!•
of the
American Mathematical Society
Number 542
Second- Order Sturm-Liouvllle
Difference Equations
and Orthogonal Polynomials
Alouf Jirari
January 1995 • Volume 113 • Number 542 (second of 4 numbers) • ISSN 0065-9266
American Mathematical Society

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{Continued in the back of this publication)

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MEMOIRS
-LIT A of the
American Mathematical Society
Number 542
Second-Order Sturm-Liouville
Difference Equations
and Orthogonal Polynomials
Alouf Jirari
°<^DED
January 1995 • Volume 113 • Number 542 (second of 4 numbers) • ISSN 0065-9266
American Mathematical Society
Providence, Rhode Island

1991 Mathematics Subject Classification.
Primary 39A10.
Library of Congress Cataloging-in-Publication Data
Jirari, Alouf, 1965-
Second-order Sturm-Liouville difference equations and orthogonal polynomials / Alouf Jirari.
p. cm. - (Memoirs of the American Mathematical Society, ISSN 0065-9266; no. 542)
Based on the author's thesis (Ph.D.: Pennsylvania State University, 1993).
Includes bibliographical references and index.
ISBN 0-8218-0359-X
1. Sturm-Liouville equation. 2. Difference equations. 3. Orthogonal polynomials. I. Title.
II. Title: 2nd-order Sturm-Liouville difference equations and orthogonal polynomials. III. Series.
QA3.A57 no. 542
[QA431]
510s-dc20
[515\.625] 94-35341
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Table of Contents
Page
List of Figures viii
Acknowledgements ix
Chapter 1. Introduction 1
1.1 The Vibrating String 1
1.2 Network Theory 2
1.3 Random Walk With Discrete Time Process 4
Chapter 2. Regular Sturm-Liouville Problem 7
2.1 Set Up 7
2.2 Preliminary Results 9
2.3 Orthogonality, Eigenfunction Expansion,
Spectral Function, and Green's Function 17
Chapter 3. Singular Sturm-Liouville Problem 28
3.1 Definition 28
3.2 CV Circles 28
3.3 Ca' Circles 35
3.4 Existence of Boundary Conditions 38
3.5 Singular Boundary Value Problems 40
3.6 Green's Function 41
3.7 Self-Adjointness 43
3.8 A-Independence of Boundary Conditions 45
3.9 Green's Formulas 50
3.10 Spectral Resolution 53
3.11 Limit-Point and Limit-Circle Tests 70
Chapter 4. Polynomial Solutions 74
V

vi CONTENTS
4.1 Formal Self-Adjointness 74
4.2 Polynomial Solutions 77
4.3 Orthogonality of Eigenfunctions 80
4.4 Eigenfunction Expansion 82
Chapter 5. Polynomial Examples 85
5.1 Classification 85
5.2 Recurrence Relations 92
5.3 Weight Functions and Self-Adjoint Forms 93
5.4 Orthogonality 96
5.5 Evaluation of the ||.||2 99
5.6 Zeros 100
Chapter 6. The Four Representative Examples 101
6.1 The Generalized Tchebyshev Polynomials 101
6.2 The Generalized Laguerre Polynomials 108
6.3 The Krawtchouk Polynomials 115
6.4 The Charlier Polynomials 118
Chapter 7. Left-Definite Spaces 124
7.1 Finite Intervals 124
7.2 Infinite Intervals 133
References 137

Abstract
It is the aim of this thesis to investigate singular second-order boundary value
problems involving the difference equation
V[p(n)A2/(n)] + q(n)y(n) = \w(n)y(n) .
Chapter 1 gives some account of problems with a rather physical character,
described by this difference equation.
Chapter 2 presents a discussion of the Sturm-Liouville boundary value
problem on an interval (a, b) where both a and b are "regular" points.
Chapter 3 derives self-adjoint difference operators when a and/or b are
"singular" points. It also takes up the abstract spectral resolution for such operators.
Chapter 4 provides necessary and sufficient conditions for a second-order
difference operator to be formally self-adjoint and have orthogonal
polynomials as eigenfunctions.
Chapter 5 shows that these sets of polynomials fall into four categories. It
also surveys their properties, which are familiar in the context of orthogonal
polynomials.
These four classes of polynomials are then illustrated in Chapter 6 by four
representative examples: the generalized Tchebyshev polynomials, the
generalized
Laguerre polynomials, the Krawtchouk polynomials, and the Charlier
polynomials.
The closing Chapter 7 is devoted to showing that these polynomials are
associated with difference operators which are self-adjoint in a left-definite setting
as well.
Vll

List of Figures
Page
1.1 Vibrating String 2
1.2 LC-Network 3
1.3 Random Walk 4
5.1 Classification Tree 88
viii

A cknowledgement s
Throughout my work I have received valuable assistance from a number of
individuals. It is a pleasure to express now my warm appreciation to each one
for their time and efforts.
First, I am truly grateful to Allan M. Krall, for his assistance. I extend my
appreciation to David M. Bressoud for his perceptive suggestions, and William
L. Harkness and Peter Morris for their helpful comments. Thanks also go to
Kathy Wyland for accurately typing the manuscript.
MACECE, the Moroccan American Commission for Education and Cultural
Exchange, deserves special mention for extending my educational experience in
the United States. This experience has been a most enjoyable one, thanks to the
direct and constant assistance of the Fulbright program staff at the Amideast
office in Washington, D.C.
ix

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Chapter 1
Introduction
This opening chapter presents Atkinson's three different interpretations of the
second-order difference equation
(1.0.1) V\p(n)Ay(n)] + q(n)y(n) = Xw(n)y(n) , n = a,..., b,
found in Ref. 2, where A is the forward difference operator (A/(n) = f(n +
1) — /(n)), and V is the backward difference operator (A/(n) = /(n) — f(n —
1)). They are different in flavor, ranging from mechanics, to network theory, to
probability theory. However, they are all confined to problems which are finite.
Moreover, they are all open to extensions, including the infinite discrete case,
which illustrate the closeness between discrete and continuous.
1.1 The Vibrating String
Consider a weightless string, stretched between two fixed points, executing
small harmonic vibrations and bearing a discrete sequence of particles with
masses ao, • • •, flm-i- Let — be the distance between an and an+i(n = 0,..., m—
2), and let un be the displacement of the particle an at time t. Suppose further
that the string extends to length beyond am-i and beyond a0) and
cm-i c_i
denote the tension of the string at an by Tr and Tt on the right and the left,
respectively (see Figure 1.1).
By Newton's second law of dynamics,
(1.1.1) a„< =Trsin0r +Ttsm0t.
Also, since the particle an does not move horizontally,
(1.1.2) TrcosOr =TicosOt = T0.
Received by the editor October 19, 1992.
1

2
ALOUF JIRARI
1/C-l 1/Cn.j 1/Cn 1/Cm-l
Figure 1.1: Vibrating String
On the other hand,
(1.1.3) tanfl/ = Cn_i(un - un_i) and tanflr = Cn(un - un+i) ,
so that, for small oscillations, (1.1.1) and (1.1.3) imply that
(1.1.4) an< = -To[cnAun - Cn-iAiin-i] = -ToV(cnAun) .
If un = Vnetwt, where yn is the amplitude of oscillation of an, then, from
(1.1.4), it follows that
-anw2yn = -T0V(cnAyn) •
If, in addition, the string is stretched to unit tension; that is, if To = 1, then
(1.1.5) V(cnA2/n) = anw2yn ,
which is of the form of (1.0.1), with p(n) = cn, q(n) = 0, w(n) = an, and A = u;2.
Remark 1.1.1
Suppose that Fetwt is applied transversely to the particle qq. The result then is
a nonhomogeneous problem described by
(1.1.6) V(c0A2/0) = a0w2y0 - F
and

STURM-LIOUVILLE DIFFERENCE EQUATIONS
(1.1.7) V(cnAj/„) = anu;2yn , n = l,...,m-l.
1.2 Network Theory
1
Consider a cascade of LC circuits with inductance a„, capacitance —, and
loop current u„ in the successive meshes (see Figure 1.2).
«-.
1At V<
<S>
• • •
V<^=
Vc,
/MM^
>>-*
• • •
V
tH
Figure 1.2: LONetwork
1 .
The current in the branch containing — is un — tin+i in the sense of un.
cn
Assume that un is of the form ynetwt, where yn is a complex constant, and, for
the nonhomogeneous problem, assume that a generator in the first mesh supplies
a voltage Eeiwt. Then, by Kirchoff's rule, it follows that
(1.2.1) E + c-iyo(iw)~l + a0y0iw + c0(y0 - y^iw)-1 = 0 ,
and, for n = 1,..., m — 1,
(1.2.2) cn-i(yn - yn-x^iw)-1 + anyniw + cn(yn - yn+x)(iw)-1 = 0.
Note that y_i = ym = 0, so that multiplication by iw in (1.2.1) and (1.2.2) gives
V(c0Ay0) = a0w2yo - E

4
ALOUF JIRARI
and
V(c„Ay„) = anw2yn, n= l,...,m- 1,
which are equivalent to (1.1.6) and (1.1.7), respectively.
1.3 Random Walk With Discrete Time Process
At time t = 0, a particle is at one of m places 0,1,..., m — 1. At successive
instants t = 1,2,..., it can move one place to the right or to the left or can
remain fixed. Suppose that the particle is in position n at some t = <o- There is
a probability an that it will be in position n -f 1 at to -f 1. There is a probability
/?n that it will be in position n— 1 at <o +1, and therefore a probability 1 — an — /3n
that it will remain in position n.
At the endpoints, the particle is considered lost if it moves to the left of 0 or
to the right of m— 1.
ft* <*0 Pn an Prn-l Qrn-1
0 n m -1
Figure 1.3: Random Walk
If a particle starting initially at r is at s when t = n -f 1, then, when t = n, it
must have been at s or s — 1 or s -f 1 with respective probabilities 1 — as — /?,,
a*-i, ft+i of moving then from these positions to s. Hence, if prs(n) represents
the probability of the particle being in position s at time n, starting in position
r at t = 0, then

STURM-LIOUVILLE DIFFERENCE EQUATIONS 5
(1.3.1) prs(n + 1) = (1 - as - 0s)prs(n) + aa_ipr>a_i(n) + 0s+ipr,s+i(n).
Setting a_i = /?m = 0 to indicate that there is no return from the right of m — 1
or the left of 0, it follows that (1.3.1) is also satisfied for s = 0 and s = m — 1.
In addition, pr*(0) = <5ra.
So, if P(n) = (pr5(ra))o<r,5<m-i,
/-<*o-/?o <*o \
1 «mr-2 I
and E = m x m unit matrix,
then (1.3.1) can be written equivalently
P(n + l) = P(n)(£ + T).
Since P(0) = E, P(n + 1) = (E + T)n.
Now, let A be an eigenvalue of T, with associated eigenvector (t/o, • • •, 2/m-i)-
Then,
At/o = «o(2/i - 2/o) - PoVo ,
Aj/n = «n(2/n + l ~ 2/n) ~/?n(2/n ~2/n-l) , n=l,...,m-2,
At/m-1 = —<*m-l2/m-l — /?m-l(2/m-l — 2/m-2) •
If a„ = — and /?„ = ——, then it follows that
an an
(1.3.2) V(cn Ayn) = a„A2/n , n = 0,..., n - 1 ,

6
ALOUF JIRARI
which is equivalent to (1.1.5).
Remark 1.3.1
Though the parameters involved are different, the mathematical modelling of
the previous three problems led to essentially the same difference equation,
V\p(n) Ay(n)] + q(n)y(n) = Xw(n)y(n) , n = a,..., b.
More examples can be found in Ref. 1, such as the vibration of a regular
atomic chain and a chain containing impurity.
What follows is an investigation of boundary value problems involving this
difference equation, inspired from the theory already established for second-order
differential equations of Sturm-Liouville type.

Chapter 2
Regular Sturm-Liouville Problem
In this chapter, the regular Sturm-Liouville problem for difference equations is
formulated. Then, preliminary results are presented concerning self-adjoint ness,
orthogonality of eigenfunctions, and eigenfunction expansions.
2.1 Set Up
Consider the difference equation
(2.1.1) p(n)y(n + 1) + p(n - l)y(n - 1) = [Xw(n) + q'(n)]y(n) , n = a,..., 6,
where A is a parameter, p(n), qf(n), and w(n) are real numbers subject to w(n) >
0 for n > a, and p(n) > 0 for n > a — 1, and a and b are finite integers with
a < b.
A few remarks are in order before the problem is stated. Firstly, the change
of variables
(n) = ti;(n)1/,2t/(n)
(2.1.2) (n) = w(n)-1/2q'(n)
(n) = w{n)-1/2p(n)w(n + l)"1'2
shows that one could assume without loss of generality w(n) = 1. Secondly,
(2.1.1) is a difference equation of Sturm-Liouville type since it can be expressed
as
(2.1.3) V[p(n)A2/(n)] + q(n)y(n) = \w(n)y(n), n = a,..., 6,
where q(n) = p(n) + p(n — 1) — ?'(w). Finally, boundary conditions are needed
for a boundary value problem. Notice that boundary conditions of the form
7

8
ALOUF JIRARI
c
cos ay(a) — sin a(p(a)Vy(a)) = 0
coB0y(b) + sin/?(p(6)Vy(6)) = 0,
where 0 < a, f3 < 7r, are equivalent to
•<-i)+(St-,),(')=0
•(»-»)+(^-0«W"°-
or
fty(a) = 0
(y(a-l) + i
where h and & are real numbers.
In addition, for (2.1.3) together with (2.1.4) to have a nontrivial solution, it
is clearly necessary that y(a — 1) / 0. Now, in £2{ayb\w)y the Hilbert space
of sequences of complex numbers 2/(a),..., y(b) with the inner product (y, z) =
b
y] u;(n)2/(n)T(n), let £ be the difference operator given by
nzza
(,„\(n\ _ V[p(n)Ay(n)] + g(n)y(n)
(iy)(n) = — , n = a,..., 6,
w(n)
set
£>L = {2/ € ^2(a, 6; w):£y £ £2(a, 6; ti>), 2/(a - 1) + ft y(a) = 0
and y(6 + 1) + iy(6) = 0},
and define L by setting Ly = ^/ for all y in D^.
Definition 2.1.1
The regular Sturm-Liouville problem is the problem of showing, when [a, 6] is of
finite length, w(n) and p(n) are finite at a — 1 and 6, and p(n) is nonzero at
a — 1 and 6, that L is self-adjoint, classifying the spectrum of L (that is, the set
of eigenvalues of L, the values of A for which
(2.1.5) Ly=Xy
has nontrivial solutions called eigenfunctions), and deriving the spectral
resolution of L (an eigenfunction expansion).
Remark 2.1.2
—ct
Setting y(a — 1) = a ^ 0, which implies that y(a) = —-, and using (2.1.3), it is
h
easy to see that y(n) is a polynomial of degree (n — a) in A. The eigenvalues of
L could then be taken as the solutions of y(b -f 1) + ky(b) = 0, in other words,
the roots of a polynomial of degree (6 — a + 1). The objective for the rest of
this chapter is to show that, in this case, the spectrum is real and discrete, the
eigenfunctions are orthogonal and it is possible to give eigenfunction expansions.

STURM-LIOUVILLE DIFFERENCE EQUATIONS
2.2 Preliminary Results
Theorem 2.2.1
L is self-adjoint.
The proof relies on the following lemma, otherwise known as Green's formula.
Lemma 2.2.2
For any y and z in £2(ai 6; w),
(2.2.1) (ty, z) - (y, tz) = p(a - l)[y(a - l)z(a) - y(a)J(a - 1)]
-?(»)[»(W+l)-y(l+lW)],
.-(7(a), rfa-lWa-l))(; ^ ) (rfj$fl_l) )

10 ALOUF JIRARI
Proof of the lemma
For y and z in £2(a, b; w),
b
(*V>*) = X>(*)('y)(*)*(*),
n=a
b
= l]{V[p(n)A2/(n)] + (?(n)t/(n)}J(n),
b
6 6
= 5^P(n)y(n + l)*(n) - ^[P(n) +P(n - !) - tf(n)Mn)*(n)
6
+ Ylp(n - i)y(n -1)*(*0-
n=a
Replace the first sum by
b
Y,P(n ~ l)y(*)*(n - 1) - K<* - l)y(a)«(a - 1) + p(b)y(b + l)z(6),
and the last one by
b
^2p(n)y(n)J(n + 1) + p(a - l)y(a - l)z(a) - p(b)y(b)-z(b + 1),
nz=a
so that
b b
(*y>z) = X^Kn)s/(n)*(n +!) - ]C^n) + p(n "x)" ?(n)Mn)J(n)
b
+ &(* " l)y(n)«(n - 1) +p(a - \)[y{a - l)z(a) - y(a)l(a - 1)]
-p(6)[y(6)2(6 + l)-y(6+1)2(6)].
That is,
(ly, *> = (y, '*) + p(« - i)[y(« -1)*(«) - y(a)^(a -1)]
- P(b)[y(b)-z(b + 1) - y(6 + 1)2(6)]. ■

STURM-LIOUVILLE DIFFERENCE EQUATIONS
11
Proof of the theorem
Observe that
(! V-
-1
1
"♦(srnO'
p(a-l)
and
p(a - 1) / \ p(a - 1)
/^± 1 W 1 *
p(a - 1)
-ft
1 +
So by (2.2.1), (£y,z) - (y,£z) equals
-1
pW
p{b)I
,p(b)
P(*>)
-1
ft2
-^(a), p(a-l)J(a-l)) p(a " J>. x x
U«-i)J
A rf^l)V y(a) \ 1
J_ _ft U«-i)»(«-i)/ , /
a -1) 1 1 + ~
p(a-l)
p(a-l)
(z(6 + l), p(6)z(*))
p(6) |
1 P^
VpW/
-ttA /v(* + i)\
1 p(t) U
,p(6)
-l / \p(%(&)
•
That is,
fc2 +
1 (_EL
H)
■z(a) -f /ip(a—l)z(a— 1), hz(a) -f z(a—1) ) x
->)
Ma) + 2/(a~l)
+ ■
\p(*)/
/"Zfc \ / 2/(6 + 1) + *#) \
(W)Z{b+1) + p(b)z(b)j ,(6+l) + M6)) ^J/(6 + 1)_p(%(,)J.

12
ALOUF JIRARI
For convenience, set
a = j , /? = j j
M\(u) = hu(a) + u(a — 1), M^u) = — rrw(a) — hp(a — l)u(a — 1),
p(a - 1)
^i(ii) = u(6 + 1) + iti(6), JV2(ti) = -£v u(6 + 1) - p(6)u(6).
p(6)
Then
(ty,z)-{v,tz)
..(-B5PJ.1TP5) (^!) + "(-^^) (SO
= a (M^)M2(2/) - Aft^)Mi(y)) + /3 (lh(z)N2(y) - Thf/JN^yfj
Also, y is in D/, means that Ly = £y and M\{y) = N\(y) = 0.
Now, let z be in D/,* where X* is the adjoint of L. Given y in Dl such that
y vanishes at a and 6, it follows that
(y,L*z) = (Ly,z) = (£y,z) = (y,£z).
Therefore, L*z = £z, since y is arbitrary. On the other hand, for y arbitrary in
D/,, M\{z) = JVi(z) = 0 by Green's formula. Hence,
DL* ={ze l2{a, 6; w) : £z G ^2(a, 6; u>) : z(a-l)+ftz(a) = 0, z(b+l)+kz(b) = 0}
and L*z = fo for all z in D/,*. In other words, L is self-adjoint. ■

STURM-LIOUVILLE DIFFERENCE EQUATIONS 13
Theorem 2.2.3
Let y be a solution of Lu = Xu and z a solution of Lu = fiu, both satisfying the
condition u(a — 1) + hu(a) = 0. Then, for a < n <b,
n
(2.2.2) (A - /i)X>0')yO>0) = P(n)[y(n + l)z(n) - y(n)z(n + 1)].
j=a
Proof
By (2.1.3), V[p(i)Ay(i)] = [Xw(j) - q(j)]y(j)
and ^\p(j)Az(j)]=z\pw(j)-q(j)]z(j).
Multiplying these equations, respectively, by z(j),—y(j), and adding gives
V\p(j)Ay(j)]z(j) - V\p(j)Az(j)]y(j) = (A - p)w(j)vU)*U),
or equivalently
(2.2.3) p(j)[y(j + l)z(j) - y(j)z(j + 1)]
= (A - fi)w(j)y(j)z(j) + p(j - l)[y(j)z(j - 1) - y(j - l)z(j)].
Then
n
£{p(i)[y(j + i)*(i) - 2/0>(i +1)] - Ki - i)[2/0>0' -1) - y(i - i)*0')]}
n
= (A-Ai)53w(i)y(iMi).
That is,
p(n)[2/(n + l)*(n) - y(n)z(n + 1)] - p(a - l)[y(a)z(a - 1) - y(a - l)2(a)]
n
which reduces to
n
p(n)[2/(n + l)z(n) - y(n)z(n + 1)] = (A - ^)^w{j)y{3)z{j),
j=a
thanks to the boundary conditions at a. ■

14 ALOUF JIRARI
Corollary 2.2.4
Let y be a solution of Lu = Xu where Im(X) ^ 0, satisfying y(a — 1) + hy(a) = 0.
Then, for a < n <b,
(2.2.4) E^)|y0)|2=Kn)MfcH^
Proof
This is due to the fact that if A is complex and y is a solution of Lu = Xu, then
y is a solution of Lu = Aw. The conclusion of (2.2.4) is immediate by setting
z = y and /i = A in (2.2.2). ■
Corollary 2.2.5
Let y\ be a solution of Lu = Xu, where Im(X) ^ 0, satisfying
yx(a — 1) + hyx(a) = 0. Then, for a < n <b,
(2-2.5) X>0'){yA0)}2
[(^2/A(n + 1)J V\(n) - yx(n + 1) (J^yx(n)JJ .
= p(n)
In particular, for real A,
(2.2.6) (^(n + J)) *>*(n) " **(n + X) (^2/A(n)) > °'
Proof
From Remark 2.1.2, 2/(n) is a polynomial of degree (n — a) in A. Fixing A in
(2.2.2), dividing by (A — /i), and letting /i approach A gives
wOHlfcO)}' = limp(n)—^ *-*^ ^ *,
= p(n)y*(w+*) (&**(")) - »*(") (&**("+*»
by l'Hopital's rule. This proves (2.2.5). If now A is real, (2.2.6) is true since
p(n) > 0 for a < n < b. ■

STURM-LIOUVILLE DIFFERENCE EQUATIONS 15
Theorem 2.2.6
Ifyx is a solution of Lu = Xu, satisfying yx(a — 1) + hyx(a) = 0, the polynomial
(2.2.7) V\(n) + kyx(n-l)
has exactly (n — a) real and simple zeros.
Proof
Suppose (2.2.7) has a complex zero A. Then A is also a zero of (2.2.7).
Multiplying yx(n) + k yx(n - 1) = 0 by y^(n - 1), multiplying Vx(n) + kyj(n - 1) = 0
by —yx(n — 1), and adding yields
y\(n)Vx(n " !) " Vx(n)y\(n " 1) = 0,
or equivalently
yx(n)yx(n - 1) - yx(n)yx(n - 1) = 0.
n —1 2
Then, Im[yA(n)yA(n - 1)] = 0. But, £>(j){y(j)}2 > |2/(a)|2 = (J) > 0
according to Remark 2.1.2. This clearly contradicts (2.2.4). For this reason, the
zeros of (2.2.7) must be real.
Analogously, suppose (2.2.7) has a multiple zero A. Then, yx(n)+kyx(n — 1) =
0 and —-2/A(rc) + &-—-yx(n — 1) = 0. Multiplying these equations by —-yx(n — 1)
dX d\ dX
and —yx(n — 1), respectively, then adding gives
lfc(n) f ^lfc(n " !)) " \7\yx(nn yx(n " X) = °'
thus contradicting (2.2.6) (note that A is real). Consequently, the zeros of (2.2.6)
must be simple.
Finally, the fact that (2.2.7) is a polynomial of degree (n — a) in A insures
that it has exactly (n — a) zeros. ■
Corollary 2.2.7
Ify is a solution of Lu = Xu satisfying the boundary conditions (2.1.4)f then y
is real.
Proof
The boundary condition y(b + 1) + ky(b) = 0 yields (6 — a + 1) real distinct
eigenvalues A. This forces y to be real since, as a function of A, y is a sequence
of polynomials with real coefficients. ■

16 ALOUF JIRARI
Theorem 2.2.8 (Wronskian-Type Identity)
Let y and z be solutions of Lu = Xu. Then, for a < n <b,
W\y, z](n) = p(n - l)[y(n)Az(n - 1) - z(n)Ay(n - 1)],
= -p(n - l)[y(n)z(n - 1) - y(n - l)z(n)]
is constant (in particular equal to W[y, z\{a)).
Proof
Setting A = /i in (2.2.3), it follows that
A{p(n - l)[y(n)Az(n - 1) - z(n)Ay(n - 1)]} = 0.
Therefore, for every n = a,..., 6,
p(n-l)[y(n)Az(n-l)-z(n)Ay(n-l)] = p(a-l)[y(a)Az(a-l)-z(a)Ay(a-l)]
and W\y,z](n) = W\y,z](a). M
These results constitute the background necessary for the characteristic
properties developed in the next section.

STURM-LIOUVILLE DIFFERENCE EQUATIONS 17
2.3 Orthogonality, Eigenfunction Expansion, Spectral Function,and
Green's Function
Theorem 2.3.1 (Orthogonality)
Let Ao,...,A&_a be the (b — a + 1) real eigenvalues of (2.1.3) and (2.1.4).
Let 2/o(rc),.. .,2/6_a(n) be their associated eigenfunctions, respectively. Then
yo(n),..., yb-a(n) are orthogonal in the sense that
(2.3.1) (2/,(n), yj(n)) = Mj , 0 < i, j < 6 - a,
vv/jere
(2.3.2) w = ||2/,(n)||2 = p(b)yi(b){±yi(b + 1)4- *^w(6)} .
Proof
It is broken into two cases:
1) i*j:
(2.2.2) for n = b, A = Aj, y = 2/,(n), /i = Ay, and z = j/j(n) reads
(^ " A;)5>(*)l*(*)yi(*) = *<*)[»(* + l)»W-w(%(Hl)].
Also, by (2.1.4), yt(fr+l)+Jbyt-(fr) = 0 and Vj(b+l)+kyj(b) = 0 which, combined,
imply that
Vi(b + l)yj(b)-iH(b)yj(b+l) = 0.
This can be accomplished by multiplication of the two equations by yj (b) and
b
—y%(b), respectively, and addition. Thus, for i ^ j, ^J w(n)yi(n)yj(n) = 0, or
nzza
equivalently (yi(n),yj(n)) = 0, the reason being that yo(n), • •. , 2/&_a(n) are real
sequences
(n = a,...,6).
2) i = i:
(2.2.5) for n = 6 and A = A, reads
£>(»){w(»)}2 = PW [(^»(* + !)) »(») " W(» + 1) Qx»<*))] •
Moreover, by (2.1.4), y,-(6+ 1) = ~&2/»(&), so that
6 r d .. . . d
Ew(nHwH>2 = i<6)wW
^2/i(6+l) + fc^2/i(6)
The next result deals with dual orthogonality, that is the orthogonality of the
polynomials y\(n) with respect to the distribution of weights p^}a at the points
A,_a, i = a,..., b. For further details on this concept, Ref. 2 is a good reference.

18
ALOUF JIRARI
Theorem 2.3.2 (Dual Orthogonality)
For a < i, j < b,
(2.3.3) ^Ur-a(i)Vr-aU)P7-a = "0')"'«* •
Proof
According to (2.3.1), for a < r, s < b,
Y,W(r)y*-a(r)yt-a(r) = W«-a.
This indicates that the (6 — a + 1) vectors ys-a(a),..., ys-a(b), s = a,..., 6,
are linearly independent. Therefore, an arbitrary vector u(a),.. .,u(b) may be
expressed in the form
(2.3.4) U(i) = YyrPr-aVr-aii) , i = d,...,b
and so
^tx;(2>(i)2/a_a(2) = 5^ti;(i)y,_fl(i)53»rPria^-a(Oi
= Zl^^ia^^COl/r-aW^-aW,
r=a »=a
= >^tV/9ria/9r-a<Sr
That is,
(2.3.5) &(«>(0y.-«(0 = ^-

STURM-LIOUVILLE DIFFERENCE EQUATIONS
Substitution for vr in (2.3.4) then gives
19
*(0 = J^Prial^-aWE^OXiJyr-aO')!
j=a
= E,l;y)tiy)2^-flWyr-a(i)Pr-ai
j=a
= X^c?)
$Jfr-a(0yr-aO')Pr-<i
Simple identification shows that
b
u(j) .
or equivalently,
uU)^2yr-a(i)yr-a(J)Pr-a = **i >
5^1fr-fl(0yr-aO')Pria = WU) ^O"-
Theorem 2.3.3 (Eigenfunction Expansion)
Ifu(a),..., ti(6) is any sequence in ^2(a, 6; w), then
(2.3.6) ti(n) = ^2,viyi-a{n)pi}a , n = a,..., 6, wnere
(2.3.7) vt- = (ti(n),yt-_fl(n)) = ^u;(n)ti(n)jfc_fl(n)
Parseval's equality also holds. That is,
(2-3.8) IMI^EMV-V

20
ALOUF JIRARI
Proof
Note that (2.3.6) and (2.3.7) are exactly (2.3.4) and (2.3.5) from the previous
theorem. The only part remaining to prove is Parseval's equality:
H|2=£>(n)Kn)|2
n=a l»'=a ) \j=a J
b b ( b \
= Y32viVjPT-aPj±a \ EW(B)*-'(B)W-«(B) f '
i=aj=a Kn=a )
b b
= YlYlViv'pi-*pj-*pi-aSi> jby (2-3-1)'
%=aj=a
= £hv.v
Definition 2.3.4
A spectral function is a real-valued, nondecreasing and right-continuous step
function, defined for A real, with jumps of amount pj2a a* ^i-a(i = ct,... ,6).
For instance,
£ PT-a > ^>°
A<A;_o<0
is a spectral function.
Two important observations are in order here.
Remarks 2.3.5
(1) The spectral function defines a Stieltjes measure dp(X) where sets not
including any of the A,_a's are of measure 0.

STURM-LIOUVILLE DIFFERENCE EQUATIONS 21
Equalities (2.3.6), (2.3.8) and (2.3.3) can then be restated, respectively, as
follows:
/oo
v(n)yx(n)dp(X),
-OO
/OO
\v(X)fdp(X),
-OO
and
/OO
yx(i)yZ(j)dp(\) = w(j)-16ij.
-OO
(2) In future use, one might desire for convenience to normalize the solutions
— 1/2
2/o, • • •, Vb-a] that is, replace yi-a(n)Pi-a by 2/*-a(n)> » = a,. •., 6. In this case,
(2.3.6), (2.3.7), (2.3.8), and (2.3.3), respectively, reduce to:
b
(2.3.12) u(n) = Y^i>j2/,-_a(n), n = 1,..., 6, where
b
(2.3.13) Vi = (ti(n),2/,_a(n)) = ^w(n)ti(n)2/,_a(n),
(2.3.14) IN'= EN2,
i=a
and
(2.3.15) ^yr-a(OPr-aO') = "0T ^i , Cl<iJ<b.
Moreover, the spectral function can then be defined as having jumps of amount
1 at A,_a, such as:
{J2 H(\-\i-a) , A>0
- J2 H(-\ + \i-a) , A<0
A<A,_o<0
where H is the standard Heaviside function.

22
ALOUF JIRARI
Definition 2.3.6
Two sequences y = (y(a),..., y(b)) and z = (z(a),..., z(b)) are linearly
independent if whenever a2/ + /?z = 0,a = /? = 0. In other words, whenever
a2/(n) + Pz(n) = 0 for n = a,...,&, a = /? = 0. y and z are linearly independent
otherwise; that is, if one is a constant multiple of the other.
Theorem 2.3.7
Let y and z be two solutions of tu = Xu on [a, b]. y and z are linearly dependent
on [a, b] if and only ifW[y, z](n), as defined in (2.2.8), is zero for every n in [a, b].
Proof
Assume y and z are linearly dependent. If either y or z is identically zero, the
conclusion is clear. So assume without loss of generality that neither is identically
zero. Then, each is a constant multiple of the other. In particular, y = jz for
some constant 7, and for every n = a,..., 6,
w[y> zKn) = -p(n - l)[y{n)z{n -!) - y(n - iM*0],
= —p(n — l)[jz(n)z(n — 1) — jz(n — l)z(n)],
= 0.
Conversely, assume W[y> z](n) = 0 for n = a,... ,6. The fact that p(n) ^ 0
on [a, 6] implies that y(n)z(n — 1) — y(n — 1)2(71) = 0 for n = a,..., b.
Hence , { = —7 -7 = K, for some constant K. (Note that, if z(n) =
z(n) z(n — 1)
z(n — 1) = 0, it follows from the difference equation that z is identically zero,
which insures the linear dependence of y and z. It is therefore safe to assume
without loss of generality that z is not identically zero). The linear independence
of y and z is then clear. ■
This chapter will be closed by two theorems concerning the Green's function
and some of its properties.
Theorem 2.3.8
Let y be a solution of(L — X)u = 0 satisfying the boundary condition y(a — 1) -f
hy(a) = 0. Let z be a solution of(L — X)u = 0 satisfying the boundary condition
z(b+l) + kz(b) = 0.
Ify and z are linearly independent and if X is not an eigenvalue, the solution of
(L — X)u = f, where f is a prescribed sequence, is given by
b
(2.3.16) ti(n) = Y,G(\J,n)w(j)f(j)
j=a
where

STURM-LIOUVILLE DIFFERENCE EQUATIONS 23
(,,17) oa^»=K!; »
v W[y,z](j)
is the Green's function.
Proof
u satisfies (L — X)u = /; that is,
(2.3.18) V[p(n)Ati(n)] -f q(n)u(n) = Xw(n)u(n) -f w(n)f(n), n = a,..., 6.
In order to determine it, use variation of parameters. Let
u(n) = A(n)2/(n) -f £(n)z(n).
Then,
Ati(n) = A(n)Ay(n) + B(n)Az(n) + 2/(n + l)AA(n) + *(n + l)AJB(n).
Set
y(n + l)AA(n) + z(n + l)A5(n) = 0,
so that
p(n)Ati(n) = A(n)[p(n)Ay(n)] -f B(n)\p(n)Az(n)]y
and
V[p(n)Au(n)] = A(n)V[p(n)Ay(n)] + £(n)V[p(n)A*(n)]
+[p(n - 1) Ay(n - 1)]Vi4(n) + |p(n - l)Az(n - l)]VB(n).
Substitution in (2.3.18) together with the fact that both y and z satisfy (L—\)u =
0 give
|p(n - l)Ay(n - l)]VA(n) + |p(n - l)Az(n - l)]VB(n) = ti;(n)/(n).
A system of two equations in A(n) and -B(n) is thus formed:
f j/(n + l)AA(n) + z(n + l)A5(n) = 0
| [p(n - l)Ay(n - l)]VA(n) + [p(n - l)Az(n - l)]VB(n) = ti;(n)/(n),
n=a, ... ,b.
y(n + \)VA{n + 1) + z(n + l)V5(n + 1) = 0

24 ALOUF JIRARI
can be substituted for the first equation, making the system:
y(n)VA(n) + z{n)VB{n) = 0 , n = a + l,...,6+l
\p(n - l)Ay(n - l)]Vi(n) + \p(n - l)Az(n - 1)]
x V5(n) = u;(n)/(n) , n = a, ...,6.
Setting u;(a — 1) = tx;(6 -f 1) = 0, one can see that both equations hold for
n = a— 1,..., 6 -f 1. Now, by Cramer's rule,
V4(n) = ; // V / \ ' and V5(n) = v " v } v '
W[y,z](n)
WTy,*](n)
(Note that W[y, z](n) is nonzero for every n by linear independence of y and z.)
Hence,
A(n) = A(n+1)+ £
6+1 »WM) B(n) = 5(a_1)+ £ yOWW)
j=n+l
wfo^Ki)
i^li ^IMO') '
and
ti(n) =
H-i
J/(n)+
B(a-1)+J2
w(j)f(j)z(j)
Now, for u to satisfy the boundary condition u(a — 1) + /iw(a) = 0,
*(»)•
[j/(a -1)4- &y(a)]+B(a-l) [z(a - 1) + &z(a)] = 0
must hold. So, due to the fact that y(a — l)-f hy(a) = 0, B(a — 1) = 0. Otherwise,
W[2/, z](a) = 0. By similar argument, it is possible to show that for u to satisfy
the boundary condition ti(6-f l)-f ku(b) = 0, A(6-f 1) = 0 must hold. As a result,
u(n) =
or equivalently,
w(j)f(j)z(j)
itLi wh>*W)
z(n) +
b+i
j=n+l
W[y,z](j)
y(n)
<n) = Y^G{\,j,n)w(j)f(j)
j=a

STURM-LIOUVILLE DIFFERENCE EQUATIONS
25
where
G(\,j,n) =
W[y,z](j)
y(n)z(j)
W[y,z](j)
a < j < n <b
a < n < j <b.
Theorem 2.3.9
(1) G(Xyjy n) is symmetric.
(2) G(X,j, n) satisfies (2.1.3) in both variables j and n, with a nonhomogeneous
term when j = n.
(3) G(Xyj, n) satisfies the boundary conditions of (2.1.4) in the sense that
G(A, a - 1, n) -f hG{Xy a, n) = 0 , G(A, b -f 1, n) -f *G(A, 6, n)= 0,
and G(A,j,a-l) + /iG(A,j,a) = 0 , G(A, j,&+ 1) + kG(Xyjyb) = 0.
Proof
(1) That G(A,i,n) = G(A,n, j) is obvious from (2.3.17).

26 ALOUF JIRARI
(2) (L — X)u = / can be written in the matrix form
M
r u(a)
u(a + 1)
u(b)
w(a)f(a)
w(a + l)/(a + 1)
w(b)f(b)
where
M
-[Xw(a)+p(a)+(l+h)p(a-l)-q(a)]
p(a)
p(a) -[Xw(a+l)+p(a+l)+p(a)-q(a+l)] p(a+l)
p(n - 1) - [Xw(n) + p{n) + p(n - 1) - q(n)] p(n)
p(b-2) -{Xw(b-l)+p(b-l)+p(b-2)-q(b-l)} p(b-l)
p(b-l)
- [Xw(b) + (1 + k)p(b) + p(b - 1) - q(b)\
If A is not an eigenvalue, the matrix M is invertible and its inverse is a matrix
G with entries G(\>j, n). Therefore, from MG = GM = /, it follows that
(2.3.19) p(j - l)G(X,j - 1, n) - [Xw(j) + p(j) + p(j - 1) - q(j)]G(X,j, n)
+p(j)G(X,j+l,n) = 6jn,
which indicates that (2.1.3) holds for the variable j. Interchanging j and n gives
p(n - 1)G(X, n-l,j)- [Xw(n) + p(n) + p(n - 1) - q(n)]G(X, n, j)
+p(n)G(X,n+l,j) = 6jn.
This can be written, using the symmetry, as

STURM-LIOUVILLE DIFFERENCE EQUATIONS 27
p(n - 1)G(XJ - 1, n) - [Xw(n) + p(n) + p(n - 1) - q(n)]G(XJ, n)
+p(n)G(A,j + l,n) = <5in,
which shows that (2.1.3) also holds for the variable n.
(3) (2.3.19) for;* = a reads
p(a - 1)G(A, a - 1, n) - [Au;(a) -f p(a) + p(a - 1) - q(a)]G(X, a, n)
+p(a)G(A,a+l,n) = <5an.
But, from the actual form of the first row of M,
-[Xw(a) -f p(a) + (1 -f /i)p(a - 1) - ?(a)]G(A, a, n) + p(a)G(A, a + 1, n) = «an.
For the two expressions to agree, it must be true that
G(A, a - 1, n) -f /iG(A, a, n) = 0.
By similar argument, it follows that
G(A,6+ l,n) + JbG(A,6,n) = 0.
Finally, using symmetry, it is easy to see that similar boundary conditions hold
for j. ■

Chapter 3
Singular Sturm-Liouville Problem
In many of the most interesting examples, the treatment of Chapter 2 fails
to apply because the Sturm-Liouville problem is singular. The objective of this
chapter is to give an extensive discussion of this situation, which was examined
from different points of view in Refs. 1 and 2. The argument is similar to that
in the theory of differential equations, which was described early on in Ref. 17
and later on in Ref. 5.
3.1 Definition
The Sturm-Liouville problem of Definition 2.1.1 is singular when p(n), q(n)
or w(n) become infinite at a — 1 or 6, when p(n) approaches 0 at a — 1 or b, or
when the interval [a, b] is infinite in length.
This is more general than the definition given in Ref. 2 where a singular
problem is ultimately one where b is infinite. In order to cover the problem in
its generality, both a and b are assumed to be singular.
3.2 Cb' Circles
Consider the interval [e, b'] where a < e < b' < b and let 0\ and 02 be solutions
of (2.1.3) satisfying the conditions
f eUe) = 0, 0i(e - 1) = "* ,
(3.2.1) 4 P(c-l)
I 02(e) = 1, 02(e-l) = O.
Then 0\ and 02 are linearly independent. In fact, for every n,
(3.2.2) W[0u02](n) = -p(e - l)[01(e)02(e - 1) - 0i(e - l)02(e)] = -l.
Also, the problem of finding the solutions of (2.1.3) satisfying the boundary
conditions
/ y(e-l) = 0
(3'2'3) I y(V + l) + *y(V) = 0.
is a regular self-adjoint boundary value problem on [6,6'].
Now, for (A) ^ 0, let Vv = 0i+rnb'02 be the solution of (2.1.3) satisfying the b'
boundary condition of (3.2.3). Clearly, Vv and 02 must be linearly independent.
Otherwise, Vv satisfies both boundary conditions, which forces A to be real.
28

STURM-LIOUVILLE DIFFERENCE EQUATIONS 29
Theorem 3.2.1 (Characteristics of the Cb' Circles)
IfIm(X) ^ 0, the solution Vv = 0\ + m&/02 satisfies the V boundary condition of
(3.2.3) if and only ifrnb' lies on a circle Cb' in the complex plane whose equation
is
(3.2.4) W[^M(b'+ 1) = 0,
with radius
(3.2.5) iv HWfMaK&' + l)!"1,
and center
(3.2.6) f^ = -M^ .
Proof
Since i/>bi(b' + l) + k Vv(&') = 0, k = - / * K But k is real, and so k-k = 0;
V>b'(o')
that is, xl>h'{b' + l)lM&') - Vv(&' + l)Vv(&') = 0, which is equivalent to (3.2.4).
This is the equation of a circle because i^b'{b' + 1) + k ipb'(bf) = 0 is equivalent
gi(y + i) + *gi(y)
m*' w + i) + ko2(vy
which describes a circle Cb' in the complex plane as k varies. What are its center
and radius?
Using the definition of Vv, (3.2.4) can be written in the expanded form
(3.2.7) \mbi\2W[e2M^ + l) + rnb'W[e2ie1](b,+ 1)
+rnhiW[0u02](b' + 1) + WVuOW + 1) = 0.
Moreover, if one sets m&/ = u + iv,
W[O2i02](b'+l) = 2iAy
-W[02,01](b'+l) = B + iC ,
which implies
and
W^W+l^B-iC,
Wr[ffi,?i](6/+l) = 2iD.

30 ALOUF JIRARI
Then (3.2.7) becomes
(2i A){u2 + v2) - (u + iv)(B + iC) + (u- iv)(B -%C) + 2iD = 0
or „
2 C 2 B D n
A A A
By completing the squares, it follows that
2 / „N2 B2 + C2-AAD
(-£) +(-s) ■
442
It is easy to see then that Cj» has center
~ , = C + iB _ (B-»C) _ W[tfi,g2](y + 1)
"**' ~ 2A ~ 2iA ~ W[02,02](b'+ 1)
and radius
\B2 + C2-4AD
n> =
\A2
1/2
= \(B + iC)(B -iC) + (2iA)(2i D)\L/' \2iA\~1.
i/Zio; 41-1
But
(B + iC)(B -iC) + (2iA)(2iD) = -W[02,0i](b'+ l)W[0i,02](b'+ 1) ,
+ W[02,Oi](b'+ 1)W[0!,0i}(V+ 1) ,
= W\0u~02]{b' + \)W[0u02]{b' + \) ,
= W^i,h](e)W[$lt02](e) , by Theorem 2.2.8,
= 1,
by (3.2.2). Therefore,
rb, = \W[02,02}(b'+l)\ \ ■
The following theorem is in order before the nesting property of the CV circles
is established.

STURM-LIOUVILLE DIFFERENCE EQUATIONS 31
Theorem 3.2.2
nib' is inside or on Cv if and only if
2 s Im(my)
(3.2.8) 5>(»)IM»)la * "^
Proof
From (3.2.4) and (3.2.7), it is immediate that mj,< is inside or on CV if and only
if
(3.2.9) W[^M(b' + l)<
K ' w[e2,e2](b' + i) ~
On the other hand, using (2.2.3) for y and z solutions of (2.1.3) and summing
from e to b', it follows that
b'
W[y,J](b' + 1) = W[y,J](e) + 2tIm(A)^u;(n) y(n)J(n).
n=e
In particular,
b'
W[02,02](b'+ 1) = O + 2rtm(A)^™(n)|02(n)|2 = 2%A ,
n=e
b'
W[0l902](b' + 1) = -1 + 2rtm(A)^™(n)01(n)02(n) = B-iC ,
n=e
which implies
&'
W[*2, W + 1) = 1 + 2iIm(A)]Ttx;(n)01(n)02(n) = -(£ + iC),
n=e
and finally,
&'
W[0i,0i](b'+1) = 0 + 2iIm(A)^tx;(n)|^1(n)|2 = 2iD .
n=e
As a result, using the right-hand side of (3.2.7),
_ b'
W[i>h.M(b' + 1) = m6/ - m6/ + 2flm(A)2ti;(n)|^#(n)|2 .

32 ALOUF JIRARI
Substitution in (3.2.9) then shows that m&/ is inside or on CV if and only if
b'
mil — rnp + 2fIm(A) Yj w(n)\xpb'(n)\2
p-^ *°.
2iIm(A)£>(n)|02(n)|2
n=e
which is equivalent to (3.2.8). ■
Remark 3.2.3
From the previous calculations, it also follows that CV has center and radius,
respectively,
b'
1 - 2iIm(A) ^2w(n)01(n)02(n)
(3.2.10) mh, = ^
b
2iIm(A)^™(n)|02(n)|2
and
I b'
(3.2.11) r6* = ~
b'
2rtm(A)^™(n)|02(n)f
Also, from (3.2.8), it is easy to see that Im(m&/) and Im(A) have opposite signs.
Theorem 3.2.4 (Nesting Property of the Cb' Circles)
If b" < V < b, then Cb" contains Cb' in its interior.
Proof
From Theorem 3.2.3, m^ is inside or on Cb' if and only if
If b" <b' <b, then
h" h'
£>(») |M«)|2 < !>(») l^'(")|2 ^ JjSw '
n=e n=e ^ '

STURM-LIOUVILLE DIFFERENCE EQUATIONS 33
and so m&/ is inside or on Cb" as well. Therefore, the CV circles are nested. ■
Remarks 3.2.5
In a discrete setting such as this one, lim /(n) is simply /(no) if no is finite,
n-*no
but it is lim /(n) if no is infinite.
n—*-oo
Now, as V—►&, the Cb' circles contract to a "limit-point" or a "limit-circle".
fr T / \
Let rrib be in the limit and let ^6 = ^1+^6^2- Then J^ti;(n)|^&(n)| <— ,
n=e ^ '
and so V>6 is in l2(e, b; w).
If the Cb' circles converge to a circle, its radius has to be strictly positive. By
b
(3.2.11), this means that 2^ w(n)\^2(n)\2 < °°- Both xfrb and 02, and therefore
n=e
all solutions of lu = Xu, are in l2(e, b; w).
If the Cb' circles converge to a point, the radius r&/ approaches 0 as b' ap-
b
proaches b. This means that YJ w(n)|#2(rc)|2 diverges, making V>6 the only
n=c
linearly independent solution in l2(e,b;w). Moreover, m&/ approaches rrib
independently of k. In particular, by setting k = 0, it follows that
(3.2.12) mt = -lim ^%±^- .
Definition 3.2.6
The limit-point case (abbreviated LP) holds at b when the Cb' circles contract to
a point. Then only one solution of lu = Xu is in £2(e, b; w).
The limit-circle case (abbreviated LC) holds at b when the Cb' circles contract
to a circle. Then every solution of lu =■ Xu is in £2(e, b; w).
Theorem 3.2.7 (LC Independence of A)
Once the limit-circle case holds at b for some value A, it holds for any value /i,
real or complex.
Proof
Suppose that for some A every solution of lu = Xu is in l2(eyb;w) and let /i
represent any number, real or complex. Let y and z be two linearly independent
solutions of lu = pu and let us show that both y and z are in l2(ey b; w).
UA(j) = W[y,02](j) and B(j) = W[y,0i](./). Then, by (2.2.3),
(3.2.13) -A(j + 1) + A(j) = (A - fi)w(j)y(j)02 (j)

34 ALOUF JIRARI
and
(3.2.14) -B(j + 1) + B(j) = (A - ii)w(j)vU)0i 0) •
Multiplication of (3.2.13) and (3.2.14) by 9\{j) and ^0), respectively, gives
[-A(j + 1) + A(j)]0iV) = [-B(j + 1) + B(j)]fl2(i),
or equivalently
B(j)02(j) - AiWiJ) = B(j + l)02(j) - A(j + 1)0!(j) ,
= w[y, W +1)^2(i) - w[y, e2](j + l)^i(i),
= W[62,01](j + l)y(j),
= -W[0u02](j+l)y(j),
= -W[0U 02](e)y(j) , by Theorem 2.2.8 ,
= y(j).
by (3.2.2). As a result, (3.2.13) and (3.2.14) can be written, respectively,
(3.2.15) A(j + 1) - A(j) = (A - ii)w(j)[AU)OiU) ~ B{j)62{j))e2(j)
and
(3.2.16) B(j + 1) - B(j) = (A - /i)w(j)[AU)6iU) - B(j)e2{j))e,(j).
It remains to show that the solutions A and B of these difference equations are
uniformly bounded as j —*• b. For, then, there exists a constant c, independent
of j, such that \y(j)\ < c|#i(j)| + c|02(.?)|- Therefore, y is also in £2(e,b;w). By
a similar argument, it can be shown that z is in £2(e,b;w). Consequently, the
limit circle case at b holds for /i.
Now, to prove the uniform boundedness of A and B, it is convenient to
combine (3.2.15) and (3.2.16) into the matrix equation
(3.2.17) (£) (j + 1) - Q) (j) = (A - n)H{j) {^j (j)

STURM-LIOUVILLE DIFFERENCE EQUATIONS
35
where
'wUWiVMV) -y(JWUi)
vUWKj) -wWiUfaU)
)■
m = (
Regardless of the norm used, it will always be true that
IO+i>M(£H+|(2)«+i>-(*)«
and so
= [l + |A-/i|||^(j)||]JQ)(i)
(b)°'+1)||-( A [1 -H IA - ^lll^(«)li]^ || (^) (e + 1)
\n=e+l
Since 2^ wOOI^iO)!2 < oo and 2^ w(j)|02(.;)|2 < oo, by Schwartz's inequal-
j=e j=e
b b
ity, ^2w(j)\0!(j)\ \62(j)\ < oo. Consequently, |A - p\ ^ \\H(n)\\ < oo and
j=e
JJ [l + |A — /i| ||i7(n)||] converges as j —* b. This means that f J
— «_L1 II \ /
(j)\\ has
n=e+l "
a fixed upper bound, or equivalently, that A(j) and B(j) are uniformly bounded
as j —* 6. The proof of the theorem is thus complete. ■
3.3 Ca' Circles
In this section, the calculations are similar, if not identical, to the ones
performed in the previous section. To avoid repetition, only the results will be
mentioned, unless there are differences.
Consider the interval [a', e — 1] where a < a1 < e — 1<6. The problem of
finding the solution of (2.1.3) satisfying the boundary conditions
(3.3.1)
y(a'-l) + hy(a') = 0
y(e) = 0
is a regular self-adjoint boundary value problem on [a', e —1]. Now, for Im(A) ^ 0,
let Vv = Oi-\-mai02 be the solution of (2.1.3) satisfying the a1 boundary condition
of (3.3.1). Here again, tf>ai and 6i must be linearly independent. Otherwise, Vv
satisfies both boundary conditions, which forces A to be real.

36 ALOUF JIRARI
Theorem 3.3.1 (Characterization of the Ca/ Circles)
Iflm(X) ^ 0, the solution Vv = #i + ma*02 satisfies the a' boundary condition of
(3.3.1) if and only ifmai lies on a circle Ca> in the complex plane whose equation
is
(3.3.2) W[Vv,<M(a') = 0,
with radius
(3.3.3) tv = |W[02,02](<Or\
and center
w[oud2](*')
(3.3.4) ma/ = -
Theorem 3.3.2
ma/ is inside or on Ca> if and only if
w[o2M(«'Y
(3.3.5) g^j^n),^^).
Proof
ma' is inside or on Ca> if and only if
(3.3.6) W^MJa')
K ' W[02,02](a') ~
On the other hand, using (2.2.3) for y and z solutions of (2.1.3) and summing
from a' to e — 1, it follows that
e-1
W\y,z](a') = W\y,z](e) - 2flm(A) £ ti;(n)y(n)z(n) .
n=a'
In particular,
e-1
W[02,02](a') = -2rtm(A) £^n)|02(n)|2,
e-1
W%,02](a') = -1 - 2Hm(X)'^2 w(n)0x(n)02(n)9
n=a'
e-1
W[02,0i](a') = l-2Hm(X)^2w(n)01(n)02(n)9

STURM-LIOUVILLE DIFFERENCE EQUATIONS 37
and
e-1
W[0i,0i](«') = -2flm(A)£ti;(n)|0i(n)|2.
Then
e-1
^Vv, Vv](<0 = ma, - ma, -2iIm(A)^tx;(n){|ma/|2|^(n)|2
+ ma>O1(n)02(n) + ma,O1(n)$2(n) + \O1(n)\2} ,
e-1
= 2ilm(ma/) - 2iIm(A) ^u;(n)|^fl/(n)|2.
Substitution in (3.3.6) then shows that ma/ is inside or on Ca' if and only if
c-l
Im(ma/) -Im(A) ^ ti;(n)|^fl/(n)|2
<0,
c-l
-Im(A) £ u(n)|02(n)|2
which is equivalent to (3.3.5). ■
Remark 3.3.3
From the previous calculations, it also follows that Ca» has center and radius,
respectively,
c-l
1 + 2iIm(A) Y, ™(nWi(n)Mn)
(3.3.7) ma, = 2g
-2iIm(A) J3 ^(n)l^(n)|2
n-a'
and
c-l '-1
(3.3.8) tv
2iIm(A)53«'(n)|02(n)|'
Moreover, from (3.3.5), Im(ma/) and Im(A) have identical signs. This together
with Remark 3.2.3 implies that mai and m&/ have opposite signs.

38
ALOUF JIRARI
Theorem 3.3.4 (Nesting Property of the Ca' Circles)
Ifa < a' < a", then Ca" contains Ca' in its interior.
Remarks 3.3.5
As a! —► a, the Ca' circles contract to a "limit-point" or a "limit-circle". Let ma
be in the limit and let tpa = $i + ma02- Then ^ w(n)\i/>a(n)\2 < ^ * , and
n=a ^ '
so V'a is in £2(a, e — 1; iu).
If the Caf circles converge to a circle, all solutions of tu = Xu are in
£2(a, e — l\w).
If the Ca' circles converge to a point, r/>a is the only linearly independent
solution in £2(a, e — 1; w). Moreover, the limit point is
(3.3.9) ma = -lim fg^|.
a'-+a V2\a' — 1)
Definition 3.3.6
The limit-point case holds at a when the Ca' circles contract to a point. Then
only one solution of lu = Xu is in £2(a, e — 1; w).
The limit-circle case holds at a when the Ca' circles contract to a circle. Then
every solution of tu = Xu is in I2(a, e — \\w).
Theorem 3.3.7
Once the limit-circle case holds at a for some value A, it holds for any value /i,
real or complex.
3.4 Existence of Boundary Conditions
Ref. 18 provides a constructive means of determining a boundary condition at
oo for difference equations of limit-circle type, by making use of known solutions
of a certain first-order system. This section shows the existence of boundary
conditions at both ends, a and 6, in all limit cases.
First, the existence of boundary conditions is established at the singular end
6. Then analogous results that follow by similar argument are stated for the
singular end a.
Let Ao be a complex number (possibly real in limit-circle cases, but with
nonzero imaginary part in limit-point cases) and let $2 be a solution of
(3.4.1) ty = X0y.
Let V>& be a solution of (3.4.1) in £2(e,b;w), a < e < b' < b. Also, suppose
Wtyby02] = -1.

STURM-LIOUVILLE DIFFERENCE EQUATIONS 39
Theorem 3.4.1
For all y such that ly = /, where y and f are in £2(e, 6; w),
a) lim W[y> 02](b' + 1) exists when b is in the limit-circle case, and
b'—*b
b) \imW[y, tpb](b' + 1) exists in all limit cases,
b* —+b
Proof
1) By assumption,
(3.4.2) V[p(n) Ay(n)] + q(n)y(n) = w(n)f(n)
and
(3.4.3) V[p(n)A02(n)] + q(n)02(n) = \0w(n)02(n).
It follows from multiplying (3.4.2) by 02(n), (3.4.3) by — 2/(n), and adding that
V[p(n)Ay(n)]02(n) - V[p(n)A02(n)]y(n) = ti;(n)/(n)02(n) - \ow(n)02(n)y(n)9
or equivalently,
Wfy, h](n) - W[y, 02](n + 1) = w(n)f(n)02(n) - w(n)\o02(n)y(n).
Summation from e to b' then gives
b'
W[yy 02](e) - W[y, 02](b' + 1) = £[-Aoii;(n)y(n)02(n) + u;(n)/(n)tf2(n)].
n=e
So
W[y, W + 1) = W[y,92](e) + Ao£u;(n)y(n)02(n) - £>(n)/(n)02(")-
n=e n=e
In the limit-circle case at 6, 02 is in <2(e, 6;u;). Therefore, both sums converge
as V -► 6. Thus, lim W[y, 02](6' + 1) exists.
2) Similarly, from
V[p(n)Ay(n)] + g(n)t/(n) = iu(n)/(n)
and
V[p(n)AV»6(n)] + q(n)xpb(n) - \0w(n)xpb(n),

40
ALOUF JIRARI
it follows that
b' b'
W[yy rpb)(b' + 1) = W[y, ifc](e) + A«>X>(n)ih(n)y(n) - $>(n)ih(n)/(n).
n=e n=e
Since ipb is in ^2(e,6;tx;), both sums converge in all limit cases at 6, as b' —» 6.
And so, lim W[y, V^K^' + 1) exists in all limit cases. ■
b'—+b
Analogously, let ij)a be the solution of (3.4.1) in £2(a, e — 1; w), a < a' < e < 6,
and suppose W[if>a,02] = —1. Then, by copying the argument of Theorem 3.4.1,
it is easy to prove the following theorem for the singular end a.
Theorem 3.4.2
For all y such that £y = f, where y and f are in £2(a, e — 1; w),
a) lim W[y, O^a!) exists when a is in the limit-circle case, and
a'—+a
b) lim W[y, V'aK0') exists in all limit cases.
3.5 Singular Boundary Value Problems
Definition 3.5.1
£2(ay b] w) is the Hilbert space of sequences y(a),..., y(6) with the inner product
e-1 b b
(3.5.1) (y, z) = ^w(n)t/(n)7(n) + ^Tw(n)y(n)-z(n) = ^ w(n)t/(n)z(n).
n=a n=e n=:a
Remarks 3.5.2
From Definitions 3.5.1, 3.2.6, and 3.3.6, it follows that £y = Ay, where Im(A) ^ 0,
has
a) all solutions in £2(ayb\w) when both a and b are in the limit-circle case,
b) only one solution in £2(a,b;w) when a is in the limit-circle and b is in the
limit-point case,
c) only one solution in £2(a, 6; w) when a is in the-limit point case and b is in
the limit-circle case,
d) or no solution in £2(a,b\ w) when both a and b are in the limit-point case.
In fact, in this last case, V>a is the only solution in £2(a, e — 1; w) and fa is the
only solution in £2(e,b\w). But ipa is distinct from fa since ma is distinct from
nib, as pointed out in Remark 3.3.3.
Definition 3.5.3
Let ipa and xpt, be solutions of £y = Ay, with ipa in £2(a, e — 1; w), ipb in £2(e, b\ w),
and
(3.5.2)
W[il>a,il>b] = ma - mb

STURM-LIOUVILLE DIFFERENCE EQUATIONS 41
(since xpa = #i + rna02iipb = #i + mb62 and W[0i,02] = -1).
Set
DL = J y e £2(a, 6; w) : £y e £2(a, 6; iu), for all A for which
ipa is in £2(a, e - 1; u;) , lim W[y, V'aK**') = 0'
a'—*a
V>& is in £2(e, 6; u;) , lim W[y, rpb](b' + 1) = 0
b'—*b
The operator L is defined by setting
(/*)(„) = (*,)(«) = Vb(n)Ay(n)]^(nMn)|
for all y in D^.
3.6 Green's Function
Theorem 3.6.1
Let X be a complex number such that ipa is in £2(a, e — 1; w), ipb is in £2(e, 6; w),
and W[rffa, V>&] = rna — ro&. Then X is in the resolvent of L and the solution of
(L — X)y = /, where f is a prescribed sequence, is given by
(3.6.1) y(n) = £g(A, j, n)w(j)f(j)
where
J=a
( fpa(j)tpb(n)
(3.6.2) G(XJ,n)= {
fna — rrib
j>a(n)ipb(j)
ma — rrib
a < j < n <b
a < n < j < b .
Proof
(L — X)y = / means that, for a < j < 6,
V[p(i)Ay(i)] + q(j)y(j) = Xw(j)y(j) + w(j)f(j).
Variation of parameters must be used to determine y. Following the proof of
Theorem 2.3.8,
Vti) = A(j)1>a(j) + B(j)il>h(J) >

42
where
ALOUF JIRARI
Ua(i)Vi4(i) + ^(i)VB(i) = 0>
\ \p(j - 1)A^0' - l)]VA(j) + [p(j - l)A^(i " l)]VB(i) = f(i)/(i).
for j = a — 1,..., 6 + 1. By Cramer's rule,
VA(j) = -W{J)fiJ)MJ) and VB(j) = -™W(^(j)
ma — mi
Simple summations then show that
™a — Mb
A(n) = A(e-l)-±W^f(j]^j) and B(n) = B(e-1)- g "WOW.fr")
j=e
ma — mi
Therefore,
(3.6.3)
^_i)^^)/(^0')
y(n) =
+
e-1
j=n + l
w(j)f(j)rpa(j)
Wla — ™>b
j=n + l
ma — m&
where A(e — 1) and B(e — 1) remain to be determined using the fact that y should
satisfy
(3.6.4)
and
(3.6.5)
limW[y,tf»](n+l) = 0
n—*b
\imW[y^a](n) = 0 .
where W[y, ipb](n + 1) = -p(n)[y(n + l)i>b(n) - y(n)i>b(n + 1)] and W[yy ipa] is
similarly expressed. Using (3.6.3) to expand y(n + 1) and y(n)> then simplifying
give
W[y, rl>h](n + 1) = Wtya, ^](« + 1) < A(e - 1) - £
J=C
Wla — ™>b
)

STURM-LIOUVILLE DIFFERENCE EQUATIONS
43
which forces A(e - 1) to equal ^ w(Mti)M) for (3 g 4) to hold By simiiar
calculations, it follows that
j=e
ma - mb
e-l
W[y, rpa](n) = Wfya, r/,b](n + 1) I -B(e - 1) + £
*>(j)f(j)il><
j=n
™>a — ™>b
lb I
e-l
Hence B(e - 1) = ^ W^^^^ for (3.6.5) to hold. Substitution in (3.6.3)
shows that
j=a
™>a — ™>b
y(n)
j=n + l
™(j)f(j)j>b(j)
™>a — Mb
Mn) +
j=a
*u>U)f(j)il>< 'i)
ma — rr
^b{n),
or equivalently,
»(n) = ^G(A>i>n)u;(y)/(i)
j=a
where
G(\J,n) =
il>g(j)j>b(n)
Wla — ™lb
i>a(n)i>b(j)
ma — mi
; a < j < n < b
; a < n < j <b
3.7 Self-Adjoint ness
Theorem 3.7.1
The operator L of Definition 3.5.3 is self-adjoint.
Proof
Let Rf = (L- A)"1/- Then rVg = (L* -J)-Xg and {Rf,g) = {f,R*g). But,

44 ALOUF JIRARI
{f,R*g) = y£w(j)f(j)(R*g)(j),
= {Rf,g),
b
= 2>(»)(JJ/)(n)?(n),
= I>(") \i2G(\,j,n)w(J)f(j) ) ?(»). by Theorem 3.6.1
= I>(i)/0") (EG(A-j»u;(«)ff(")) •
j=a
Therefore,
or
(R*9)U) = £^n)G(A,i,n)£(n),
(«**)(i) = £>(n)G(A, j, n)g(n) = £>(n)G(A, j, n)«/(n).
n=a n=a
This jshows that ((Z, - A)"1)* = (L - A)"1, or equivalently ((L - A)*)"1
(L-A)-i. _
Hence L* — A = L — A, which implies that L* = L. ■
Theorem 3.7.2
Tie resolvent operator (L — A)*"1 is a bounded operator and
(3.7.1) ll(i-A)"1!) < *
IMA)|
Proof
Let (L — X)y = f and suppose A = a + i /?• Then
((L - A)y, y> = ((L - (a + i/3))y, y) = ((L - a)y, y) - i/3(y, y)

STURM-LIOUVILLE DIFFERENCE EQUATIONS
45
and
\{(L-X)y,y)\>m\\yf.
This implies that
ll(£-A)y|||M|>|/?|||2,||2,
or
ll(i-A)y||>|/?||M|.
Then
11/11 > l/?l IP-A)"1/!!,
or equivalently
\\(L-X)-lf\\<^;
that is,
Before moving on to the next section, it is worth mentioning that, in Ref.
19, to characterize all self-adjoint extensions of the minimal closed operator
associated with a Jacobi matrix of limit-circle type, boundary conditions are
constructed using elements which are solutions of a homogeneous second order
difference equation.
3.8 A-Independence of Boundary Conditions
The proofs of the next three results are given at the endpoint b only. They
carry in virtually the same way for the endpoint a.
Theorem 3.8.1
If the limit-point case holds at b (resp. a), then the Wronskian boundary
condition
(3.8.1) \imW[y,xl>b](b'+l) = 0 (resp. lim W[yy^a}(a') = 0)
b'-+b a'—Ki
holds automatically for all A such that Im(X) ^ 0.

46 ALOUF JIRARI
Proof
The general solution of
(L-X)y = f
is
y- yp + <xi/>a + l3tl>b,
where
b
yP(n) = J2G(X,j,n)w(j)f(j).
In the limit-point case at 6, rpa is not in £2(e, 6; w). For this reason,
y = yP + <*^&
and
wr[y,^](»/ + i) = wr[*i^K'/ + i).
From Theorem 3.6.1, \\m.W[yp>tl)b\{b' + 1) = 0, by construction of the Green's
b'~-*b
function, so that (3.8.1) holds independently of A. ■
Theorem 3.8.2
Assume the limit-circle case holds at b (resp. a) and let tpb (resp. ipa) be the
solution ofiy = A0 y satisfying lim W[y, ^(AoJK&'+l) = 0 (resp. lim W[y, ^a(A0)](a') =
0) for a particular value of \$. Then,
lim My, MW + 1) = 0 (resp. lim W[y, V>a(A)](a') = 0),
b'-+b a'—* a
where ipb(X) (resp. rpa(X)) is the solution ofiy = Ay given by the LP-LC
derivation for a given A different from Ao.
The proof of this theorem relies on the following lemma, which represents the
analogue of Titchmarsh's lemma for the continuous case (see Ref. 17).
Lemma 3.8.3
For any fixed Ao and A,
(3.8.2) lim W^(Ao), V>&(A)](&' + 1) = 0 (resp. lim W[V>a(A0), ^«(A)](a') = 0).
6'—*b a'—*a
Proof of the lemma
By construction in 3.2, the functions rpb' satisfy
(3.8.3) Vv(Ao, b' + 1) + k Vv(A0,6') = 0

STURM-LIOUVILLE DIFFERENCE EQUATIONS 47
and
(3.8.4) Vv(A, b' + l) + k Vv(A, 6') = 0 .
If follows from multiplying (3.8.3) by Vv(A,6')> (3.8.4) by -Vv(A0,&')> and
adding that
Vv(A0, V + l)Vv(A, V) - ^(Ao, &')Vv(A, 6' + 1) = 0,
and so
(3.8.5) W[Vv(Ao), MA)](*' + 1) = 0.
But, since ipv = 0i -f m&/02 = ^i + m^2 + (m&/ — m&)02 = V>& + (™&' — rn&)02,
(3.8.5) can be expanded into
(3.8.6) W[^(A0),^(A)](6/ + 1) =
^m6,(A)-m6(A))^6(A0^^^
-(m6i(A0) - m6(A0))(m6i(A) - m6(A))W[02(Ao),02(A)](&' + 1).
Now,
\mbi(X) - mb(X)\ < 2rh, =
b'
|Im(A)|X>(n)|02(A,n)|2
by (3.2.11), and
b'
W*(Ao), 02(A)](&' + 1) = W[M*o), 02(A)](e) + (A- A)^ti;(n)^(A0> n)02(A, n),
n=e
by (2.2.3), so that
|(m>-(A) - mhCX))W[MX0),02WW+l)\ < W^f^6^^
|Im(A)|[5>(»)|0a(X,n)|2
\ n = e
/»' \1/2/>' _ xl/2
|A - A0| I £ w(n)IV-6(Ao, «)|2 £ «;(n)|tf2(A, n)|2
+ \2f! / \"=e
|Im(A)|[5>(n)|0a(X,»)|a

48 ALOUF JIRARI
Similarly,
|(mt,(Ao)-m6(A0))^2(Ao)>^(A)](6'+l)|< W[fr(Ao),,fafl)](c)
\lm(\o)\(jrw(n)\02(\o,n)f
in = e
{»' Y'2(b' _ ^1/2
|A - Aol MT »(n)l^(Ao, n)|2 ) ( £ t»(n)|^(A, n)|2
+ \2f!
|Im(Ao)|(i>(n)|02(AO)n)|2
and
|(m»,(A0) - m6(A0))(mM(A) - mt(A))W[02(Ao), 02(A)](6' + 1)|
< W[02(\0),02(\)](e)
|Im(A)||Im(A0)| [X>(n)|02(Ao,n)|2 J I £>(n)|02(A,n)|2
|A-Ao| lJ2w(n)\e2(X0,n)\A f 5>(n)|02(A,n)|2 j
|Im(A)||Im(A0)| [£ tx,(n)|02(AO)n)|2 ] (£ w(n)\62(\,n)f
On the other hand, W[02,#2] = 0» and ^ *s possible to assume W[^&,02] = 1
since 62 and tpb are linearly independent.
All that remains to do now is use the triangle inequality in (3.8.6) to draw a
conclusion.
In the limit-circle case at 6, all the solutions are in £2(e> b; w) but all the factors
of the form (m&/ — ra&) approach 0 as b1 approaches 6. (3.8.2) then follows.
/»' \
In the limit-point case at 6, only I 2_\ iy(n)l^,&(n)|2 J converges as V —► 6,
\n=e /
leading to the same conclusion. ■
Proof of the theorem

STURM-LIOUVILLE DIFFERENCE EQUATIONS 49
If y is in Dl, then for Ao fixed, by Theorem 3.6.1, the solution of (L — Ao)y = /
satisfies
(3.8.7) y(n) - I ^ —— 1 ^(A0,n)
+ {±WU)f(J)MX0'J)\M^n)
I *-** rna — rrih '
\J=a
and
(3.8.8) *» + 1) = ( £ ^-)/0>*(Ao,i) I ^ (A „ + 1}
For this reason,
Wfo, V»(A)](n + 1) = -p(n)[y(n + 1)V-»(A, n) - y(n)^(A, n + 1)] ,
= j £ M;W(A0,i) | ^a(Ao)>^(A)](n + 1)
+ I E"0')/0>a(A0,i) ) W*(Ao),MA)](n + 1) •
As n —► 6, the first sum reduces to 0 (two terms for n = 6 — 2, one term
for n = b — 1, no term for n = b). Also, as n —► 6, the second sum exists and
\imW[M\0), ^&(A)](n+l) = 0 by the lemma. Hence limW^, ^6(A)](n+l) = 0.
n-*b n—*6

50
ALOUF JIRARI
3.9 Green's Formulas
Theorem 3.9.1
Let y and z satisfy Ly = / and Lz = g, together with the boundary conditions
at a and b.
a) If lim W[yy 4>a](a') = lim W[z, iM(a') = 0, then lim W\y,z](a') = 0.
b) lflimV[y>^](6/ + l)fl=rHmW'[z>^](6/ + l) = O^then KmW\y,J](V + l) =
b'—+b b'—*b b'—+b
0.
Proof
By Green's formula,
0 = {Ly, z) - (y, Lz) = - lim My,7](6' + 1) + lim W[y,l]{a').
b —*b a'—>a
If y and z are modified so they vanish as a' —► a or b' —► 6, then W[y, 2] = 0
at the other end. ■

STURM-LIOUVILLE DIFFERENCE EQUATIONS 51
Remark 3.9.2
The independence of boundary conditions on A can be used to replace z by ~z in
Theorem 3.9.1. The conclusion is then that
lim W\y, z){a') = 0 and lim W[yy z](b' + 1) = 0.
a'—*a b'—+b
The following shows the variations of Green's formula,
b'
(3.9.1) £ [z(n)(£y)(n) - {Tz){n)y{n)}w{n) = -W[y,J](b' + 1) + W\y,z](a'),
n=a'
in the four possible limiting cases as a! —► a and b' —► 6.
1) If the limit-point case holds at both a and 6, the right-hand side of (3.9.1)
vanishes as a1 —* a and b' —* 6, reducing (3.9.1) to
(3.9.2) £>(n)(4/)(") " (EX")*")] = °-
n = a
2) If the limit-circle case holds at a but the limit-point case holds at 6, then
lim W[j/,7](&' + 1) = 0. On the other hand, since the limit-circle case holds at
b'—*b
a, 0\ and 02 are both in £2(a, e — 1; w) for all A, in particular for A = 0. Also, 0\
and 02 can be assumed real-valued.
Finally, observe that
where
(0 -A __fp(n-1)01 (n) p(n-l)02(n)\(O -l\
\1 0J~ V -*i(n-l) -02(ni) Al 0/
/p(n-l)^(n) -0!(n-l)\
XVp(n-l)02(») -02(n-l)J '
so that
-WM(») = (W[zA](n),W[z,(>2](n)) (j „*) (%£]$)

52 ALOUF JIRARI
Therefore, (3.9.1) can be written
b
(3.9.3) ^[z(n)(£y)(n)-(^)(n)y(n)]w(n) =
n=a
/ \/0 -1\ /limW[y,0i](a')\
(^w%w),^w%<wj)(; ;) [^w[yM{al)),
where all these limits are known to exist.
3) If the limit-point case holds at a but the limit-circle case holds at 6,
lim W[2/,7](a/) = 0. An argument similar to that in the previous case shows
a'-*a
that (3.9.1) changes to
b
(3.9.4) X>(")(4/)(") " (Tz){n)y{n)}w{n) =
n=a
( \ /0 -1\ /lim^fo.W'+l)
-(Um^^ + l^lim^^ + l)) (I Q ) (jg^W'+l)
4) If the limit-circle case holds at both a and 6, then a combination of cases
2) and 3) can be used. In fact, for a < a! < e < b* < 6, (3.9.1) is equivalent to
b' _
EW(/y)(n)-(/z)(n)y(n)] =
n=a'
-W[y,l]{b' + 1) + W[y,J](e) - W[y,J){e) + W[y,l](a').
Also, note that, in this case, 6\ and 02 are in both £2(a,e— 1; u>) and £2(e, b]w).
Therefore, from (3.9.1), using (3.9.3) and (3.9.4), it follows that

STURM-LIOUVILLE DIFFERENCE EQUATIONS 53
b
(3.9.5) ]T>(")(4/)(") " (E){n)jf(n)]w{n) =
nz=a
( \ /o -i\ /limwMiKV + m
- (jm^,*](* + 1), lirn ^, *](* + 1)) (? ^) (^gf^^(y + J} j
+ (i™^^1](6' + l)>iim^^2](a0) (J V) ()§
limPy[y,^1](a')>
li^V[y,02](a')
and J
"(J o1)-
Then (3.9.5) shortens to
b
(3.9.6) £ [*(»)('»)(») " (E)(n)y(n)Mn) =
n=a
(«.(»)••*(•)•)(J -,) (SS») ■
where * represents the conjugate transpose.
3.10 Spectral Resolution
This section takes up the spectral resolution of L in the sense of linear
operators in a Hilbert space.
From classical Hilbert space theory (see for instance Ref. 9), it is known that
a self-adjoint operator L with domain Dl in a Hilbert space H can be written
as
/oo
\EL{dX) .
-OO
ElW i —oo < A < oo, is a collection of projections, which are strong limits of
the Cayley transform of L, satisfying
(1) EL{\) < EL{p) when A < /i,

54
ALOUF JIRARI
(2) El{X) —► 0, the zero operator, when A —► —oo,
(3) #l(A) —► J, the identity operator, when A —► +oo,
(4) #l(A) is continuous from above.
The section essentially follows Levinson's treatment of the differential case,
which can be found in Ref. 5.
If (yk(n))k are the eigenfunctions of the boundary value problem on (a'yb')y
then, for every k, there exists complex numbers s* and tk such that
(3.10.1) yk(n) = **0i(n,A*) +t*02(n, A*) ,
where #i(n, A*) and 02(n, Xk) are the solutions of £y = Ajty, satisfying conditions
(3.2.1).
Definition 3.10.1
P(a',b')(^) is the 2x2 matrix-valued function given by
(3.10.2) /V,M)(A) =
C y, Rl > A>0
0<At<A
- J2 Rl , A<0
V A<Afc<0
where, for every ky
d2 _ fs*S* $ktk\
k~ UkSk tktkj

STURM-LIOUVILLE DIFFERENCE EQUATIONS
55
Remark 3.10.2
From (3.10.2), it follows that P(a',6')(A) is a real-valued and right-continuous step
function, defined for A real, with jumps of amount Rl at A*. Also, P(a/,6/)W(^+) =
0.
Theorem 3.10.3
There exists a nondecreasing 2x2 matrix-valued function />(A), deRned on
(—oo, oo), sa tisfying
p(0+) = 0
and, for A > fi,
p(X) - p(n) = Jim (p(a',6')(A) " P(*',b')(l*)) •
Proof
By Theorem 2.3.8, if one uses the solutions Vv of Section 3.2 and Vv of Section
3.3, it follows that the Green's function is given by
G(a>yb')(*>j>n) =
*l>a'U)il>b'(n)
. v I ma/(A)-m6/(A)
; ol < j < n < b'
i>a>(n)ipb'(j)
raa/(A) — m&/(A)
If j = 6 and A = Aq (with Im(Ao) ^ 0), this becomes
; a' <n<j <b' .
(3.10.3)
Hi(n) = {
raa/(A)Vv(n)
™>a'{ty ~~ rn&/(A)
m6/(A)Vy(n)
I ma/(A) — rrib'(X)
e < n <bf
a' < n < e
Also, note that Hi (resp. Hi) satisfies the equation £y = Xoy (resp. Iy = X§y)
with the conditions Hx{d_- 1) + ft #i(a') = Hi(bf + 1) + ib#i(&') = 0 (resp.
77i(a' - 1) + tiHi{a') = Hi(b' + 1) + iff i(V) = 0).

56 ALOUF JIRARI
By (2.2.2),
e-1
|2
2ilm(\0)Y^\Hi(n)\w(n) =p(c-l) [i/1(e)i/1(e~l)-i/1(e-l)i/1(e)] ,
p(e-l)|m6/(A0)|2
|ma/(A0)-m&/(Ao)|2
|m6,(A0)|2
|ma/(Ao)-m6/(A0)|:
__ 22lm(ma/(A0))|m6/(A0)|2
" |ma/(A0)-m6/(A0)|2
[Vv(e)Vv(e-!)- Vv (e-l)tp a/(e)],
r[ma/(A0)-ma/(A0)],
so that
e-l
(3.10.4) £ \Hl(n)\M») = T ^-q^lm^Ao)!2
v ; 4v Im(Ao)|ma/(A0)-m6/(A0
2
Similarly,
2rfm(A0)£|#i(n)|2tx;(n) = -P(c " 1) [#i(<tfffi(e " 1) " #i(« " l)JaTi(c)] i
n = e
\mat(Ao)-mbi{Ao)\z
-|ma/(A0)|2
[mbi{Xo) - m6/(A0)J,
|ma/(Ao)-m6/(A0)|2
—2iIm(m&/(A0))|ma/(A0)|
|ma/(A0)- m&/(A0)|2
so that,
(3.10.5) f>l(«)|M«) = ^(,mt'("0\)|ma,(A°)|2M2
v } 4r Im(Ao)|ma/(Ao)-m6/(Ao)|2

STURM-LIOUVILLE DIFFERENCE EQUATIONS
Adding (3.10.4) and (3.10.5) then gives
E\u <mi3.„<m - Im(*MAo))|"H'(Ao)l2 - Im(mj,<(Ao))|ma<(Ao)|2
_ ,^l(n)i ^ ~ Im(A)|ma-(A0)-m6,(Ao)|' •
57
Im(A0)
Im
ma/(Ao)m&/(Ao)m6/(Ao) —ma/(Ao)ma/(Ao)mfc/(Ao)
Im(Ao)
Im
(ma/(A0) - mb'(X0))(mat(X0) - ma/(A0))
mai (\o)m,bi (\q)
™>a'(^o) — ^*6'(^o) J
or equivalently,
(3.10.6)
^|J1(n)|Vn) = ?-jrrIm
ma/(Ao)mfe/(Ao) 1
™>a'^o) — ^6;(Ao)j
Im(Ao)
Let us now compute the Fourier coefficients of H\, given by
b'
2^i7i(n)2/fc(n)ii;(n) , for every k .
On one hand, by (2.2.2),
e-1
(A*-A0)£ffi(n)y*(n)^
n=a'
sjfem&/(Ao)ma/(Ao)—tffcm&/(Ao)
^a;(A))—m&/(Ao)
by (3.10.1) and (3.10.3). On the other hand,
(A* - A0)5^iTi(n)y*(n)ti;(n) = -p(e - lfty^H^e - 1) - y*(e - l)#i(e)]
_ «sjbma/(Ao)m5/(Ao) — <jbma/(Ao)
ma/(A0) - rrib'(Xo)
Therefore, for every ky
b'
Y^ Hi(n)yk(n)w(n)
tk
A* — A0

58
ALOUF JIRARI
and, by Parseval's equality,
b'
(3.10.7)
\hf
From (3.10.6) and (3.10.7), it follows that
oo N2 Im
El1*! _
t=1|At-A0p
m<t' (Ao)mj/ (Ao)
[ m0/(Ao) — mj'(Ao)
which can be written
(3.10.8)
J-oo |A-
Im
Im(Ao)
mai (Xo)mi,i (Xq)
00 <**$,»*)(*) _ [ mfl/(A0) - m»*(A0)]
A0|2
Im(Ao)
where p?%, 6/\(A) is the lower right component of P(a',6')(A) defined in (3.10.2).
By similar argument, if we set
H2(n) = <
Vv(™)
^a;(Ao) — m&/(Ao)
Vv(rc)
then
and
so that
(3.10.9)
e-l
K ma/(Ao) — m&/(Ao)
Im(rfv(A0))
; 6 < n < V
; a1 < n < e ,
^JH2(n)\ W{n) - Im(Ao)|ma/(Ao)_mt/(Ao)|2
Eiw ^M2,„/^\ - -Im(m£7(Ao))
\H2(n)\ «,(„) - Im(Ao)|ma/(Ao)_m6/(Ao)|2
E I rr / \|2 / \ Lma,(^o) — ^6;(Ao)J
|tf2(n)| «,(«) = ±-±-r .
n=a'
For the Fourier coefficients of H^
e-l
Im(Ao)
(A* - Ao) 2^ #2(n)y*(n)ti;(n) = a , m (\ \
~, ma'(Ao) - m&/(A0)

and
so that,
STURM-LIOUVILLE DIFFERENCE EQUATIONS
(A* - A0)£>2(n)yt(„Mn) = J\~?"ffi>
5^ff2(n)y*(n)ti;(n) = ** ,
59
A* — A0
and, by Parseval's equality,
(3.10.10)
oo i |2
From (3.10.9) and (3.10.10), it follows that
Im
y M2
i^lAit-Aol2
[ma/(Ao) — mj,/(Ao)J
Im(Ao)
or equivalently,
(3.10.11)
L
Im
«, |A-Ao|2 Im(Ao)
raa/(Ao) — m&/(Ao)
where pi?fe/\(A) is the upper left component of P(a',b')(^) given by (3.10.2).
Finally, analogous calculations show that
e-l
2iIm(A0)^^i(n)^2(n)tx;(n) =
roa/(Ao)m&/(Ao) — ma/(Ao)m&/(Ao)
|ma/(Ao)-m6/(A0)|2
and
o-t /\ \V^lw \Tw \ / \ rna'(Xo)mbi(X0) - ma/(A0)m&/(Ao)
2tIm(A0)2^#i(n)#2(n)ti;(n) = v y v ^
*-? |ma/(A0) - mb>(\o)r
so that,
Im
Eur \TTr \ ( \ L |roa'(A0) - m&'(A0)|2J
ma/(Ao)m6/(Ao)
Im(Ao)

60
ALOUF JIRARI
which is equivalent to
1
J2Hi(n)lh(n)w(n) - ^^l^W-^MP
raa/(Ao) + rrib'(Xo)
Im(Ao)
In addition, a combination of Parseval's equality and the polarization identity
(see Ref. 10) leads to
Sktk
Therefore,
that is,
(3.10.12)
^^(n)^(nMn) = ^_^.
fma/(Ao) + mfe/(Ao)
£IA*-A0I2
-Im
ma'(\o) — mt,i(\Q)\
f
J — (
00 <»"',>')(*) _ 2
Jim
Im(A0)
ma'(X0) + mt,'(X0)
ma'(^o) — rat'(Ap)J
|A-A0|
Im(Ao)
where pP, j,;\(A) is the lower left component of P(a',b')(ty-
Let
(340.13) «<.'«M=(£&i £|t)
where
Mn(X0) = Im
^21 (Ao) = Im
M22(A0) = Im
1
ma'(\o) - rnb'(\0)\
1 ma'(Ao) + m&/(A0)
[2 ma/(Ao) — ra&/(Ao)J
ma/(A0)m6/(A0)
= M12(A0)
wv(Ao) — m6/(Ao)J

STURM-LIOUVILLE DIFFERENCE EQUATIONS 61
Equations (3.10.8), (3.10.11), and (3.10.12) can then be combined into
{3101) 7-co |A-A0p " Im(Ao) •
By choosing Ao = i, since A is real, it follows that
17-co |A-Ao|» I " LL, A' + l I - lM«".>')(0| < * .
where K is a constant. Now, choose /i > 0 and let A be in (—/i,/i). Then,
Therefore,
P(a>,b')(») ~ P(a',b')(-») < K(l + /i2)
and, since P(a',b') 1S nondecreasing with P(a'tb')(Q+) = 0,
/V,&')(A) " ^(«'.»')("^) - ^C1 + ^)
and
P(a',»')(A)</f(1 + ^2).
This shows that P(a',6')(A) is uniformly bounded on compact subsets of the
real line. Helly's first convergence theorem (see Ref. 10) guarantees the existence
of a subsequence of p matrices, which converges weakly, that is, at all points of
continuity, to p(X) with the desired properties. ■
Theorem 3.10.4
Iff is in £2(a, b; w), there is a function G(X) in L2(—oo, oo) with inner product
and norm, respectively,
{G,H)P= j" H*dpG,
/OO
G*dpG,
■oo
such that, if
^) = g(a)-x;(^;a> )/(»h»),

62
ALOUF JIRARI
then
and
/oo
E*dpE = 0 ,
-co
& rOO
*£\f(n)\2w(n) = G*dpG
n-n J-OO
Proof
1) Let / be in Dl. Also, let / and A/ vanish at a and 6 if a and b are finite,
and vanish for n sufficiently large if a or 6 are infinite. If {a'yV) is sufficiently
near (a, 6) and Lf = h, then, by Parseval's equality,
£|(l/)(»)I>»('0 = D*m|'»m
= £
* = 1
S(fli(n>A*)ltf2(n,A*))&(n)u;(n)
#*
But
£ (%$) "<">"<"> = £ («&$) ivi"<"»a«"»]+ ""«">!
which equals
b' b
[ftt:^)^iH..j?pMa/MKU:A4)+?;<<X;tO/(")
or equivalently
hK^^Hl^^'K^^^t''^^)^

STURM-LIOUVILLE DIFFERENCE EQUATIONS
63
Therefore,
b'
i:.(^)^-'-i:.^(^!)]+«<-<^)H
A*
A*)
A*)
/(n)tx;(n),
where
and so,
= A* G(a/i6/)(Ajb) ,
G(^)(ao=e(£(;;J:))/(bHb)'
6 *oo
(3.10.15) ^|(L/)(n)|2«;(n)= / A2G^a,|60(A)<ip(a,,60(A)G(a,,60(A)
Let G(Xk) = £ (£[£ J*!)/(n)u;(n). Then, for JV > 0,
0<
/—N aoo
+ /
-oo ./iV
G'(A)dp(a,,6,)(A)G(A) ,
- W2
AT /.oo
/ +/
./-oo JN
A2G*(A)<tya*|6*)(A)G(A)
1 f°°
<-^j oo^G*(X)dp(al,bl)(\)G(\)
Since, Parseval's equality applied to / reads
6
/oo
G'(A)<f/,(a,,,0(A)G(A) ,
-oo

64 ALOUF JIRARI
if follows that
0 < £|/(n)|2™(n) " / G*(\)dP{a,y)(\)G(\)
n=a J~N
1 r°°
-^/ooA2G*(A)^(a''4')(A)G,(A)
so that, by (3.10.15),
\J2\f(n)\2w(n)- I G*{X)dpia,,bl){\)G{\)\ < ^f^\{Lf){n)\2w{n) .
Now, let (a', V) —► (a, 6). Using Theorem 3.10.3 and Helly's second
convergence theorem (see Ref. 10), it follows that
J2\f(n)\2w(n)~ f~NG*(X)dp(X)G(X)\ < -L]T|(L/)(„)|2«,(n) ,
which, as N —► oo, shows that
6
/oo
G*(\)dp(\)G(\) .
■°°
2) Let / be such that / and A/ vanish at a and 6. There exists a sequence
(fj)fLi m £*L; with /j and Afj vanishing at a and b for all j, such that
6
y-+oo
lim£|(/i-/)(n)|2ti;(n)=0.
Also, by Parseval's equality,
6
where
/oo
(Gj-Gky(X)dp(X)(Gj~Gk)(X),

STURM-LIOUVILLE DIFFERENCE EQUATIONS
65
and
*■«-£(!&: JO «■>■*■>■
Since lim fj = /, (Gj)j^zl form a Cauchy sequence in L2{—00,00). Therefore,
j—+oo
there is a G in £?(—00, 00) such that lim Gj = G. Moreover,
o,w-i (#:$) «.h-)|=|e (#:$) w -/x-w-
£|(/j-/)(n)|2*(n)
<
EW(n,A) + «l(n,A))«i(n)
by Schwartz's inequality. Hence,
and so
we (s&jd «»)»(»)•
Thus, if / vanishes at a and 6,
£|/(n)|2™(n) = .lim £|/;(n)|2ti,(n)
/OO
G|(A)dp(A)G,(A)
-00
/OO
G'(A)dp(A)G(A) .
-OO
3) Now, if / is arbitrary in £2(a, b; w)y define f(a',b') by
/(«',»')(»)=| 0 , otherwise,

ALOUF JIRARI
and let
n=a x ' ' n=a' x 7
Since, for (a', 6') C (a", 6"),
/OO
(Gf(a//,6//)-G(a/j6/))*(A)dp(A)(G(a//j^)-G?(a/j6/))(A) = ^ j/(n)| u;(n),
"°° rG(a",&")-«*')
(G(a',b')) is a Cauchy sequence as (a', &') —» (a, 6). This implies that there exists
a G(A) in L2(—oo,oo) such that lim ^/^(A) = G(A). Letting (a7, V)
approach (a, 6) and using (3.10.16) shows that
£|/(n)|2ui(n) = / G*(A)^(A)G(A) .
4) Finally, since (?(<,/,&/) —► G in L2p(—oo,oo),
/I[°w-E(^:a!)w»w]^)[GW-e(^;^)/(»W"
must approach 0 as (a',6') approaches (a, 6), which completes the proof of the
theorem. ■
Theorem 3.10.5
IfG(X) is the limit of ]£ f J1**' ^\ f(n)w(n) in L*(-oo,oo), then
/OO
(01(n,X),02(n,X))dp(X)G(X)
■OO
in^2(a,6;u;). T/jafc is,
6 I r
r ,lim ,Ek(n)" / {Oi(nA),02(n,\))dp(\)G(\)
w(n) = 0 .
Proof

STURM-LIOUVILLE DIFFERENCE EQUATIONS
Let /= (,i,i/) and /,(n) = J (^(n, A),02(n, A))dp(A)G(A).
If (a',6')c (a, 6), then
£ (/^(^//("M") = J2 CF70 (») f / (»i(n,X)Mn, X))dp(X)G(X)
n — nl n — nl »■•//
67
w(n),
■/
£ (f:rfP)(n)(Oi(n,\),02(n,X))w(n)
dp(X)G(X)
Likewise,
b'
/c
so that subtracting gives
£|(/-//)(«)l2««n) = /
„=a' «/(-oo,oo)-J
E (T3^) W^i(». A)> ^(n, X))w(n)
dp(X)G(X) ,
^(FTD^XlMn,*))™®
dp(X)G(X) .
Now, /J ( ^ / ' /)(/—//)(n)tx;(n) is the transform of a function in ^2(a,6;tx;),
which vanishes outside (a', 6'). Consequently, the sum from a' to V in brackets
is in L2{—oo, oo). Applying Schwartz's inequality, it follows that
52\(f - fif(n)\w(n)
[/-,w(t(«-.J0(/-/'X"W"))*(i)
* (£(K:*>)</-/')("W"))] [/__,_, G*(A)*(A)G(A)

68 ALOUF JIRARI
By Parseval's equality the first term of the product on the right-hand side is
b'
less than or equal to V^ |(/ — //)(n)| w(n), which implies that
n=a'
£ K/ " //)H|2^H ^ / <T (A)dp(A)G(A)
n=a/ ^(-00,00)-/
Let (a', &') -+ (a, 6) then J —» (—00,00). The result is that
/(n) = lim / (^(n, A), 02(n, A))<fp(A)G(A) in *2(a, 6; «,).
J—(-00,00) ,//
The following two theorems establish the intimate connection between the
matrix M, defined explicitly in the proof of Theorem 3.10.3 in terms of ma(X)
and m&(A), and the spectral matrix p(A), whose existence only is guaranteed by
the same theorem.

STURM-LIOUVILLE DIFFERENCE EQUATIONS
Theorem 3.10.6
Let p(X) be the limiting spectral matrix of Theorem 3.10.3 and let
M(A0) = lim M(a/fc/)(A0),
(3.10.17)
where M(a/&/)(Ao), the matrix given by (3.10.13). Then,
(3.10.18)
f°° dp{\) = M(Ap)
Proof
This follows immediately from (3.10.14) by letting (a', 6') approach (a, b).
Theorem 3.10.7
If Ai and A2 are reai, t/ien,
1 /*A2
p(A2) - p(Xi) = lim - / M(/i + iv)dn .
i/—0+7T JXl
Proof
(3.10.18) for Aq = /i + iV reads
dp{\) _ M(fi + iv)
|A — /i — iv\2 v
Integrating both sides with respect to /i, from Ai to A2, gives
rAa '°° dp{\) rAa
-oo(A-/i)2 + i
Now, the left-hand side is
*C£j>=$*?«=£"<>+'*»'
c[tv&?]«» -j:[j:
dp,/v
((/, - A)/i/)' + 1
dp{\)
■£h-(^)--*(^)]
But, as I/-+0+,
0 if A < Ai ,
7r if Ai < A < A2 ,
0 if A2 < A .

70
ALOUF JIRARI
Therefore,
lim / /
^-o+y.oo \jx
i/dp
lXl (A-/i)* + i/
dp(\) = *(p(\2)-p(X1)),
and finally,
p(X2)-p(Xi)
M(fi + zV)d/i
3.11 Limit-Point and Limit-Circle Tests
The purpose of this section is to present sufficient conditions for either the
limit-point or the limit-circle case to hold when the right singular end, 6, is oo.
The following theorem is the discrete analog of LeVinson's limit-point criterion
treated in Ref. 5.
Theorem 3.11.1 (Mingarelli's Limit-Point Criterion)
Let M(n) be a sequence of strictly positive numbers and assume that for all
sufficiently large n, there exists positive constants K\ and Ki such that
(3.11.1)
q(n) < KxMin)
(3.11.2)
p(n- iy'2VM(n)
M(n)!/2M(n - 1)
<K2,
and
(3.11.3)
Y, (P(n " l)M(n))~1/2 = oo.
n=N
Then V[p(n)Ay(n)] + q(n)y(n) = Aj/(n) is limit point at oo.

STURM-LIOUVILLE DIFFERENCE EQUATIONS
71
Proof
It can be found in Ref. 14, provided p(n) and — q(n) are substituted for cn and
6n, respectively.
Remark 3.11.2
Note that Mingarelli's test only treats the case where w(n) = 1. So, in order to
use the test, it is necessary to find a way of transforming a second order difference
equation of Sturm-Liouville type with w(n) ^ 1 into an equivalent one where
w(n) = 1. This is accomplished in the following theorem.
Theorem 3.11.3
V[p(n)Ay(n)] + q(n)y(n) = \w(n)y(n)
is equivalent to
V[p(n)Ay(n)] + \p(n)+p(n - 1) + w(ny^2q(n)]y(n) = Ay(n),
where
y(n) = w(n)1/2y(n),
p(n) = w(n)-^2p(n)w(n + l)"1/2 ,
and q(n) = w{n)~ll2[—p(n) — p(n — 1) + q(n)] .
Proof
From V\p(n)Ay(n)] + q(n)y(n) = \w(n)y(n), it follows that
(3.11.4) p{n)y(n+l)+p{n-l)y{n-l) = [\w(n) + p{n) + p(n - 1) - q(n)]y(n).
Now, let
y(n) = w(n)1/2y(n)y
p(n) = w(n)-^2p(n)w(n + \)~lf2 ,
and
q(n) = w(n)~1/2[-p(n) - p(n - 1) + q(n)].
Then,
y(n) = w{n)~lf2y{n),
p(n) = w(n)1/2p(n)w(n+l)1/2 ,

72 ALOUF JIRARI
and
p(n) + p(n — 1) — q(n) = — w(n)1' 2q(n).
(3.11.4) then becomes
p(n)y(n + 1) +p(n - l)y(n - 1) = [A - w(n)~1/2if(n)]y(n).
Subtracting p(n)y(n) + p(n — l)y(n) from both sides implies that
V[p(n)Ay(n)] = [A - w(n)-1/2q(n) - p(n) - p(n - l)]y(n).
Thus,
V[p(n)Ay(n)] + \p(n) + p(n - 1) + w{n)~lf2q{n)]y{n) = Ay(n). ■
The second half of this section is devoted to Atkinson's tests. Their proofs
are all available in Ref. 2. Though, it is worth mentioning that V[p(n)Ay(n)] +
q(n)y(n) = Xw(n)y(n) is transformed into a matrix equation. In fact, it is
equivalent to
(3.11.5) p(n)[y(n + 1) - y(n)] = [Xw(n) - q(n)]y(n) + p(n - l)[y(n) - y(n - 1)].
Also, if
(3.11.6) v(n) = p(n)[y(n + 1) - y(n)],
then
v(n - 1) = p(n - l)[y(n) - y(n - 1)]
and
v(n — 1)
*„) = y(„-i) + -L_i.
(3.11.5) and (3.11.6) can be combined into
(31L7) U»)J - [Xw(n)_q(n) 1+Mn)-ig(n)j Un - 1)) '
Theorem 3.11.4 (Atkinson's Limit-Circle Test)
X) 1 oo
P --— < oo, and \" ,
The proof, as can be found in Ref. 2, relies on the following lemma.
oy ou ou ^ I / M
#!>(») < oo, £|«(n)| < oo, £-7^7 < co, and ^ J„ _ L < °°. tben
n=e n=e n=e
the limit-circle case holds at oo.

STURM-LIOUVILLE DIFFERENCE EQUATIONS 73
Lemma 3.11.5
oo oo
For a fixed complex X with Im(X) # 0, ^ iu(n)|0i(n, A)|2 and ^ w(n)\02(ny A)|2
n=e n=e
converge or diverge together.
Theorem 3.11.6 (Atkinson's Limit-Point Test)
oo
If y^ w(n) = oo where N is sufficiently large and, for some real A, Xw(n) —
n=N
q(n) > 0 for n> N, then the limit-point case holds at oo.

Chapter 4
Polynomial Solutions
In this chapter, several tasks are accomplished. First, it is shown that every
second order linear difference equation can be put in self-adjoint form. Then,
necessary and sufficient conditions are given for the existence of polynomial
solutions. Finally, the orthogonality of the eigenfunctions associated with the
difference operator is established. Essentially, the treatment of sections 3 to 5 of
Ref. 11 is adopted here, with minor differences, namely in the step size and the
form of the difference equation.
4.1 Formal Self-Adjointness
Let L be the operator given by
(4.1.1) (Ly)(n) = A(n)VAy(n) + B(n)Ay(n) + C(n)y(n) .
Then z(n)(Ly)(n) =z(n)A(n)y(n + 1) - 2A(n)z(n)y(n) + A(n)z(n)y(n - 1)+
z(n)B(n)y(n + 1) — z(n)B(n)y(n) + z(n)C(n)y(n).
Adding and subtracting y(n)A(n + l)z(n + 1), y(n)A(n — l)z(n — 1), and
y(n)B(n — l)z(n — 1), it follows that
z(n)(Ly)(n) =V[z(n)A(n)Ay(n) - y(n)A(A(n)z(n)) + z(n)B(n)y(n + 1)]
+ y(n)[VA(A(n)z(n)) - V{B{n)z{n)) + C(n)z(n)].
Therefore,
z(n)(Ly)(n) - y(n)[VA(A(n)z(n)) - V(B(n)z(n)) + C(n)z(n)]
= V[z(n)A(n)Ay(n) - y(n)A(A(n)z(n)) + z(n)B(n)y(n + 1)].
74

STURM-LIOUVILLE DIFFERENCE EQUATIONS 75
Definition 4.1.1
The operator L given by
(4.1.2) (Lz)(n) = VA(i4(n)z(n)) - V(B(n)z(n)) + C(n)z(n)
is the formal adjoint operator of L.
In this case,
(4.1.3) z(n)(Ly)(n)-y(n)(Lz)(n) =
V[z(n)A(n)Ay(n)-y(n)A(A(n)z(n) + z(n)B(n)y(n + 1)].
Theorem 4.1.2 _
The formal adjoint of L is L.
Proof
By expanding (4.1.1) and (4.1.2), it is clear that the operator L given by
(4.1.4)
(Ly)(n) = (A(n) + B(n))y(n + 1) + (C(n) - 2A(n) - B(n))y(n) + A(n)y(n - 1)
has formal adjoint the operator L given by
(4.1.5) (Lz)(n) = A(n + \)z{n + 1) + (C(n) - 2A(n) - B{n))z{n)
+(A(n-l) + B(n-l))z(n-l).
Applying this to L, it is easy to see that, in turn, the formal adjoint of L, L, is
given by
(4.1.6)
(Ly)(n) = (A(n) + B(n))y(n + 1) + (C(n) - 2A(n) - B(n))y(n) + A(n)y(n - 1).
Comparing (4.1.4) to (4.1.6), it is clear that L = L. ■

76 ALOUF JIRARI
Definition 4.1.3
A second order linear difference operator L is formally self-adjoint if L = L.
Theorem 4.1.4 (Criterion for Self-Adjointness)
L is formally self-adjoint if and only if AA(n) = B(n) for every n.
Proof
From (4.1.1) and (4.1.2), L = T if and only if
A(n)VAy(n)+B(n)Ay(n)+C(n)y(n) = VA(A(n)y(n))-V(B(n)y(n))+C(n)y(n).
This is equivalent, after simplification, to
(AA(n) - B(n))y(n + 1) - (AA(n - 1) - B(n - l))y(n - 1) = 0.
But, since y is arbitrary, a necessary and sufficient condition for this to hold is
that AA(n) = B(n) for every n. ■
Remark 4.1.5
From (4.1.4) and (4.1.5), the formal adjoint of L given by
(Ly)(n) = P(n)y(n + 1) + Q(n)y(n) + R(n)y(n - 1)
is L given by
(Lz)(n) = P'(n)z(n + 1) + Q'(n)z(n) + R!(n)z(n - 1),
where P'(n) = R(n + 1), 0'(n) = Q(n), and R'(n) = P(n - 1). in this case, the
formal self-adjoint ness criterion is that P(n) = R(n + 1) for every n.
Theorem 4.1.6
Every second-order linear difference operator can be put in self-adjoint form.
Proof
Observe that any second order linear difference operator of the form V[p(n)At/(n)]+
q(n)y(n) is self-adjoint. This is because
V\p(n)Ay(n)]+q(n)y(n) = p(n)y(n+l)+[q(n)-p(n)-p(n-l)]y(n)+p(n-l)y(n-l).

STURM-LIOUVILLE DIFFERENCE EQUATIONS 77
It is therefore clear that the self-adjointness criterion of Remark 4.1.5 is
satisfied. The question now is to find a function w(n) such that
(4.1.7) w(n)A(n)VAy(n) + w(n)B(n)Ay(n) = V\p(n)Ay(n)].
Since y is arbitrary, this would mean by identification that
{
w(n)A(n) + w(n)B(n) = p(n)
w(n)A(n) = p(n — 1) ,
or equivalently
(4.1.8)
{w(n)A(n) = p(n —
w(n)B(n) = Vp(n
-i)
)•
Therefore, w(n+l)A(n-{-l) — w(n)A(n) = p(n)—p(n — l) = Vp(n) = w(n)B(n),
or
w(n + 1) A(n) + JB(n)
This implies that
and so
w(n) A(n + 1)
ttKj + 1) = yrA(j) + B(j)
i=o
4.2 Polynomial Solutions
First, observe that the second-order linear difference operator L of (4.1.1)
should be of the form
(4.2.1) (Ly)(n) = (an2 + bn + c)VAy(n) + (dn + /)Ay(n) ,
where a, 6, c, d, and / are constants, for the equation Ly = Xky to have
polynomial solutions of degree k, for every k > 0. In fact, for & = 0, some nonzero
constant, say a, is a solution. This implies that (C(n) — Xo)a = 0 and so C(n) = Ao.
Without loss of generality, it is possible to take C(n) = 0. Otherwise, substitute
C(n) - Ao for C(n).

78
ALOUF JIRARI
Similarly, for k = 1, some an + /3 is a solution. By substitution, it follows
that B(n)a — Ai (an + /?) = 0, which forces B(ri) to be of the form dn + f.
Finally, for k = 2, some an2+/3n+y is a solution. This implies that 2A(n)a +
(dn + f)(2an + a + /?) — A2(an2 + /?n + 7) = 0. It is therefore necessary that
A(n) take the form an2 + bn + c.
The goal now is to set necessary and sufficient conditions for
(4.2.2) (an2 + bn + c)VAy(n) + (dn + f)Ay(n) - Xky(n) = 0
to have polynomial solutions 2/*(n) of degree k. This is achieved in the following
theorem.
Theorem 4.2.1
i A,
Let 0(m) = m! — +
l(m — 2)! (m—1)! m!J
If (4.2.2) has a polynomial solution of degree k, then 9(k) = 0.
Conversely, if m is the smallest positive integer such that 6(m) = 0, then
(4.2.2) has a polynomial solution of degree m and there is no solution of degree
less than m.
Proof
k
Assume that (4.2.2) has a polynomial solution 2/*(n) = ^bjn*. Then, simple
i=o
substitution in (4.2.2) shows that

STURM-LIOUVILLE DIFFERENCE EQUATIONS 79
(an2-+bnr+c) x
k ,. Jb—1
-2bknk -26^-1 n*"1 - ... - 26xn -260
Jb ,. Jb —1
+»*E7i^H)w^^«E ji§^H)w^»/+ • • • + Mn-i)+>o
+(<fn+/)x
^Ti^F' +^E7j7tn)!»i +... + 61(n+l)+6o
f^3\{k-3)\ Uj-(k~l-j)-
-A,
— bkTl —bk-lTl — ... — &i7l —6o
&*n* +&Jb-m*-1 +... + 6m +&0
equals zero. In particular, the coefficient of nk must equal zero; that is,
abkjr^y.+ dbkJk^ly. ~ Xkh = 0,or bkkWW = o-
Since 6* ^ 0, this is equivalent to 0(&) = 0.
Conversely, it is not hard to see that the coefficients of n*"1, n*~2,... in the
left-hand side of (4.2.2) are ,respectively, bk-i0(k — 1) + cjb~i, &jb_2#(& — 2) +
cjb-2j--j where the Cj's are functions of &*, a, 6, c, d, /, A*, and m. For
instance, the coefficient of n*""1 is
6fcfc(F^2)!+a6t-1(irr3)i+<i6t(Jfe - 2)!2!+d6fc-1(Fr2)t+/6fc(ifezli)!"At6fc"1'
which is equal to bk-iO(k~l)+Ck-i where ct-i = bkk\
b d f
;+„ .».... + ■
L(jfe-l)! (fc-2)!2! (ifc-l)!j

80
ALOUF JIRARI
Now, let k be the smallest positive integer such that 0(m) = 0. Then, 0(ra-r) ^
0
for r = 1,2,..., k. By setting the coefficients bk-iO(k—l)+cjb-i, bk-20(k—2)+cjb_2,...
equal to zero, it is possible to determine (up to the coefficient &*, which could
be set without loss of generality equal to 1) the coefficients bj. This shows that
there is at least one polynomial solution and one of the solutions is of degree m.
But since 0 must vanish for the degree of that polynomial, it follows that there
is no solution of degree less than m. ■
Remark 4.2.2
(1) From Theorem 4.2.1, (4.2.2) has a polynomial solution of degree k if and
only if k is the smallest positive integer such that Xk = ak(k — 1) + dk.
(2) When d ^ —ma, where m is a positive integer, for each positive integer k,
there exists a polynomial solution of degree k for (4.2.2) where A* = ak(k — 1) +
dk.
However, when d = —ma, it may happen that Ami = Am2 for mi ^r«2. In fact,
when
d = —(mi + rri2 — l)a,
(mi - m2)[a(mi + m2) - a + d\ = 0,
and
a mi(mi — 1) + mid = a m2(m2 — 1) + m^d.
4.3 Orthogonality of Eigenfunctions
Theorem 4.3.1
If ni andri2, with ni < n^, are two real roots of p(n) = 0, then the eigenfunctions
yi of L, defined by (4.2.1), are orthogonal in the Hilbert space £2(ni + 1, n^', w),
provided yi(n) and Ayi(n) are finite for ni and 712, for every i.

STURM-LIOUVILLE DIFFERENCE EQUATIONS 81
Proof
Consider the self-adjoint form of equation (4.2.2). By Theorem 4.1.6, it is
w(n)(a n2 + bn + c)VAy(n) + w(n)(dn + f)Ay(n) — Xw(n)y(n) = 0,
or
(4.3.1) V(p(n)Ay(n)) = Xw(n)y(n)
where, by (4.1.8) and (4.1.9),
p(n) = (a(n + l)2 + b(n + 1) + c)w(n + 1),
Now, as pointed out in the beginning of the proof of Theorem 2.2.3, if A,-
and Aj (i ^ j) are two eigenvalues, with corresponding eigenfunctions yi and yj,
respectively, then
Ajp(n - l)[yj(n - l)Vifc(n) - yi(n - lJVy^n)] J = (A,- - AyMnJy.-fnJy^n) .
The summation from (ni + 1) to n2, where n\ < n2, gives
(4.3.3) p(n2)[2/i(n2)V2/,(n2 + 1) - 2/,(n2)V2/i(n2 + 1)]
n2
-P(ni)[%(ni)v2/*(ni + !) - 2/*(^i)V%(ni + 1)] = (A,- - A,) ]T w(n)yi(n)yj(n).
n=ni + l
If, p(ni) = p(n2) = 0, if 2/,(ni), 2/,(n2), %(ni), %(n2), A2/,(ni), A2/,(n2), Ay^ni),
n2
and Ayj(n2) are finite, then (4.3.3) implies that jT^ w(n)y%(n)yj(n) = 0>
n=m + l
which shows that y,- and yj are orthogonal in ^2(ni + 1, n2; iu). ■
Remark 4.3.2
Under the same finiteness conditions on the eigenfunctions and their differences
at ni and n2, (4.3.2) shows that the conclusion of Theorem 4.3.1 follows if, in
particular, n\ < n2 are two real roots of a(n + l)2 + b(n + 1) + c = 0.

82
ALOUF JIRARI
4.4 Eigenfunction Expansion
In this closing section, it is shown that for polynomial solutions of
L(y) = Ay, the rather abstract eigenfunction expansion given for y by
Theorem 3.10.4 is necessarily a series expansion.
Theorem 4.4.1
The polynomial eigenfunctions of L, (2/jb(n)), k = 0,..., b — a, are complete in
£2(ayb;w).
Proof
It consists of showing that if / is a function in £2(a,b; w) such that (/, 2/a?) = 0
for every fc = 0,...,& — a, then / = 0. But since for every fc, there exists a*,
i
k = 0,..., b — a, such that n-7 = yj (*kyk(n), j = 0,..., b — a, it suffices to show
that
(/(n), nj) = 0 for j = 0,..., b - a, implies that / = 0 .
This will be accomplished by the Fourier transform technique that Keener used
in Ref. 8 for the completeness of the Hermite and the Laguerre polynomials.
f°° .
The function F(z) = I etznd(wfp)(n) is the Fourier-Stieltjes transform of
J—oo
wfp. Since z = Re (z) + ilm(2r), it can also be written
/oo
eiRe (*)»e-Im(*)"d(ti;/p)(n) ,
-OO
= XXRe (z)ne~lm(-z)nw(n)f(n) ,
and so
Iwl = (Ee~2Im(i)n»2(»)l/(")l2
1/2
|2 \
< ^-N.).^) 2>(»)|/(»)|a
|2
Now, y^tx;(n)|/(n) | <oo since / is in t2(a,b;w). Also, if 6 is finite, £V2Im(*>nu>(n)<
n=a n=a
OO
for z bounded, making F(z) an entire function.

STURM-LIOUVILLE DIFFERENCE EQUATIONS 83
However, as Chapter 6 will show, there are two cases where b is infinite. The
generalized Laguerre polynomials in £2 [ A, oo; e"~7n——. — 1, where j>
V T(n-A+1) J
0,
a > — 1, and A is an integer, and the Charlier polynomials in £2 [q, oo; —= J,
V pnT(n-q + iy
where p > 1 and q is an integer.
b
In the first case, Yj e~2Im^^ntx;(n) < oo, and therefore F(z) is an analytic
function, in the half-plane Im(z) > — — which contains 0.
oo
In the second case, yj e~2Im^nw(n) < oo, and F(z) is an analytic function,
n=q
lojr n
in the half-plane Im(z) > — which also contains 0.
This shows that, in all cases,
j=o J'
But, for all j,
/oo
njd(wfp)(n) ,
-OO
-OO
6
= iJ' yjnJtx;(n)/(n)
since (/(n),n') = 0. Therefore, F{z) = 0.
By Ref. 3, F determines wfp up to a constant, and so wfp = 0. Thus,
/ = 0. ■

84 ALOUF JIRARI
Corollary 4.4.2
Let (yk(n)) he the normalized polynomial eigenfunctions ofL, given by Definition
3.5.3. For f in £2(a, b; w),
b-a
f(n) = *52(f(n)>yk(n))yk(n)
and (ParsevaVs equality)
ii/ii2 = £l</(»)-y*(«)>l2-
*=0

Chapter 5
Polynomial Examples
In this chapter, all orthogonal polynomial sets which satisfy a second order
difference equation of the form
(5.0.1) (an2 + bn + c)VAy(n) + (dn + f)Ay(n) - [ak(k - 1) + dk]y(n) = 0
are classified. There are four categories, each of which will be extensively
discussed in turn.
5.1 Classification
By setting h = 1 throughout Ref. 11, one can see that Lancaster proved the
following:
(5.1.1) (An2 + Bn + C)A2y(n) + (Dn + F)Ay(n) + fiy(n + 1) = 0 ,
where \i = k(k — 1)A + kD, has the polynomial solution of degree k
(5.1.2)
Vk(n) =
n-l
n
A? + {B-D)j + C-F
}J-0A(j-l)* + B(j-l) + C^
A(j - l)2 + B(j -1) + C
n-l
n
jii ^0 - *)2 + (B - C)(j - t) + C - f
Also, if T denotes the usual Gamma function, if a\ya2 are the roots of A(j —
1)2 + B(j -1) + C = 0, and if /?i,/?2 are the roots of Aj2 + (B-D)j + C-F = 0,
then there are four possible situations:
1) If A j 0, (5.1.2) reduces to
(5.1.3) yk(n) =
r(n^Qr(n-ft)^
r(n - ai)r(n - a2)
(n - p1 - 1).. .(n - A - t)(n - ft - 1) .. .(n - fo - k)
T(n — a\)T(n — c*2)]
r(n-/?!)r(n-/?2)j •
2) If A = 0, B j 0, and fl - D j 0. (5.1.2) becomes
85

86 ALOUF JIRARI
3) If A = 0. B = 0, D ^ 0 and C 1 0. it follows from (5.1.2) that
(5.1.5) yk(n)=T(n-&)(-£-} x
Afc[(n_A_1)...(n_A_fc)_l_(:^)n
4) If A = 0. B d 0. B - D = 0, and C - F £ 0. then (5.1.2) yields
<516> »<•> = rjsrhsy (^9"A* [r("" "'> (sh)"
Remark 5.1.1
The expanded form of (5.1.1) is
[An2 + Bn + C]y(n + 2)
(5.1.7) +[-2An2 - (2B - D)n - (2C - F) + Ai]2/(n + 1)
+[An2 + (fl - D)n + (C - F)]y(n) = 0.
On the other hand, the expanded form of (5.0.1), where (n + 1) is substituted
for n is
[an2 + (6 + d + 2a)n + (c + / + a + 6 + d)]2/(n + 2)
(5.1.8) +[-2an2 - (26 + d + 4a)n - (2c + / + 2a + 26 + d) - X]y(n + 1)
+[an2 + (2a + 6)n + (c + a + b)]y(n) = 0.
Then, by simple identification, it follows that (5.1.7) and (5.1.8) are equivalent
if and only if

STURM-LIOUVILLE DIFFERENCE EQUATIONS
(5.1.9)
( A = a
B = b+d+2a
C=c+f+a+b+d
( D = d
F = f + d
This means that, under conditions (5.1.9), a solution y(n) in Lancaster's setting
is a shifted solution y(n + 1) in this present setting. Therefore, it must be true
that (5.0.1) has the polynomial solution of degree k
(5.1.10) yk(n) =
ll/ifi-l
QJ2 + (& + 2q) J + (c + a + b)
f}0<3-l)2 + (6+<f+2a)(j-l) + (c+f+a+b+d)
nff«ti-l)2 + (t+d+2a)(j-l) + (c+/+a+6+d)
M a^ ■ *)2+(6+2a)(j - ^)+(c+a+6)
In addition, thanks to (5.1.9), cases 1) to 4) now read:
1) a#0
2) a = 0, b + d + 2a ^ 0, and b + d + 2a - d # 0
3) a = 0, 6 + d + 2a = 0, d # 0, andc + / + a + 6 + d#0
4) a = 0, 6 + d + 2a ^ 0, 6+d + 2a-d = 0, andc + / + a + 6 + d-/-d^0,
or equivalently
1) a#0
2) a = 0, 6 ^ 0, and 6 + d # 0
3) a = 0, 6 ^ 0, b + d = 0, and c + / # 0
4) a = 0, b = 0, d ^ 0, and c # 0.
But is this classification complete?
The tree in Figure 5.1 explores the possible combinations of cases involving the
coefficients a, 6, d, 6 + d, c, and c +/. The right-hand side shows what situation
each combination leads to.

ALOUF JIRARI
0
1)
^^e + / = 0 Ml
- M2
\ ^e + / = 0 MZ
0 ^^c + /^o M4
'i + d^O Impossible
,6 + <4 = 0 Impassible
e + / = 0 Af5
Af6
4)
k + d = 0 Impossible
b + d^O 2)
c + / = 0 Ml
X ^c + /^0 3)
rb + d = 0
^e + / = 0 M8
0 ^c + /#0 3)
'b + d^O 2)
Figure 5.1: Classification T>ee

STURM-LIOUVILLE DIFFERENCE EQUATIONS 89
Ml through M8 are miscellaneous cases which must be examined separately.
Ml: This case is trivial since all the coefficients are zero.
M2: fAy(n) = 0 with / # 0 imply that Ay(n) = 0. Therefore, y(n) is
constant.
M3: cVAy(n) —cAy(n) = 0 with c ^ 0 imply that Ay(n — 1) = 0. Therefore,
y(n) is constant.
M4: cVAy(n) + fAy(n) = 0 with c + / ^ 0. Consequently,
(c + f)Ay(n) - cAy(n - 1) = 0,
Ay(n) __ c
Ay(n-1) " c + f
" Ay(j) /c\"
i=iiAt/(i-i) \,c+/; '
At/(n) = At/(0)(^)n,
j=o j=o \C^JS
so that
n-l
y(n) = y(0) + Aj O^-i^y.
M5: dnAy(n) — dky(n) = 0 with d 0. Therefore,
nAy(n) — &y(n) = 0,
ny(n + 1) - (n + Ar)y(n) = 0,
y(n + 1) __ n + k
y(n) n
n-l / . , .v n-l
so that
j/(n)=KG) n (L^) ■

90 ALOUF JIRARI
M6: (dn + f)Ay(n) - dk y(n) = 0. Then,
(dn + f)y(n + 1) - [d(n + k) + f]y(n) = 0,
y(n + 1) _ d(n + k) + f
y(n) dn + f '
so that
n-l
y(n) = y(o)H
j=0
M7: bnVAy(n) — bnAy(n) — dky(n) = 0. As a result,
d(j + k) + f
dj + f
—bnAy(n — 1) — dky(n) = 0, (bn + dk)y(n) = bn y(n — 1),
t/(n) _ bn
y(n — 1) 6n + dk'
so that
*>-*>n(ij&»)
M8: (bn + c)VAy(n) - (bn + c)At/(n) + 6Ary(n) = 0. So,
-(bn + c)Ay(n - 1) + 6Ary(n) = 0,
(fen + c)t/(n - 1) = [b(n - k) + c]t/(n),
S/(w)
so that
6n + c
y(n - 1) " 6(n-jb) + c'
6j + c
2/(") = 2/(0)11
&(i - *) + cj'
Thus, the classification is complete and the solution of (5.0.1) given by (5.1.10)
becomes:
CASE 1: If a 1 0,
(n-A -2) ... (n-ft -*-l)(n-/?2 -2) ... (n-ft -4-1)
r(n-Qi-l)r(n-a2-l)l
r(n-/?i-l)r(n-^2-l)J

STURM-LIOUVILLE DIFFERENCE EQUATIONS
91
where <*i, <*2 are the roots of aj2 + (6 + d)j + c + / = 0, and /?i, /?2 are the roots
ofa(j'+l)2 + 60'+l) + c = 0.
CASE 2: If a = 0. b ± 0, and 6 + d 4- 0.
(5.1.12)
. r(n-ft-l) / 6 N""1
^)=r(n-ai-i)y x
f « o^ /■ a l ^lXn-ai-1) /6 + d\
n-i
where «i is the root of (6 + d)(j — 1) + (c + / + b + d) = 0, and /?i is the root of
bj + (c + 6) = 0.
CASE 3: If a = 0 6 ^ 0, 6 + rf = 0. and c + f ^ 0.
(5.1.13) W(„)=r(n-/31-1)(^) x
n-1
where /?i is the root of bj + (c + 6) = 0.
CASE 4: If a = 0, 6 = 0, d 4 0, and c £ 0,
<5I14> »W■ rc.-L.-i) (3)""a* fr(""*"])(£f'
where c*i is the root of d(j — l) + c + / + d = 0.

92 ALOUF JIRARI
5.2 Recurrence Relations
The procedure used to recover the recurrence relation that each one of the
four polynomial sets of solutions given by (5.1.11) through (5.1.14) satisfies is
the one used by Lancaster in Ref. 11. The idea is to find A(k), B(k), C(k) and
D(k) such that
(5.2.1) A(k)yk+2(n) + [B(k)n + C(k)]yk+1(n) + D(k)yk(n) = 0,
for every k and every n. This is accomplished by substituting yk(n) in (5.2.1)
and "summing" k times according to the formula
(5.2.2) A-k[u(n)v(n)] = u(n-k)A-kv(n)-kAu(n-k + l)A-^k^v(n) + ... .
After simplification, this leads to a polynomial in n which has to be identically
zero. A(k)y B(k)y C(k)y and D(k) are determined by setting the coefficients of
the different powers of n involved equal to zero. Carrying out all the tedious
calculations involved in the method outlined leads to the following derivations:
CASE 1: If a 1 0:
^>=-(H(;+2*)'
B(k) = (^+2k + l\(^ + 2k + 2\ (1 + 2k) ,
C(k) = 2(fc + 1)^ + (6*2 + 8Jfc + 2)~ + ^l + ^ + (3Jfc2 - 2k - 2)^
(JL (JL (JL (JL (JL
+ Uk3 + 6*2 + 2*)- + (2* - 1)% + (2k3 - 9k2 - Ilk - 2)- - (4* + 2)^
a a2 a a
- (Sk3 + 12ib2 + 4k) ,
and
D(k) = (k + l)C-£ -(k + 1)^ - k(k + 1)*^ - k(k + 1)^
a* a* a* a*
+ (6k2 +8k + 2)^- + 2k(k + 1)^- - *(3*2 + 5* + 2)^- + k2(k + 1)^
a3 a3 a3 a3
+ (k + 1)^ - 2(k + l)2^- - *(3*2 + 5* + 2)^ + 2Jfc2(2fc + 1)(* + 1) J
+ 4Jfc(3*2 + 5* + 2)^ + 2Jfc(3*2 + 5* + 2)% - 2k2(k + l)2^
a2 a1 a1
- 2k2(k + 1)24 + 2(* + 1)24 + *3(5* + 4)(* + 1)-
+ 8k2(k + l)2- + 4fc2(* + l)2^ + 2*4(ifc + l)2 .
a a

STURM-LIOUVILLE DIFFERENCE EQUATIONS 93
CASE 2: If a = 0, 6 ^ 0, and 6 + d # 0,
A(k) = -b ,
B(k) = d ,
C(k) = f + kd + 2{k + l)b,
and
D(k) = (k+ l)~ - (k + 1)/ - k(k + 1)(6 + d) .
CASE 3: If q = 0. 6^0. 6 + d=0. and c+ f ^ 0.
A(k) = B(k) = 1 ,
C(*) = -(* + 2)-|,
and
CASE 4: If a = 0, 6 = 0, d ^ 0 and c £ 0,
A(*) = -c ,
B(Jb) = <f ,
C(k) = f + kd,
and
£>(i) = (Jb + l)d .
5.3 Weight Functions and Self-Adjoint Forms
Theorem 4.1.6 is used to put each equation
A(n)VAy(n) + B(n)Ay(n) = C(n)y(n)
in the self-adjoint form
V[p(n)Ay(n)] = iu(n)C(n)y(n) ,
where the weight function
-"-AW
by (4.1.9), and
p(n) = to(n)[i4(n) + B(n)],
by (4.1.8).
CASE 1: If a 4 0. by (5.0.1),
(5.3.1) (an2 + bn + c)VAy(n) + (dn + /)Ay(n) = [a*(Jfc - 1) + kd]y(n).

94
ALOUF JIRARI
Therefore,
n-l
a/2 + 6; + c + <# + /
n-l
(i - Q?i)(i - ^2)
n^
j1=!10«O +1)1+ *(* + !) + « fJiU-Mi-M
where ari,c*2,/?i, and /?2 are as defined in (5.1.11), and
p(n) =
T(n — ai)T(n — 0C2) r 9 /» i\ / r\i
r(n-/?!)r(n-/?-2)1
— fl(n ~ ai)(n — <^2)r(n — ai)r(n — a2)
" r(n-A)r(n-/?2) '
__ aT(n + 1 - ai)r(n + 1 - a2)
r(n-/?i)r(n-&)
so that the self-adjoint form of (5.3.1) is
[ar(n + 1 - ai)r(n + 1 - a2)
(5.3.2)
V
r(n-/?i)r(n-/fc)
Ay(n)
r l/l i\ . l^i T(n-ai)r(n-a2) , .
CASE 2: If q = 0,6^0, andfc + ci ^ 0. by (5.0.1),
(5.3.3) (fen + c)VAy(n) + (dn + f)Ay(n) = fccfy(n).
Therefore,
n-l .
i=o
where ai and /?i are as defined in (5.1.12), and
„(„\ - ri(b + d)j+(c + f) _ (b + d\n r(n-ttl)
w"Ai V + (c + 6) "UJ r(n-A)'
/6 + <i\nr(n-ai).. ...
(6 + d)"+1 r(n + 1 - ai)
6" T(n-/?i) '

STURM-LIOUVILLE DIFFERENCE EQUATIONS
95
so that the self-adjoint form of (5.3.3) is
(5.3.4) vr(* + *)"+1II
n+l-ari)A J A+<f\"r(n-ai)
6" r(
CASE 3: If a = 0. b £ 0. b + d = 0. and c + f 4 0. by (5.0.1),
(5.3.5) (6n + c)VAy(n) + (-6n + /)Ay(n) = -k b y(n) .
Therefore,
"~ bj + c- bj +
w(n)=n
f-J-0 Hj + i) + c
L = (lLLY l = (HIV *
~~[ b ) "ffa-A)"^ b ) r(n_/?l)
;=o
where /?i is as defined in (5.1.13), and
t'H^Ywhrf+n
(c+f)
n + 1
bn
r(n-A)'
so that the self-adjoint form of (5.3.5) is
\(c + f)n+1 1
(5.3.6)
-Vy(n)
-»(^)"ct<->-
6n T(n-/?i)
CASE 4: If q = 0,6 = 0, d^O, and c ^ 0, by (5.0.1),
(5.3.7) cVAy(n) + (<fn + f)Ay(n) = *<f y(n) .
Therefore,
n —1 ,. , j. ., n—1 ., n —1
It/ ]
where ai is as defined in (5.1.14), and
/<An fd\n dn+1
p(n) = f - J r(n-ai)(c+dn+/) = f - J r(n-ai)<f(n-ai) = -^-r(n-ai + l) ,
so that the self-adjoint form of (5.3.7) is
(5.3.8) V
dn+1 1 dn+1
—^-r(n - ax + l)Ay(n) = fc——r(n - ai)y(n) .
Remark 5.3.1
These self-adjoint forms together with boundary conditions assigned where
appropriate make up singular boundary value problems. Thus, they generate
self-adjoint operators on Hilbert spaces. This will be discussed in detail in the
next chapter.

96
ALOUF JIRARI
5.4 Orthogonality
From (4.3.3), if p(n) vanishes at n\ and n2, then the polynomials yi(n) and
yj(n) are orthogonal in £2{n\ + 1, n2;iu), provided 2/,(n), 2/j(n), Ayi(n), and
Ayj(n) are finite for ni and n2. In this section, the interval of orthogonality,
if any, is investigated for each one of the only four possible cases established in
Section 5.1.
CASE 1: If a ± 0:
From (5.3.2),
_ r(n-tti)r(n-q2)
WW~ r(n-ft)r(n-ft)'
and
. v _ ar(n + 1 - at)r(n + 1 - a2)
PW~ r(n-^)r(n-/%)
where <x\, a2 are the roots of aj2 + (6 + d)j + c + / = 0, and ft, ft are the roots
ofa(i + l)2 + 6(i+l) + c = 0.
w(n) has zeros at ft, ft — 1,..., ft, ft — 1, • • •, and poles at c*i, c*i — 1,..., a2,
a2— 1,..., while p(n) has zeros at ft, ft — 1,..., ft, ft — 1,..., and poles at a\ — 1,
oc\ — 2,..., a2 — 1, a2 — 2, There are a lot of possibilities, all of which lead
to an interval of orthgonality of length one at most, except for one:
If there exist positive integers j and £ such that a2 = ft — j and a\ ■=. f32— ly
then the polynomials are orthogonal in l2{f3\ + 1, ft — ■£; w) if ft — ^ > ft + 1.
They are orthogonal in £2(/32 + 1, ft - j; w) if ft -.; > ft + 1.
Likewise, if a2 = ft — j and c*i = ft — ^, then the polynomials are orthogonal
in ^2(ft + l, ft-*; w) if ft -* > ft + l. They are orthogonal in *2(ft + 1, ft-j; u;)
ifft-j>ft + l.
Remark 5.4.1
Clearly, from this discussion ft,ft and therefore c*i,a2 must be integers.

STURM-LIOUVILLE DIFFERENCE EQUATIONS
97
CASE 2: Ua = 0.b±0.ajidb + d±0:
From (5.3.4),
and
where
. fb + d\nT(n-ai)
P(n) = (6+ «)(_- J J___
-ori)
)
Firstly, if
6 + d
«* =-JT1 and * = "S " L
< 1; that is, —2 < — < 0, then w(n) has zeros at oo,/?i,
6
Pi — 1,..., and poles at ot\, c*i — 1,..., while p(n) has zeros at oo, /3\, /?i — 1,...,
c c + /
and poles at ai — 1, ql\ — 2, Therefore, if /?i -f 1 > a?i; that is, - <
Secondly, if
> 1, then w(n) has zeros at oo, /?i, /?i — 1,..., and poles at
then the polynomials are orthogonal over the interval [/?i + 1, oo) =
|6 + <*
6 + d'
cti, ori — 1,..., while p(n) has zeros at —oo,/?i,/?i — 1,..., and poles at ot\ — 1,
ai,—2, Therefore, if oc\ = /?i — r for some positive integer r; that is, if
c + f c
= 7 + 1 + r, then the polynomials are orthogonal over the interval
6 + a b
(-oo,---r-l].
Otherwise, using the formula r(z)r(l — z) =
and
(fc + <f)n + l
sin7T2r
-, it follows that
p(n) =
)r(ai - n + lX-l)"-1 sin tt(1 - ax) '
1 7T
6"
T(n - /?i)r(ai - n)(-l)n sin tt(1 - ax) '
7T
Dividing by the constant — — - the self-adjoint form of (5.3.3) becomes
(5.4.1)
sinir(l — ai)'
\(b + d)n+1 1
[ (-l)"6n r(n-/?!)r(ai-n)
AY*(n)
( b J (_l)»-ir(n-/?Or(ai-n+l)Mw(n)-
l)r(a1-n+l)
The zeros ofp(n) are /?i,/?i —1,..., and ai,ai + l,..., while the zeros of w(n)
are /?i,/?i — 1,..., and c*i + l,ai+2, Therefore, {ni,ni} — {ai,/?i}, and so

98
ALOUF JIRARI
the polynomials are orthogonal in £2(ai +1, f3\\ w) if a\ < f}\> or £2(/3\ +1, ot\\ w)
if f)\ < oc\. Moreover, in this case,
n-l
(5.4.2) yk(n)=T(n-l31-l)r(a1-n + 2)(-^j x
' I (JL.)
T(n - fa - k - l)r(ai - n + 2) \b + dj
n~l
Remark 5.4.2
From the various intervals of orthogonality this discussion led to, it follows that
- has to be an integer in case 2.
CASE 3: If a = 0, b ^ 0, b -f d = 0, and c H- f # 0:
From (5.3.6),
and
K.,.(«+/)(£+/)*T^.
where /?i = —- — 1.
6
If
c + f
< 1, tx;(n) and p(n) have zeros at oo,/?i,/?i — 1, Therefore, the
polynomials are orthogonal over the interval [f3\ + 1, oo) = — -, ooj.
If
> 1, u)(n) and p(n) have zeros at —oo,/?i,/?i — 1, For this
reason, any interval of orthogonality has to be of length one, say [/?i — 1, /?i].
The same is happening if
ft-1,....
Remark 5.4.3
As in case 2, - must be an integer.
6
1, where w(n) and p(n) have zeros fli,

STURM-LIOUVILLE DIFFERENCE EQUATIONS 99
CASE 4: If a = 0, b = 0, c ^ 0, and d ^ 0:
From (5.3.8), w(n) = ( - ) T(n - ax) ,
and P(n) = d(-J T(n + 1 - ax) ,
where ai = — .
a
TAe polynomials in this case are not orthogonal, since p(n) does not have two
zeros.
5.5 Evaluation of the || . ||2
Assume that the polynomials {yfc(rc)}£L0 are orthogonal in £2(ni + l,ri2; w).
If (5.2.1) is multiplied by t/*(n) w(n)> then the summation is taken from (ni + 1)
to 712, it follows that
(5.5.1) B(k) ^ nyk+1(n)yk(n)w(n) + D(k) £ lA(n)w(n) = 0 .
ni+l ru + 1
Similarly, if (5.2.1) is written for (k — 1), multiplied by yk+i(n)w(n), then the
summation is taken from (ni + 1) to 712, it follows that
fl2 7l2
(5.5.2) A(k - 1) £ yj+1(n)ti;(n) + fl(* - 1) £ ny*(n)y(4+1(n)ti;(n) = 0 .
ni+l ni+l
Now, if A(* - 1)JB(*) 7^ 0 for every ib, (5.5.1) and (5.5.2) imply that
or
^ B(jb _ 2)£>(jb - 1) ^
ni + l v ' v ni-f-1
A(* - 2)B(* - 1) A(* - 3)B(* - 2) 2-, »*-a("Mn)
B(k - 2)D(k - 1) B(k - 3)D(k - 2)
'n,+l
and so on, so that
(,,3) iLWH^^i,^;,;;.^"^-

100
ALOUF JIRARI
Remark 5.5.1
The condition A(k — l)B(k) ^ 0 for every k does not impose any additional
requirements. For, the three cases where orthogonality holds are:
CASE 1: a # 0, where A(k) = - (- + k J (- + 2k)
and B(k) = ( - + 2* J (~ + 2ib + 1J (- + 2k + 2 J .
CASE 2: a = 0, 6 ^ 0, and b + <f ^ 0, where A(ib) = -b and £(ib) = <f.
CASE 3: a = 0, 6 ^ 0, 6 + d = 0, and c + / ^ 0, where ;4(jfe) = £(ib) = 1.
Clearly, in case 3, A(k — l)B(k) ^ 0 for every k. For cases 1 and 2, recall that
the condition d ^ —ma was established in Remarks 3.2.2. This guarantees that
A(k) ^ 0 and B(k) ^ 0 in case 1. In case 2, since a = 0, it guarantees that
d ^ 0. Therefore, A(ib - l)B(k) ^ 0 for every k.
5.6 Zeros
As mentioned in Ref. 15 and Fejer's notes at the end of Ref. 6, if in the
recurrence relation (5.2.1), A(k)D(k) > 0 for every &, then the polynomials
{2/*(n)}jbLo f°rm a generalized Siurmian sequence. As a result, the zeros of yjb(n)
are real and distinct. They also lie on the interval of orthogonality of the y^s
and separate the zeros of 2/jb-i(rc).

Chapter 6
The Four Representative Examples
This chapter is a survey of the polynomials available in the literature, which
satisfy difference equations of the form (5.0.1). There are actually four
representative examples, each of which deserves a section of its own, listing all the
characteristic properties.
6.1 The Generalized Tchebyshev Polynomials
The generalized Tchebyshev polynomials are defined in Lesley's papers (see
Refs. 12 and 13) as the polynomials satisfying
[n2 + (p + 3 - A - B)n + (B - 1)(A - p - 2)]A2Pk(n)
(6.1.1) +[(a + /3 + 2)n-(A- l)(a + 1) - (B - \){p + 1)]Aft(n)
-Jb(Jb + a + p + l)Pk(n + 1) = 0 ,
where a, p are positive integers, n = A>..., B — 2, and k = 0,..., B — A. By
(5.1.9), this is equivalent to
[n2 - (A + B + a + l)n + A(B + a + 1)]VAT*(n)
(6.1.2) +[(a + /? + 2)n - A(a + 1) - B(p + l)]ATk(n)
=k(k + a + P+l)Tk(n),
where a, p are positive integers, n = A+lf...fB — 1, and Ar = 0,..., B — A.
Clearly, this is an example of case 1 since a = 1 ^ 0. From (4.3.1), c*i
and a2 are the roots of (j — B)(j — (A — P — 1)) = 0. Therefore, c*i, a2 = £,
A — P—l. Also, /?i and /?2 are the roots of (j +1 — A)(j — B — a) = 0. Therefore,
^i,A = i4-l,a + B.
Since a! =z p2 — a and a2 = /?i — /?, it follows from the discussion of case 1 in
Section 5.4 that these polynomials are orthogonal over the interval [A, B], By
(5.3.2), the self-adjoint form of (6.1.2) is given by
101

102
ALOUF JIRARI
(6.1.3)
But
T(n + 1 - B)T(n + 2-A + /3)
T(n + 1 - A)T(n - a - B)
ATt(n)
k(k + a + p + ^r^-B^n-fl-A + g
T(n + 1 - ff )
T(n - a - B)
T(n + 1 - A)T{n - a - B)'
= (n- B)...(n- B -a),
= (-l)«+1(B-n)...(B + a-n),
Similarly,
, na+1r(B + a-n+l)
^ ; T(B-n+l) '
fI(^L=(„-B-l)...(n-B-„),
= (-l)a(5 + 1 - n) ... (B + a - n),
r(B + a-n+l)
^ ' T(B-n+l) '
So, substitution in (6.1.3) and multiplication by (—1)" give
T(B + a - n + l)T(n -A + /3+2)
(6.1.4)
T(B - n)T(n - A + 1)
ATk(n)
= k(k + a + ^l)^B^-^fJ)n-AA^+l)Tk(n).
T(B-n+l)T(n-A + l)
From (5.1.11), the polynomial solutions T]fc(n) are given by
T(n-A+f3)T(n-B-l)
(6.1.5) r4(») - T(n_B_1)r(n_A+^
T(n-A-k)T(n-a-B-k-l)\

STURM-LIOUVILLE DIFFERENCE EQUATIONS 103
But
r(n - a - B - 1) _ 1
T(n - B - 1) ~ (n-a-B- l)(n - a - B)... (n - B - 2) '
= (zll!
(B + a+l- n)(a + B - n)... (B + 2 - n) '
T(g + 2-n)
^ ; T(5 + 2 + a - n) '
= (n-B-2)...(n-a-B-k-l) ,
Similarly,
r(n - B - 1)
T(n - a - B - k - 1)
= (-l)a+k(B + 2-n)...(B + a + k+l-n),
+kT(B + a + k + 2-n)
K ' T(B + 2-n)
Simple substitution in (6.1.5) then gives
T(n-A+/3)r(B+a+k+2-n)
<e> 1 «\ T (n\-f n* T(n-A)T(B+2-n) k
(6.1.6) Tfc(n)_(-1) r(n_A+^)r(5+2+a_n)A
T(n-A-k)T(B+2-n)
As for the zeros, it is hard to use the test of Section 5.6 since the expressions
for A(k) and D(k), as shown in case 1 of Section 5.2, are very complicated. For
this reason, a different approach is adopted: If in (6.1.1) k(k + a + 0 + l)Pjfe(n)
is added and subtracted, then
(n-A + 0 + 2)(n - B + l)A2Pk(n)
+[(a + /3+2)n-k(k + a + /3+l)-(A- 1)(« + 1) - (B - l)(j3 + l)]Aft(n)
-k(k + a + j3 + l)Pjb(n) = 0 .
When a = /? = 0, this becomes
(n - A + 2)(n - (B + 1) + 2)A2P*(n) + [2n + 3 - k(k + 1) - A - (B + 1)]Aft(n)
-k(k + l)Pjfe(n) = 0 .

104
ALOUF JIRARI
Fejer's result mentioned in footnote 17 of Ref. 11, together with his notes at the
end of Ref. 6, can be used to conclude that Pk(n) has real distinct zeros which
lie on [Ay B] and separate the zeros of Pjb_i(n), since k < (B + 1) — A.
Also, since both A and B are finite, they are both in the limit-circle case.
Finally, the boundary value problem with the generalized Tchebyshev
polynomials as eigenfunctions is
(Ly)(n) = V
T(B + a-n+l)r(n-A + /3 + 2)
T(B - n)T(n -A + l) yW
• *> fA n r(£ + a-n + l)r(n- A + /3+l)\ t
set in i£ A, B: ———— -7—; - — , where A and B are inte-
V T(B-n + l)T(n-A + l) /'
gers such that A < B and a, ft are positive integers.
The boundary conditions are
W[y, 1](A) = p(A - l)[y(A) - y(A - 1)] = p(A - l)Vy(A) = 0 ,
and
W[y, 1](B + 1) = p(B)[y(B + 1) - y(B)] = p(B)Ay(B) = 0 ,
where
T(B + a-n+l)T(n-A + 0 + 2)
PW ~ T(B - n)T(n -A+l)
Fnr f in fi (a R- T(B + <* ~ n + l)T(n - A + 0 + 1)\
F°rfm£ [A'B' T(B-n + l)T(n-A+l) )'
/(»)=E(/(»)>r»(»))J^JT5
B-A
where
\r(n-A+/3)r(B+a+k+2-n)
T rnW n* r(n-A)r(B+2-n)
r(n-A-k)T(B+2-n)
Remarks 6.1.1
1) Since p(A— 1) = 0 and p(B) = 0, the boundary conditions p(A—l)Vy(A) = 0
and p(B)Ay(B) = 0 actually mean that Vy(A) and Ay(B) must be finite.
2) According to Ref. 7, the Hahn polynomials satisfy the equation

STURM-LIOUVILLE DIFFERENCE EQUATIONS 105
(N-l- n)(p+ 1 + n)Pk(n + 1) - [(N - 1 - n)(p+ 1 + n) + n(N + q - n)]Pk(n)
+n(N + q - n)Pk(n - 1) + *(* + p + q + l)ft(n) = 0 ,
where p > —1, q > — 1, and k = 0 ..., N — 1.
Rewriting this equation with n + 1 instead of n, adding and subtracting
(N - 2 - n)(p + 2 + n)[Pk(n + 1) - Pjb(n)], it follows that
(#-2-n)(p + 2 + n)A2P*(n) + [(7V-2-n)(p + 2 + n) - (n + l)(tf + g-l-n)]
x AP*(n) + Jb(t + p + g + l)P*(n + 1) = 0 ,
which is equivalent, using (5.1.9), to
[n2_(^^)n]vA^(n)+[(JH^+2)n-(p4-l)(iV-l)]Ai/Jb(n) = *(*+p+g+l)tf*(n) .
This is clearly a special case of (6.1.2) where A = 0, B = N — 1, /? = p, and
a = g. Making these sustitutions in the generakized Tchebyshev problem shows
the following:
The boundary value problem with the Hahn polynomials as ei gen functions is
(Ly)(n) = V
T(N + q-n)r(n + p + 2) .
r(7V - 1 - n)r(n + 1) V{ \
^T(N + q-n)T(n+p+l) . x
= *(* + p+« + *> r(j?-n)r(»+?) y(w) '
• „o/« ,r , T(N+ q-n)T(n+p+l)\
set in £z I 0,iV — 1; ~-^ -r 1, where TV, a, ana <j are positive
integers.
The boundary conditions are
W[y, 1](0) = p(-l)Vt/(0) = 0 ,
ana1
W[y, 1](#) = p(tf - l)Ay(N - 1) = 0 ,
tu/iere
„r„^ = r(jv + <?-n)r(n + p + 2)
n ; r(jv - n - i)r(n +1) '
meaning thai Vy(0) ana* Ay(N — 1) are /inrte.

106 ALOUF JIRARI
torjmt\v,n 1, r(Ar _ n)r(n + 1} )>
N-\
f(n)=Yl(f(n),Hk(n))^,
where
_ (-l)*r(n)r(AT + l-n) t
kK ' T(n+p)T(N + l + q-n)
T(n + p)T(N+l + g + k-n)
T(n - k)T(N + 1 - n)
3) The analogue to Legendre polynomials referred to in Ref. 11 are also a special
case of the generalized Tchebyshev polynomials. They satisfy the difference
equation
(n - a)(n - b)VAEk(n) + [2n + (1 - a - b)]AEk(n) = k(k + l)Ek(n) ,
where n = a + 1,..., 6 — 2, and k = 0,..., b — a — 1, which follows from (6.1.2)
by setting A = a, 5 = 6—1, and a — /? = 0.
Thus, the boundary value problem with the analogue to Legendre polynomials
as eigenfunctions is
(Ly)(n) = V[(n-a+ l)(n - 6 + l)Ay(n)]
= k{k+l)y{n),
set in £2(a,b — 1; 1), where a and b are integers with a < b.
The boundary conditions are
W[y,l](a)=p(a-l)Vy(a) = 0,
and
W[y, 1](6) = p(b - l)Aj/(6 - 1) = 0 ,
where
p(n) = (n-a+ l)(n -6+1).
That is, Vy(a) and Ay(b — 1) are finite.
Forf in P{a,b- 1; 1),

STURM-LIOUVILLE DIFFERENCE EQUATIONS 107
where
Ek(n) = (-1)*A*
r(n-q)r(6 + l + ib-n)
[T(n-a- k)(b + 1 - n)
Further, case 1 of Section 5.2 shows that they satisfy the recurrence relation
-£7*+2(n)+(3+2Jb)(2n-a-6-^
The Jordan polynomials, Qk(p)> are then obtained by setting Ek(ri) = 2kk\Qk(ri).
The recurrence relation then becomes
4(*+2)Q*+2(n)-2(3+2*0(2n-a^^
A simple application of (5.5.3) then gives
iiQti|2=4H2r+i)(jv2~i2)(iv2~22)---(iv2~fc2) >whereAr=6-a-
The discrete Tchebyshev polynomials, Dk(n), mentioned in Ref. 4 as a special
case of the Hahn polynomials, are actually a variation of the Jordan polynomials.
In fact, Dk(n) = 2*Q*(n), so that ||D*||2 = -JL-(N* - 12)...(7V2 - t2),
where N = b — a.
Since the Jordan and the discrete Tchebyshev polynomials differ from the
analogue to the Legendre polynomials by factors which depend only on k, they
generate the same boundary value problem.
4) The analogue to Jacobi polynomials referred to in Ref. 11 are of no interest
to us since they're orthogonal over an interval of length 2. They satisfy the
difference equation
(_n2 _ 4n _ S)A2Pk(n) + [-(p + q + 2)n + (qp - 2p - 3)]APk(n)
+k(k + 1 + p + g)Pib(n + 1) = 0 ,
where p and <j are positive integers. By (5.1.9), this is equivalent to
[n2-(p+q)n+(qp-p+q-l)]VAyk(n)^(p+q+2)n-pq+p-q + l]Ayk(n)
= *(* + l+p+flf)y*(n) .
Following case 1 of Sections 5.1 and 5.3, it is easy to see that {c*i, a2} = {—2,0},
and {/?i, /%} = {Pi Q ~ 2}, so that the self adjoint form of the difference equation
is
V[(n + 2) ... (n - 9 + 2)n ... (n - p)Ayk(n)]
= k(k + 1 + p + q)(n + 1) ... (n - 9 + 2)(n - 1) ... (n - p)2/*(n) .

108
ALOUF JIRARI
Clearly, p(n) = (n + 2) ... (n — q + 2)n ... (n — p) has zeros —2, — 1,..., q — 2,
and 0,1,.. .,p, while w(n) = (n + 1) .. . (n — <j + 2)(n — 1) .. .(n — p) has zeros
— 1,0,..., g — 2, and 1, 2,...,p. This means that the polynomials are orthogonal
over [—2,0], provided —1 is not a zero of p(n) or w(n); that is, if q = 0. In this
case,
V[(n + 2)n .. .(n - p)Ay*(n)] = *(* + 1 + p)(n - 1) ... (n - p)yk(n) ,
and
y*(*0 =
1
(n-2)(n-3)...(n-p-l)
A*[n(rc-1) ... (n-k+l)(n-2)(n-g) ... (n-p^b-4)]
6.2 The Generalized Laguerre Polynomials
The generalized Laguerre polynomials are defined in Ref. 12 as the polynomials
satisfying the difference equation
(6.2.1) (n-A + a + 2)A2Pk(n) + [(n-A + 1)(1 - e7) + a + l]APk(n)
-*(l-e7)P*(n + l) = 0,
where k > 0, n > A, j > 0, and a > —1. By (5.1.9), this is equivalent to
(6.2.2) [ne7-Ae7]VAL*(n) + [(l-e7)n^^
Clearly, this is an example of case 2 since a = 0,6 = e7^0, and 6+d = 1 ^ 0.
From (5.3.3), ct\ is the root of j + a + 1 — A = 0; that is, c*i = A — a — 1. Also,
/?i is the root of e7j + (1 — A)ey = 0; that is, /?i = A — 1.
= e~7 < 1 and —A < a + 1 — A, it follows from the discussion
Since
ei
of case 2 in Section 5.4 that these polynomials are orthogonal over the interval
[A, oo).
By (5.1.12), the polynomial solutions are given by
(6.2.3)
M«) = J(n~*A) AS)"-1**
T(n-A + a)
T(n-A + a) /
r(n - A - k)
and the self-adjoint form of the difference equation they satisfy is
(6.2.4)
V
. ,.„r(n-i + a + 2).r ,
T(n-A + 1)
=(ow*(i-«T)r(rAttt^1)^(")
T(n-A + 1)

STURM-LIOUVILLE DIFFERENCE EQUATIONS 109
Also, based on the result of case 2 in Section 5.2, the recurrence relation is
given by
(6.2.5) L»+2(n) + [(l-e-T)n+(fc+a+l)e-T -(A+k+2)]Lk+1(n)
-K*+l)(a+l+*)e-7M») = 0 ,
and, by (5.5.3),
2 _ *!(a + k)\ f. T(n - A + a + 1)
M -"is^rV r(»-A+i) •
As for zeros, since (6.2.5) shows that A(&) = —e7 and £)(&) = — (k + l)(a +
1 + &), where 7 > 0, a > —1, and k > 0, A(fc)D(&) > 0 for every &, and the
result stated in Section 5.6 applies.
Also, since A is finite, it is in the limit-circle case. It remains to run the tests
of Section 3.11 to see if 00 is in the limit-point or limit-circle case.
_yn T(n-A + a+l)
w(n) = e 7n—K— —-?■
OO
implies that \^ w(n) < 00, since
w(n + 1) _ e"7(n + l)T(n - A + a + 2)T(n - A + 1)
n±™o w(n) ""n^o e"7nr(n- A + a + l)T(n - A+ 2)
v -yn~ A + a + 1
= lim e 7 — = e 7 < 1 .
n-+oo n-i4 + l
On the other hand,
_7n r(n - A + a + 2)
p(n) =
e
r(n - A + 1)
00
implies that yj p(^)-1 = 00, since
n=A
p(n) .. e7(n+1)r(n-A + 2)r(n-A + a + 2)
lim —: —r = lim — - ztzt, ; rr— ,
n-oop(n+l) n->oo e7nr(n - A + l)T(n - A + a + 3)
n- A+l
= lim eJ = e7 > 1 .
n-*oo n — A + a + 2

110
ALOUF JIRARI
This shows, according to Theorem 3.11.4 and 3.11.5, that for the generalized
Laguerre polynomials neither Atkinson's limit-point test nor his limit-circle test
at oo can be used. The only alternative left is Mingarelli's limit-point criterion,
which does require rewriting (6.2.4) so that w(n) = 1.
Following the procedure of Theorem 3.11.3, (6.2.4) is equivalent to
(6.2.6) V[p(n)AL*(n)] + [#n)+^^^ = *(l-^)Z*(n),
where
1 < \ ( M/2r / x ■7nr(n-i + q + l)1/2
Lk(n) = w(ny<zLk(n) = e 7a r(n - 4 + l)1/2 ^ '
p(n) = w(n)-^2p(n)w(n + l)"1/2 ,
n r(n-A + l)1/2 r(n-A + a + 2) i^±u
^n-A + a+l)1/2 T(n-A + 1)
r(n-^ + 2)1/2
X r(n-A + a + 2)!/2 '
and
3 T(n - A + 2)1/2r(n - A + a + 2)1/2
62 r(n-A + iy/2r(n-A + a + iy/2 '
= e^2(n -A+ \yl\n - A + a + l)1'2
iu(n) 1'2q(n) = — u;(n) *[p(n)-f p(n — 1)] ,
-^n-
T(n-A-fl)
T(n - A + a + 1) '
T(n-A + a + 2) (n_^T(n-A + a+1)
T(n-A+1) T(n-A)
= -(n - A + a + 1) - e7(n - A)

STURM-LIOUVILLE DIFFERENCE EQUATIONS 111
so that
Q(n) = [p(n) + p(n - 1) + w(n)-^2q(n)^ ,
= e^2{n -A + l)1/2(n - A + a + l)1'2*
e7/2(n - A)1/2(n -A + a)1/2 -(n-X + a + 1)- e7(n - 4) .
Remark 6.2.1
1) From the new difference equation (6.2.6), it is clear that the interval of
orthogonality is still [A, oo).
2) Atkinson's limit-circle test at oo given by Theorem 3.11.4 fails once again
oo oo
since Y^ w(n) = Y^ 1 = oo.
3) It is equivalent to say that yk is in P{N) oo; w{n)) or that y* is in P{N) oo; 1),
since
oo oo oo
£ |y*(n)|2u,(n) = £ \yk(n)w(n)^\2 = £ |fr(„)|2.
n=iV n=N n=N
Now that tu(n) = 1, for Atkinson's limit-point test at oo to work, it suffices
to show that for some real A, Q{n) < A or n sufficiently large. Also, note that
n=N
^p(n —1) x/2 = oo since, for N > A,
oo oo
£ p(n - l)"1/2 = £ e~T\n - A)-^(n -A + a)~1/4 ,
n=N n=N
oo
-7/4'
n^N[(n-A)(n-A + a)}^'
oo .
> e~il* V -
" ^(n-A + aY/*'
= OO
iii.. ( i. n — A + a . . v^ 1 l
by the limit comparison test hm — = 1 and > -r- = oo I.
\ n—oo n — A *-*'T (n — A)2 I
N n=Ny ' /

112 ALOUF JIRARI
This shows that if M(n) is set equal to 1 in Theorem 3.11.1, the only thing that
remains to prove for Mingarelli's limit-point test at oo to work is that Q(n) < K\
where K\ is a positive constant, for n sufficiently large. Therefore, Atkinson's
and Mingarelli's tests are in some sense equivalent. Since
Q(n) = e7/2(n -A + l)1/2(n - A + a + l)1'2
+ e^f2{n-Aff2{n-A + a)^2-(n-A + a + 1) - e7(n-A) ,
for n sufficiently large,
Q(n) 2 e^/2n + e7/2n - n - e^n = -n(l - e7'2)2 < 0 ,
since j > 0. Therefore, A = K\ = 0 and the generalized Laguerre problem is
limit-point at oo.
Finally, the boundary value problem with the generalized Laguerre polynomials
as eigenfunctions is
(Ly)(n) = V
' „ T(n - A + a + 2) '
6 r(n-A+l) Ay(w)
• ,«> / * _™ r(n - A + a + 1) \
se< in kt I A,oo;e —^7 -j rr-1 I, where A is an integer, 7 > 0, and
a < -1.
T/iere is on/y one boundary condition,
W[y, 1](A) = p(A - l)Vy(A) = 0,
where
p(n)-e-^T(n-A + a + 2)
Kj C T(n-A + 1) ■
This essentially means that Vt/(A) must be finite.

STURM-LIOUVILLE DIFFERENCE EQUATIONS
113
where
*<">-i^7&5'*",)A'
r(n-i4 + g) (w.n"
[r(n-A-ib)
Remark 6.2.2
1) The analogue to Laguerre polynomials referred to in Ref. 11 follows from
the generalized Laguerre polynomials where ^4 = 0, a = 0, and e7 = p. They
satisfy the difference equation
(pn)VAAk(n) + (1 - p)n AAk(n) = (1 - p)fc ^(n) ,
which takes the self adjoint form
V [p-n(n + l)AAk(n)} = p-»*(l - p)Ak(n) .
These polynomials are orthogonal in ^2(0, oo;p~n) and, by case 2 of Section 5.2,
they satisfy the recurrence relation
Ak+2(n) -
i4*+i(n)+ii^^i4*(n) = 0>
sothat,||^|P = ?^L
TAe boundary value problem with the analogue to Laguerre polynomials as
eigenfunctions is
(Ly)(n) = V[p-n(n+l)Ay(n)}
= p-nk(l-p)y(n),
set in ^2(0, oo; p~n), where p > 1.
T/iere is only one boundary condition,
W[y, 1](0) = p(-l)Vy(0) = 0 ,
where
p(n) = p-n(n+l).
For f in£2(0,oo;p~n),
/(») = D/(»M»(»)>FrS,
* = 0
ll^ll2

114
ALOUF JIRARI
where
— ^-U*
Ak(n) = pn-1A
i^Lpi-
[T(n-ky
2) The Meixner polynomials of the first kind surveyed in Ref. 4 are a special
case of the generalized Laguerre polynomials, the one where A = 0, e7 = -, and
c
a = f3 — 1. They satisfy the difference equation
c(n + p + 1)A2Mk(n) + [(c - l)(n + 1) + /?c]AMt(n) - k(c - l)Af*+i(n) = 0 .
Using (5.1.9), this is equivalent to
nVAMi(n) + [(c - l)n + /?c] AAf*(n) = *(c - l)Af*(n) ,
where 0<c<l,/?>0, & > 0 and n > 1. The self-adjoint form of this equation
is
c r(» + i) AMfc(n)
= cng!L±^ib(c_1)Mt(n))
r(n +1)
and the polynomials are orthogonal in t2 ( 0, oo; cn—^ -r- J.
TAe boundary value problem which generates the Meixner polynomials of the
first kind is
\Br(n+ /? + !)
(Lj/)(n) = V
c" r(n + i)'Ay(n)
se< in ^2 (0, oo; cny^——• )' where /? < 0 and 0< c < 1.
T/iere t5 on/y one boundary condition,
W[y,l](0) = p(-l)Vy(0) = 0,
where
p(n) = c1
„r(n + /?+l)
r(n +1) •
That is, Vy(0) is finite.
Mk(n)
/(») = E(/(»).^(»)>Sji5

STURM-LIOUVILLE DIFFERENCE EQUATIONS 115
where
Mk{n) =
r(») i-„A*
r(n + /? - 1)
6.3 The Krawtchouk Polynomials
r(n+ /?-!)
T(n - k)
The Krawtchouk polynomials are defined in Ref. 16 in terms of the polynomials
satisfying the difference equation
(6.3.1) p(n + 1 - B)A2Pk(n) + (n + 1 - qA - pB)APk(n) - kPk(n + 1) = 0 ,
where p > 0, q > 0, p + q = 1, k = 0,..., B — A, and n = ^4,..., B — 2. By
(5.1.9), this is equivalent to
(6.3.2) (-nq + qA)VAPk(n) + (n - qA - pB)APk(n) = kPk(n + 1) ,
where n = A + l}.. ,}B — 1.
This is another example of case 2 since a = 0, 6 = —g^O, and b + d =
—g + 1 ^ 0. From (5.3.3), ct\ is the root of (1 — q)j — pB = 0; that is, ct\ = B.
Also, /?i is the root of — qj + q(A — 1) = 0; that is, f3\ = A — 1. Moreover,
following the discussion of Section 5.2 for case 2, it is easy to conclude that these
polynomials are orthogonal over the interval [^4, B], By (5.1.12), the polynomial
solutions are
(6.3.3)
Pkin^T(n-A)T(B+2-n)^n~ *[T{n_A_k*T{B_n+2) (f)
where n = A,..., B and k = 0,..., B — A.
The self-adjoint form of the difference equation they satisfy is
(6.3.4)
n-1
I p"*1 1
V[-^T(n-A+l)T(B-n)APk^
■kPk(n) .
qn T(n-A+l)T(B-n+l)
Notice that if the weight is multiplied by —p~AqB(B — A)l, it becomes
pn-AqB-n | 5 ~ ) t and the equivalent of (6.3.4) is then
(6.3.5)
= p"-AqB^(Bn-fjkPk(n).
n-A+l Bo (B A)1 APk(n)
/ q T(n-A+l)T(B-n) k{ '
Also, a simple application of the result of Section 5.2 for case 2 shows that
the recurrence relation is
(6.3.6) qPk+2(n) + [n-(-qA-PB+k-2(k+l)q)]Pk+1(n)

116
ALOUF JIRARI
+(k + l)p(B-A-k)Pk(n) = 0 .
The Krawtchouk polynomials are given by Kk(n) = —-~^. They satisfy the
recurrence relation
(6.3.7) q(k+2)\ Kk+2(n) + [n+(^A-i>B+k-2(k+l)q](k + l)\ Kk+1(n)
+(t+l)! f<B-i4-t)tf*(n) = 0 ,
or equivalently, dividing by (k + 1)!,
(6.3.8) g(* + 2)Kk+2(n) + [n + {-qA -pB + k-2(k + l)q)]Kk+1(n)
+p(B - A - k)Kk(n) = 0.
From (5.5.3), it follows that
ii**ii
2 _ pk(B-A)...(B-A-k + 1)
qk k\
n=A V '
£-A
= g^ (B-A)\ y (B_A)_n(B-A\
qkk\(B-A-k)\^oPq \ n ) >
= Pk1-k(B-A)(p + <l)B-A,
= '<->(*->*).
since p + g = 1 . In addition, (6.3.8) shows that ^4(Ar) = q(k + 2) and D(k) =
p(£ — ^4 — Ar), where p > 0, g > 0, and k = 0,..., B - A - 2. Therefore,
A{k)D{k) > 0 and the result of Section 5.6 applies.
Further, since A and B are finite, the Krawtchouk polynomials are limit-circle
at both A and B.
Finally, the boundary value problem with the Krawtchouk polynomials as eigen-
functions is
=tp»-v-"(f:^)«(»),

STURM-LIOUVILLE DIFFERENCE EQUATIONS 117
set in £2 ( A,B)pn~AqB~n I . J J, where A and B are integers, p > 0,
q > 0, and p-\- q = 1.
TVie boundary conditions are
W[y,l](A)=p(A-l)Vy(A) = 0,
and
W[y, 1](B + 1) = p(B)Ay(B) = 0 ,
tu/iere
K„\ _ ^n-A+l^B-n (^ — A).
n)-_P q T(n-A+l)T(B-n)'
That is, both Vy(A) and Ay(B) are finite.
Forf in? ^,5;p"-V-" (f I^))>
where
KJn) - nn-A)V(B+2-n) (lY** J 1 /VT
Remark 6.3.1
The Greenleaf polynomials discussed in Ref. 11 satisfy the difference equation
(n + fl)VAGi(n) - 2nAG*(n) = -2JbG*(n) ,
where n = —B + 1,..., B — 1, and k = 0,..., 2B. They clearly represent the
special case of Krawtchouk polynomials where A = — B and p = q = -.
Therefore, the boundary value problem they generate is
(Ly)(n) = V
_L (25)! "
" 22B+i r(n + B + l)r(B - n) yW
= *2^ („%)»(»).

118 ALOUF JIRARI
set in £2 I —B, B; -r^ I _ J J, where B is a positive integer.
The boundary conditions are
W[y, 1](-B) = p(-B - l)Vy(-B) = 0 ,
and
where
W[y,l](B+l) = p(B)Ay(B)=0,
1 (2B)!
P(n) = —.
22B+1 T(n + B + l)r(5 - n)
So, Vy(—B) and Ay(B) must be finite.
ForftnP(-B,B;±(™B)),
IB
Gk(n)
/(«) = £</(")>G*(")>p^
where
Gk(n)
_r(n+B)T(B + 2-n)Ak
k\
T(n + B-k)r(B-n + 2)}
6.4 The Charlier Polynomials
The Charlier polynomials are defined in Ref. 12 as the polynomials satisfying
the difference equation
(6.4.1) A2Pk(n) + (-pn -p + pq+ l)APk(n) + pkPk(n + 1) = 0 ,
where q is an integer, p > 1, n > g, and k > 0. Using (5.1.9), this is equivalent
to
(6.4.2) (pn - pq)VACk(n) + (-pn + pq + l)ACk(n) = -p*C*(n) ,
where n > q + 1.
Clearly, this is an example of case 3 since a = 0, 6 = p ^ 0, 6 -f d = p — p = 0,
and c + / = —pg + pg + 1 = 1 ^ 0.
From (5.3.5), /3\ is the root of pj + (p — pg) = 0; that is, /?i = g — 1.
Since
c + /
< 1, it follows from the discussion of case 3 in Section 5.4
that these polynomials are orthogonal over the interval [g,oo).

STURM-LIOUVILLE DIFFERENCE EQUATIONS
By (5.1.13), the polynomial solutions are given by
119
(6.4.3)
n-l a*
C*(n) = r(n-g)pB-1A-
Lpn-1r(ra-?-ir)J '
and the self-adjoint form of (6.4.2) is
(6.4.4)
pnT(n-q + l)
ACk(n)
= -pk
pnT(n - q + 1)
Cfc(n) .
Also, using the result of Section 5.2 for case 3, the recurrence relation is
(6.4.5) Ck+2(n) +
n - (k + 2) -
pq+l
Ck+1(n) + —Cfe(n) = 0 ,

120
ALOUF JIRARI
and, by (5.5.3),
||Cfe||2 = *!p-(*+*)ei.
ib + 1
As for zeros, since (6.4.5) shows that A(k) = 1 and D(k) = , A(k)D(k) > 0
P
for every k and the result of Section 5.6 applies.
Also, since q is finite, it is in the limit-circle case. It remains to run the tests
of Section 3.11 to see if oo is in the limit-point or limit-circle case.
w(n) = p(n) = —
pn(n — q)l
implies that YJ w(n) < oo, but /_^p(rc) * = oo. For,
n—q n—q
.. ti;(n+l) 1
hm , x = hm — -r = 0,
n->oo W[n) n-+oop(n — q + 1)
and
p(n)
lim ——-—-T- = lim p(n — q + 1) = oo.
n-*oop(n + 1) n-*oo
This shows, according to Theorems 3.11.4 and 3.11.5, that for the Charlier
polynomials, neither Atkinson's limit-point test, nor his limit-circle test at oo can
be used. Now, in order to use Mingarelli's limit-point criterion, (6.4.2) must be
rewritten so that w(n) = 1. Following the procedure of Theorem 3.11.3, (6.4.2)
is equivalent to
(6.4.6)
V jp(n)ACfc(n)] + [p(n) + p(n - 1) + w(n)-l'2q(nj\ Ck(n) = -pkCk(n) ,

STURM-LIOUVILLE DIFFERENCE EQUATIONS 121
where Ck(n) = w(n)^Ck(n) = _^-2__Cfc(n) ,
p(n) = w(n)~1/2p(n)w(n + 1)~1/2 ,
_p^r(n-g + 2)1/2
p^rCn-^+l)1/2 '
and
so that
= P1/2(n-g+l)1/2,
w(n)~1/2q(n) = -i^n)"1^)+p(n - 1)] ,
= —1 — w(n)~1p(n — 1) ,
pT(n-g+l)
p^IXn-g) '
= -l-p(n-g) ,
Q(n) = p(n) + p(n - 1) + w(n)-1/2g(n) ,
= P1/2(n - q + l)1/2 + ^(n - q)1/2 - 1 - p(n - q) .
Now that w(n) = 1, for Atkinson's limit-point test at oo to work, it suffices
to show that for some real A, Q(n) < A for n sufficiently large.
Note that
oo oo
*1/2
n=q+l n=g+l
-1/2
oo 1 oo 1
n=g+l F n=l
This shows that, if M(n) is set equal to 1 in Theorem 3.11.1, Q(n) < Ki where
K\ is a positive constant is the only requirement for Mingarelli's limit-point test
at oo to work.
For n sufficiently large,
Q(n) S 2p1/2n1/2 - pn - 1 = -(p1/2^2 - l)2 < 0 .

122
ALOUF JIRARI
Therefore, K\ = 0 works and the Charlier problem is limit-point at oo.
Finally, the boundary value problem with the Charlier polynomials as eigen-
functions is
(Ly)(n) = V \——±——Ay(n)]
[pnT(n-q+l)
= — pk
pnT(n - q + 1)
V(n)
set in P ( g, oo; ——7 ~ 1, where q is an integer and p > 1. There is only
V pnr{n-q+l)J
one boundary condition,
W[y,l)(q) = p(q-l)Vy(q) = 0,
where
p(n)~ P»r(n-q + iy
which means that Vj/(g) must be finite.
For f in t2 [ q, oo; ——. — 1 ,
/(n) = D/(n),C*(n))^,
k=0
where
n-l \k
Cki^^Tin-q^-'A
1
[pn-irin - q - k)
Remark 6.4.1
The Charlier polynomials mentioned in Ref. 4 satisfy the difference equation
a2A2Pk(n) - (n + 1 - a)APk(n) + kPk(n + 1) = 0 ,
where a > 0, k > 0, and n > 0, or equivalently,
nVAPk(n) + (-n + a)APk(n) = -kPk(n) , (n > 1) .
These two equations follow from (6.4.1) and (6.4.2), respectively, by setting
p = - and q = 0. Therefore, these polynomials are a specific case of the
a
Charlier polynomials surveyed in Ref. 12. They are also called Poisson-Charlier

STURM-LIOUVILLE DIFFERENCE EQUATIONS
123
polynomials. In fact, if the weight function is multiplied by e a, it becomes
an
e a—r, which is the Poisson distribution,
n!
For these polynomials, the boudary value problem is
(Ly)(n) = V
e~a—Ay(n)
nl
k an
„/ ane~a\
set in ll I 0, oo; :— ), where 0 < a < 1.
V nl J
There is only one boundary condition,
W[y, 1](0) = p(-l)Vt/(0) = 0 ,
where
p{n) = e — ,
n!
implying that V2/(0) must be finite.
0 / ane~a\
For f in £2 f 0, oo; — J,
/(») = £(/(»), ft (»)>|7^,
k=0
ii^ii2
where
l-n \k
Pk(n) = Tiny-" A
r(n-*)J

Chapter 7
Left-Definite Spaces
The purpose of this closing chapter is to set the difference operators of the
four representative examples discussed in Chapter 6 in a left-definite space.
This is accomplished in two parts. The first one applies to the generalized
Tchebyshev and the Krawtchouk polynomials, for which the intervals of
orthogonality are finite. The second one applies to the generalized Laguerre and the
Charlier polynomials, for which the intervals of orthogonality are infinite.
This follows the treatment of the self-adjointness of the Weyl problem in a
left-definite space presented by Krall and Race in Ref. 10.
7.1 Finite Intervals
Equations (6.1.4) and (6.3.5) show that, for both the generalized Tchebyshev
and the Krawtchouk problems, the difference equation is of the form
(7.1.1) V\p(n)Ay(n)] = fiw(n)y(n) ,
where n = A,..., By with A and B in the limit-circle case, and p(n) negative for
n = A,...,£. (7.1.1) can be written equivalently
(7.1.2) V[p(n)A2/(n)] + w(n)y(n) = Xw(n)y(n) ,
where A = /i + 1.
Note that the eigenvalues A are now nonzero. In fact, they are k(k + a +
ft + 1) > 0 for the generalized Tchebyshev problem (a,/? positive integers, k =
0,..., B — A), and k + 1 > 0 for the Krawtchouk problem (k = 0,..., B — A).
124

STURM-LIOUVILLE DIFFERENCE EQUATIONS 125
Definition 7.1.1
H1(Ai B,p, w) is the Hilbert space consisting of all sequences y{A)y..., y(#), for
which
(7.1.3) llyllfp = J2 {-K» " 1) |Vy(n)|2 + «,(„) |y(n)|2} .
The inner product in H 1(A, B,p, u;) is given by
B
(7.1.4) (y, z)Hi = ^ i-P(n ~ l)Vy(n)Vz(n) + ti;(n)y(n)z(n)} .
Now let a < a! < e < V < b. The following two theorems exhibit left-definite
square integrable solutions of (7.1.2) in H1(ei 6,p, w) and H1(ai e — l,p, u;).
Theorem 7.1.2
For a77 A with Im(X) ^ 0, there is a solution i(>b(n, A) = #i(n, A) + m&(A)02(rc, A)
of (7.1.2) in H1(e) 6,p, w), the Sobolev space with inner product
b
(y, z) =^2 {~P(n ~ !)Vy(n)VJ(n) + tx;(n)y(n)2r(n)} .
Proof
In (7.1.2), multiplying by y(n) and taking the summation from e to V give
(7.1.5) ^{v[p(n)Ay(n)]y(n) + ^n)|y(n)|2} = A^tx;(n)|y(n)|2.
N JV N
By the formula ^ u^Vvk — [ukVjc]M_1 — ^2 vJb-iVtijb, it follows that
k=M k=M
b' b'
Y,y("W \p(n)Ay(n)] = [y(n)p(n)Ay(n)]f_1 - £>(n - l)|Vy(n)|2.
n=c n=e
Substitution in (7.1.5) yields
b'
(7.1.6) £ {-p(n - l)|Vy(n)|2 n + ™(n)|y(n)|2}
n=c
b'
+ [y^M^A^n)]^, = A£™(n)|y(n)|2 .

126
ALOUF JIRARI
Let tpb' = 0i + m&/(A)02, where 6\ and 02 satisfy conditions (3.2.1), be the
solution of (7.1.2) such that if>v(V + 1) + * i/>b'(b') = 0. Then,
p(e - l)1>h,(e - l)AVv(e - 1) = p(e - 1)
-1
-1
lp(e-l)J
— mt>i(\) ,
-1
[p(e - 1)
+ rrib'(X)
p(e - 1)
and, if Vv(&') = #, then A^(6') = Vv(&' + 1) - Vv(&') = -(1 + k)K, so that
p(b'Wh,(b')A1>h,(b') = -p(6')^(l + *)ff = -P(6')(l + *)|tf|2 .
If y(n) is replaced by ipb'(n) in (7.1.6), then
*'
(7.1.7) Yl {-P(n ~ 1)|V^(^)|2 + w(n)\M*)\2}
n=e
, »'
-p(6')(l + *)|#|2 + -^^3^ + mb,(\) = A£>(n)|<M")|2 •
Let A = /i + zV, i/ ^ 0. Then separating the real part from the imaginary part
in (7.1.7) leads to the equations
and
(7.1.8)
J2utn)\to(n)\* = !*&*&,
0
£ {-p(n - l)|VW(n)|2 + w(n)\Mn)\2} ~ p(b')(l + k)\K\2
- -1 -Re(mt,(A)) + /iIm(mt'(A)).
p(e - 1)
Note that \K\2=\ipb<(b')\2<oo. Otherwise, if \K\=oo, for £to(n)|Vv(n)|2<
OO
to hold, w(n) would have to have at least a zero of multiplicity 2 at &', which
is not the case for the generalized Tchebyshev and the Krawtchouk polynomials
(see Sections 6.1 and 6.2).

STURM-LIOUVILLE DIFFERENCE EQUATIONS 127
Now if k = -1, then A^/(6') = 0 and m6/(A) = - *L\> so that> by (718)>
^{-P(n-l)|V^(n)|2+«'(n)|^(n)|2} = 1f^irRe K(A)^Im(w;(A)).
Since p(n) < 0 and V < 6,
b'
^2\-p(n - l)|VVv(n)| n+iu(n)|Vv(n)| j
n=e
-^^-Re^A))-^"^11
p(e-l) v\"" ' ^ „
Now, if 6' —► 6 on the left-hand side, it follows that
£{^(n-l)|V«n)^
Theorem 7.1.3
For all A with Im(X) ^ 0, there is a solution ipa(n, A) = #i(n, A) + ma(A)02(rc, A)
of (7.1.2) in H1(ay e — l,p, u;), the Sobolev space with inner product
e-l
(y, z) = X) {"^n " 1)V2/(n)VJ(n) + w(n)2/(n)j(n)} .
Proof
Following the steps of the previous proof,
(7.1.9) £ {-p(n - l)|Vy(n)|2 +w(n)\y(n)\2}
n=a'
e-1
+ B/(n)Kn)Ay(n)]!"-i = A^u;(n)|y(n)|2 .

128 ALOUF JIRARI
Let ipai = $i + mai(\)02, where 0\ and 02 satisfy conditions (3.2.1), be the
solution of (7.1.2) such that Vv(a' - 1) + /*Vv(a') = 0. Then,
p(e - l)Vv(e - l)AVv(e - 1) = —, 77 - ma/(A),
p{e - 1)
and, if rpa'{a' — 1) = H, then
p(a' - l)^a,(a' - l)A^(a' - 1) = -p(a' - 1) (± + l) |tf |2.
Substituting Vv(rc) for y(n) in (7.1.2), letting A = p + zV with 1/ ^ 0, and
separating real from imaginary parts yield the equations
and
(7.1.10) £ {-p(n - l)|VVv(«)|2 + ^(n)|V-a-(n)r}+p(«'-l) Q + l) |# |2
1 . ^ / Mu Im(mq/(A))
+ Re (ma/(A)) - //-—- ^—^
-p(e-l) ""^^ " v
Note that # can always be chosen so that \H\ < oo. So, if/i = —1,
A^«/(a' - 1) = 0 and ma/(A) = - *; ~ ;, so that, by (7.1.10),
/\V2\CL' — \)
£ {-p(„-l)|V^(n)r+^)|^|2} = -7^Iy+Re (ma,(A))Vm(m;'(A)),
n=a'
and
c-l
p(e-l)
Y, {~P(n - l)|VVa(n)|2 + w(n)\Mn)\2} < oo.
n=a
Theorem 7.1.4
Ify satisfies V[p(n)Ay(n)] + ir;(n)y(n) = Au;(n)y(n) witn tne conditions that
p(a — l)Vy(a) = p(b)Ay(b) = 0, and y and 2: are in H1(a,b,p,w). Then,
■z(a - \)p{a - l)Vy(a) = z(6)p(6)Ay(6) = 0.

STURM-LIOUVILLE DIFFERENCE EQUATIONS 129
Proof
It suffices to show that \z(b)\ and \z(a — 1)| are finite.
z(a — 1) can actually be chosen so that \z(a — 1)| < oo.
b
For z(b), since z is in H1(a,bJp,w)J V^ «{-p(n —l)|Vz(n)| +u;(n)|z(n)| ><
n—a
OO.
In particular, iu(&)|z(&)| < oo. Because 0 < \w(b)\ < oo and |^(^)|2 < oo,
it must be true that \z(b)\ < oo. ■
Definition 7.1.5
t * o u *u j-ff ± • u <o \< \ V\p(n)Ay(n)] + w(n)y(n)
Let I be the difference operator given by (ly)(n) = —!£A-^— '/ v v .
w[n)
Set
Dc = \y S H\a,b,p,w): ly £ H1(a,bip,w)ip(a-l)Vy(a) = 0 , p(b)Ay(b) = 0}.
The operator C is defined by setting Cy = ly for all y in £>£.
Theorem 7.1.6 (The Dirichlet Formula)
For y in Dc and z in H1(a16,p, w),
(7.1.11) (£y,*)* = (y,*)iP •
Proof
(7.1.11) means that
b
^{V[p(n)At/(n)] + w(n)y(n)}-z(n)
n—a
b
- ^{~-P(n ~ l)Vy(n)Vz(n) + iu(n)t/(n)z(n)} ,
which is clearly true, given the boundary conditions in Dc, since
b b
£vb(n)Ay(n)]J(n) = [z(n)p(n)Ay(n)]*_i - X>(" ~ l)Vy(n)Vz(n) . ■
n=a n—a
Theorem 7.1.7
Dc is dense in H1(a1b1p1w).
Proof
Suppose that for some/in H1(aib1piw)1 (t/, /)#i = 0 for every t/in Ac. (7.1.11)
implies that (Ly,f)i2 = 0. If t/ is chosen in H1(a1bipiw) so that Cy — j (this
is possible if one shows that the range of C is all of H1(a1bip1w)1 which is
the objective of Theorem 7.1.9, then Ly = f in l2{a,b\w) and (/,/)/» = 0.
Therefore, / = 0 in P{a, 6; w) and / = 0 in Hl{a, 6,p, w). ■

130 ALOUF JIRARI
Theorem 7.1.8
C is symmetric.
Proof
In the Dirichlet formula, (t/, z)jji = (Lt/, z)^2, replace z by Cz. Then,
(7.1.12) (y1Cz)H^ = (Ly1Lz)l2.
Reversing the roles oft/ and z in (7.1.12) gives (z, £t/)#i = (Lz, Lyjp, and taking
the complex conjugate implies that
(7.1.13) (Cy%z)Hi = (Ly,Lz)t*.
By (7.1.12) and (7.1.13), it follows that (y, Cz)Hx = (£y, z)Hi. ■
Theorem 7.1.9
The inverse operator C~l exists and is bounded on H1(a1bipiw) . Moreover,
the range of C is all ofH1(a16,p, w).

STURM-LIOUVILLE DIFFERENCE EQUATIONS 131
Proof
1) C is one to one: For, let u and v in H1(a1b1p1w) be such that Cu = Cv.
Then Lu = Lv, or L(ti — v) = 0. Using (7.1.11) for t/ = z = u — v, it follows that
(L(ti - w), u - v)t2 =(u-v,u- v)Hi = \\u - v\\2H1,
= £ {-p(n - !)l v(" - »)rf+™(* - *)(»)l2},
= 0.
Therefore, u — v = 0 in Hl(a,b,p,w), or ti = v in H1(a1b1p1w)1 and so £-1
exists.
2) C~l is bounded: Let / be in H1(aibipi w) and use (7.1.11)for y = z = C~lf.
Then {C~lf,C-lf)m = {f,L~lf)o and
\\C-Xf\\h < WfWoWL-'fWo < \\L-%A\f\\h < II^IHI/lllf.,
from Definition 7.1.1 of H1(aibipi w). Dividing by ||/||jyi and taking the supre-
mum give
H^llm < \\l-%12.
3) The range of C is all of H1^, 6,p, iv): Let / be in /^(a, 6,p, vu)and a<a!<V<
b.
From Chapter 3, the solution of£y = f with W[y, i>a'](a>') = W[y1 Vv](&'+1) = 0
b'
is t/(n) = ]T ^/(A^n)™^)/^), where ||t/||#i(a',&') < #a'&'||/||#i(a',&')- As
i=a'
(a7, 6') approaches (a, 6), t/(n) approaches t/a6(rc), the solution of the boundary
value problem on (a, 6), and Ga'b' approaches Gab- Hence,
Ill/afrlljyriCa',^') < HyaftlljST^a,*) < Kab\\f\\w(a,b)'
Letting (a', V) approach (a, b) on the left-hand side implies that
and the proof is complete. ■
Theorem 7.1.10
C and therefore C~l are self-adjoint.
Proof

132
ALOUF JIRARI
Dc is dense in H1(a1b1p1w) by Theorem 7.1.7 and £ is symmetric by
Theorem 7.1.8. Therefore, £ is self-adjoint and, since £-1 exists by Theorem
7.1.9, £-1 is also self-adjoint. ■
Remark 7.1.11
The operator £ is positive. In fact, by (7.1.11), for y in Dc,
{^y^y)m = (y,£y)w = (Ly,Ly)t2 > 0.
This implies that the eigenvalues A of £ are positive. In fact, if y is an eigen-
function,
0 < (Cy, y)m = (At/, y)m = M\y\\m and so A > 0 .
Now, let p(X) be the spectral projection measure of £. Then,
/•OO
for all / in H\a, 6,p, w) , f(n) = / dp(X)f(n) ,
Jo
rOO
for all f in Dc , Cf(n) = / Xdp(X)f(n) ,
Jo
r°° l
and forall/intf1^,*,;?,™) , C~1f(n) = -dp(X)f(n) .
Jo X
Theorem 7.1.12
The formula for the spectral resolution in ^2(a, 6; w),
b-a
f(n) = ^2{f(n)^k(n))yk(n)
kzzO
still holds in H1(a16, p, w).

STURM-LIOUVILLE DIFFERENCE EQUATIONS 133
Proof
Let / be in H1(a1b1p1w) and g in Dc< Then,
b-a
•£<
*=0
(^, S(/(n)>yfc(n))yfc(n)),a = (Lg,f)t2,
= (9J)m , by (7.1.11),
f°°
= (g, dp(\)f(n))Hi,
Jo
= (Lg, / dp{\)f{n))t^ by (7.1.11),
Jo
so that
b-a
v" -co
(L«7, QT(/(n), yfc(n))yfc(n) - / dp{\)f{n)))t> = 0,
for all # in Ac. This holds for any element Cg of H1(aibipiw); that is, any
/•oo &-«
element L# in $?{a,b\w). Therefore, / dp(X)f(n) = Y^(/(n), yk(n))yk(n).
7.2 Infinite Intervals
(6.2.4) and (6.4.4) show that, for the generalized Laguerre and the Charlier
problems, the difference equation is of the form
(7.2.1) V[p(n)Ay(n)] = fiw(n)y(n) ,
where n = a,..., oo, with a in the limit-circle case, oo in the limit-point case,
and p(n) positive for n > a. (7.2.1) can be written equivalently
(7.2.2) V[-p(n)Ay(n)] + w(n)y(n) = Xw(n)y(n) ,
where A = 1 — //.
Here again, the eigenvalues A are nonzero. In fact, they are 1 +pk > 0 for the
Charlier polynomials (p > 1, k > 0), and 1 — k(l — e7) > 0 for the generalized
Laguerre polynomials (7 > 0, k > 0).

134 ALOUF JIRARI
Definition 7.2.1
H1(a1 oo, p, w) is the Hilbert space consisting of all sequences y(a), y(a + 1),...,
for which
oo
(7.2.3) IM&, = 52 {p(» " l)|Vj/(n)|2 + w(n)\y(n)\2} < oo.
n=a
The inner product in H1(ai oo,p, w) is given by
oo
(7.2.4) (y, z)H. = 53 {p(» - l)Vj/(n)Vz(n) + w(n)y(n)I(n)} .
Theorem 7.2.2
If y satisfies V[—p(n)Ay(n)] + w(n)y(n) = Xw(n)y(n) with the condition that
p(a — l)Vy(a) = 0, and y and z are in H1(a1 oo,p, w), then
J(a — l)p(a — l)Vy(a) = 0 and lim J(n)p(n)Ay(n) = 0 .
n—»>oo
Proof
it \ V^ / ^ f V[-p(n)Ay(n)] -f ti;(n)y(n) \ __, .
(Ly, z)/a = 2^w(n) j -^ > z(n),
n=a ^ ^ ' ^
oo
= 5Z {Vi-pi^Ayin)] + tx;(n)y(n)} z(n),
n = a
oo
= X) {P(n - l)Vy(n)Vz(n) + w;(n)y(n)J(n)} - ^(nMnJAyfa)]^ ,
n=a
= (y,*)iP - [^(nWnlAyln)]*! .
Now, y and z are in H1(ay oo,p, w). This implies on one hand that (y, z)#i <oo.
On the other hand, it implies that Ly and z are in ^2(a, oo;w), so that
(Ly,z)i2 < oo. Therefore, [z(n)p(n)Ay(n)](^_1 < oo.
Since p(a — l)Ay(a — 1) = p(a — l)Vy(a) = 0, z can be defined at a — 1 so that
\z(a - 1)| < oo and J(a - l)p(a - l)Vy(a) = 0.

STURM-LIOUVILLE DIFFERENCE EQUATIONS 135
Also, lim ~z(n)p(n)Ay(n) < oo, but it must be true that this limit vanishes.
n—+00
In fact, if lim J(n)p(n)Ay(n) = A where A ^ 0, then, for all e > 0, there exists
n—+00
iV such that, for all n > N, \J(n)p(n)Ay(n)\ > A — e, or |p1/2(n)At/(n)| >
|pi/2(n)J(n)|-
OO
y E H1(ayooypyw) implies that /_^{p(n — l)|Vy(n)|2 + tx;(n)|2/(n)|2} < 00.
n=a
OO
So, in particular, Y^P(n ~~ l)l^2/(n)|2 < °°- This means that /(n) = p(n —
n=a
l)1/2Vt/(n) is in ^2(a,oo; tx;). Consequently, g(n) = /(n + 1) = p(n)1/2Vt/(n +
1) = p(n)1/2 At/(n) is in ^2(a, 00; u;).
Now, since z is in H1(aiooipiw)i w(n)ll2z(ri) is also in^2(a, 00; u;). Therefore,
the product (p(n)ll2Ay(n))(w(n)ll2z(n)) is in ^(0,00; w). But,
00 00 .j
£ IpW^AyOOIKn^'zHl > (A - e) ^ __7__w(„)|z(„)|,
n=N n=N*^^ ' \ /I
"( It^)1'2'
00 1
\ti;(n)/
For the generalized Laguerre problem, —7—- = — -7 = n — A + a + 1,
w(n) T(n—A+a+1)
p(n)
while for the Charlier polynomials, —— = 1. Therefore, in both cases,
w(n)
00
y^ Ip(n)1 '2Ay(n)\ \w(n)l'2z(n)\ diverges. This is clearly a contradiction. Thus,
n = a
the assumption that A ^ 0 must be wrong. So, lim ~z(n)p(n) Ay(n) = 0. ■

136
ALOUF JIRARI
Definition 7.2.3
i.^. • //. w x V\—p(n)Ay(n)] + w(n)y(n)
Let £ be the difference operator given by (ty)(n) = L FK J V\ V J K .
w(n)
Set
Dc = {ye Hl{a,oo,p,w) : ^y G Hl{a,oo,p,w),p{a- l)Vy(a) = 0}.
The operator £ is defined by setting Cy = ^/ for all y in £>£.
Then, all the properties proved for C in the finite case still hold in the infinite
one.

References
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489 Jean-Pierre Gabardo, Extension of positive-definite distributions and maximum entropy,
1993
488 Chris Jantzen, Degenerate principal series for symplectic groups, 1993
487 Sagun Chanillo and Benjamin Muckenhoupt, Weak type estimates for Cesaro sums of
Jacobi polynomial series, 1993
486 Brian D. Boe and David H. Collingwood, Enright-Shelton theory and Vogan's problem
for generalized principal series, 1993
485 Paul Feit, Axiomization of passage from "local" structure to "global" object, 1993
484 Takehiko Yamanouchi, Duality for actions and coactions of measured groupoids on von
Neumann algebras, 1993
(See the AMS catalog for earlier titles)

Second-Order Sturm-Liouville Difference Equations
and Orthogonal Polynomials
Alouf Jirari
This well-written book is a timely and significant contribution to the
understanding of difference equations. Presenting machinery for analyzing many
discrete physical situations, the book would be of interest to physicists and
engineers as well as mathematicians. The book develops a theory for regular and
singular Sturm-Liouville boundary value problems for difference equations,
generalizing many of the known results for differential equations. Discussing the
self-adjointness of these problems as well as their abstract spectral resolution
in the appropriate L2 setting, the book gives necessary and sufficient conditions
for a second-order difference operator to be self-adjoint and have orthogonal
polynomials as eigenfunctions. These polynomials are classified into four
categories, each of which is given a properties survey and a representative example.
Finally, the book shows that the various difference operators defined for these
problems are still self-adjoint when restricted to "energy norms". This book
would be suitable as a text for an advanced graduate course on Sturm-Liouville
operators or on applied analysis.
ISBN 0-8218-0359-X