^i summation 11, 31-
34
Ramanujan's quarterly reports 29
Ramanujan's theta-function 18
Rangachari, S. S. 7,113, 263, 324
Rankin, R. A. 326, 328-329, 332-333,
342, 370, 373,404, 484
Rao, K. S. 79
Rao, M. B. 140
reciprocal of a modular equation 216
reciprocal relation 334
Riesel, H. 140
Rogers, L. J. 14,18, 30, 77-79,144,
163,166-168,207,398
Rogers-Ramanujan continued fraction
11,30-31,79-80,267
Rogers-Ramanujan continued fraction,
combinatorial interpretation 79-
80
Rogers-Ramanujan continued fraction,
finite form 31
Rogers-Ramanujan identities 11, 77-
79
Rothe, H. A. 14
row-column method of summation
114
Roy, R. 29

510
Index
Russell, R. 315,364,377,400,416,435,
439,444
Schlafli, L. 5, 315, 364, 377,400,416
Schoeneberg, B. 328, 342, 368,484
Schoissengeier, J. 142
Schroter, H. 5-7, 66, 72, 315, 364,411,
439-440
Schroter's formulas 6-7, 11,66-74
Schur, I. J. 77
Schwarz, H. A. 83
Scoville, R. 79
Sears, D. B. 83
Selberg, A. 79, 347
Selmer, E. S. 146-147,150
septic algorithm for calculating pi 305
Singh, S. N. 79
Sipos, P. 147, 150
Slater, L. J. 78
Sohncke, L. A. 5, 315, 364, 377,400,416
Somashekara, D. D. 39, 73, 79
squaring the circle 193-195
Srivastava, H. M. 18
Stanton, D. 32
Stark, H. M. 337, 339
Stieltjes, T. J. 144,163,166-168,207
Stolarsky, K. 10
stroke operator 326
Stubban, J. O. 146
Subbarao, M. V. 83
summation by rows or columns 113-
114
Svrakic, N. M. 80
Swinnerton-Dyer, P. 83
Szekeres, G. 79
Tannery, J. 6,45, 72
Tartaglia 354
Tasaka, T. 6, 72, 366
Tata Institute 10
taxi cab story 199
theta-functions 3-7,11 -12, 34, 98-
104,114-125,139-141, 218-219,
221-238, 249-297, 302-324, 330-
334, 337-488
theta-functions, basic identities 39-41,
43-52
theta-functions, logarithms of 38
theta-functions, values 103-104, 210
theta-functions, values in terms of
elliptic function parameters 122-
125
theta-function transformation formulas
36,43-44, 102, 208-209
Thiruvenkatachar, V. R. 8, 32, 34,104
triplication formula 238-241
University of Illinois 10
University of Madras 2
University of Mysore 86
valence formula 329, 334, 336
Vaughn Foundation 10
Venkatachaliengar, K. 8,18, 32, 34,
104
Verma, A. 78-79
Vidyasagar, M. 83
Viete, F. 197
Villarino, M. 10,150,184,190
Waadeland, H. 10, 92,146
Wall, H. S. 28
Wallis,J. 200
Watson, G. N. 6,10-11,16,19, 24, 30,
77, 83-84,162,194,198,244, 346
Weber, H. 5, 385,416,425-426,440,
444
Weierstrass, K. 83
weight of a modular form 328
Wetzel, J. 10
width of a subgroup of the modular
group at a cusp 328
Wilf, H. S. 79
Wilson, J. 29
Wilson, R. L. 78
Wolfram, J. 151
Woyciechowsky, J. 147
Wright, E. M. 36, 39,197
Zeilberger, D. 78
Zhang, L.-C. 347
Zucker, I. J. 6,10,140,142, 262