ENCYCLOPEDIA
OF MATHEMATICS AND ITS ApPLICATIONS
Editorial Board R. S. Doran, P. Flajolet, M. Ismail, T.-y' Lam, E. Lutwak
Volume 96
Basic Hypergeometric Series Second Edition This revised and expanded new edition will continue to meet the need for an authoritative, up-to-date, self contained, and comprehensive account of the rapidly growing field of basic hypergeometric series, or q-series. It contains almost all of the important summation and transformation formulas of basic hypergeometric series one needs to know for work in fields such as combinatorics, number theory, modular forms, quantum groups and algebras, probability and statistics, coherent-state theory, orthogonal polynomials, or approximation theory. Simplicity, clarity, deductive proofs, thoughtfully designed exercises, and useful appendices are among its strengths. The first five chapters cover basic hypergeometric series and integrals, whilst the next five are devoted to applications in various areas including Askey-Wilson integrals and orthogonal polynomials, partitions in number theory, multiple series, and generating functions. Chapters 9 to 11 are new for the second edition, the final chapter containing a simplified version of the main elements of the theta and elliptic hypergeometric series as a natural extension of the single-base q-series. Elsewhere some new material and exercises have been added to reflect recent developments, and the bibliography has been revised to maintain its comprehensive nature.
ENCYCLOPEDIA OF MATHEMATICS AND ITS ApPLICATIONS
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ENCYCLOPEDIA OF MATHEMATICS AND ITS APPLICATIONS
BASIC HYPERGEOMETRIC SERIES
Second Edition
GEORGE GASPER Northwestern University, Evanston, Illinois, USA MIZAN RAHMAN Carleton University, Ottawa, Canada
CAMBRIDGE UNIVERSITY PRESS
PUBLISHED BY THE PRESS SYNDICATE OF THE UNIVERSITY OF CAMBRIDGE
The Pitt Building, Trumpington Street, Cambridge, United Kingdom CAMBRIDGE UNIVERSITY PRESS
The Edinburgh Building, Cambridge CB2 2RU, UK 40 West 20th Street, New York, NY 10011-4211, USA 477 Williamstown Road, Port Melbourne, VIC 3207, Australia Ruiz de Alarc6n 13, 28014 Madrid, Spain Dock House, The Waterfront, Cape Town 8001, South Africa http://www.cambridge.org
© Cambridge University Press 1990, 2004 This book is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 1990 Second edition 2004. Printed in the United Kingdom at the University Press, Cambridge
Typeface Computer Modern 10/12 pt.
System 'lEX
[TB]
A catalogue record for this book is available from the British Library Library of Congress Cataloguing in Publication data Gasper, George. Basic hypergeometric series / George Gasper, Mizan Rahman. - 2nd edn. p. cm. - (Encyclopedia of mathematics and its applications; v. 96) Includes bibliographical references and indexes. ISBN 0 521 83357 4 1. Hypergeometric series. I. Rahman, Mizan. II. Title. III. Series. QA353.H9G37 2004 515'.243--dc22 2004045686 ISBN 0 521 83357 4 hardback
The publisher has used its best endeavors to ensure that the URLs for external websites referred to in this book are correct and active at the time of going to press. However, the publisher has no responsibility for the websites and can make no guarantee that a site will remain live or that the content is or will remain appropriate.
To Brigitta, Karen, and Kenneth Gasper and Babu, Raja, and to the memory of Parul S. Rahman
Contents
page xiii xxi xxv
Foreword Preface Preface to the second edition
1 Basic hypergeometric series
1 1 1
1.1 Introduction 1.2 Hypergeometric and basic hypergeometric series 1.3 The q-binomial theorem 1.4 Heine's transformation formulas for
8 2¢1
series
13
1.5 Heine's q-analogue of Gauss' summation formula
14
1.6 Jacobi's triple product identity, theta functions, and elliptic numbers
15
1. 7 A q-analogue of Saalschiitz's summation formula
17
1.8 The Bailey-Daum summation formula
18
1.9 q-analogues of the Karlsson-Minton summation formulas
18
1.10 The q-gamma and q-beta functions
20
1.11 The q- integral
23
Exercises Notes
24 34
2 Summation, transformation, and expansion formulas
38
2.1 Well-poised, nearly-poised, and very-well-poised hypergeometric and basic hypergeometric series
38
2.2 A general expansion formula
40
2.3 A summation formula for a terminating very-well-poised 4¢3 series
41
2.4 A summation formula for a terminating very-well-poised 6¢5 series
42
2.5 Watson's transformation formula for a terminating very-well-poised 8¢7
series
42
2.6 Jackson's sum of a terminating very-well-poised balanced 8¢7 series
43
2.7 Some special and limiting cases of Jackson's and Watson's formulas: the Rogers-Ramanujan identities
44
2.8 Bailey's transformation formulas for terminating 5¢4 and 7¢6 series
45
2.9 Bailey's transformation formula for a terminating
47
vii
1O¢9
series
viii
Contents
2.10 Limiting cases of Bailey's 1O¢9 transformation formula
48
2.11 Bailey's three-term transformation formula for VWP-balanced S¢7 series
53
2.12 Bailey's four-term transformation formula for balanced 1O¢9 series
55
Exercises Notes
58 67
3 Additional summation, transformation, and expansion formulas
69
3.1 Introduction 3.2 Two-term transformation formulas for 3¢2 series
69 70
3.3 Three-term transformation formulas for 3¢2 series
73
3.4 Transformation formulas for well-poised 3¢2 and very-well-poised S¢4 series with arbitrary arguments
74
3.5 Transformations of series with base q2 to series with base q
77
3.6 Bibasic summation formulas 3.7 Bibasic expansion formulas
80 84
3.8 Quadratic, cubic, and quartic summation and transformation formulas
88
3.9 Multibasic hypergeometric series
95
3.10 Transformations of series with base q to series with base q2 Exercises Notes
4 Basic contour integrals 4.1 Introduction 4.2 Watson's contour integral representation for 2¢1 (a, b; c; q, z) series 4.3 Analytic continuation of
96 100 111
2¢1(a, b; c; q, z)
113 113 115 117
4.4 q-analogues of Barnes' first and second lemmas
119
4.5 Analytic continuation of r+l ¢r series
120
4.6 Contour integrals representing well-poised series
121
4.7 A contour integral analogue of Bailey's summation formula
123
4.8 Extensions to complex q inside the unit disc
124
4.9 Other types of basic contour integrals
125
4.10 General basic contour integral formulas
126
Contents
ix
4.11 Some additional extensions of the beta integral
129
4.12 Sears' transformations of wen-poised series
130
Exercises Notes
132 135
5 Bilateral basic hypergeometric series 5.1 Notations and definitions 5.2 Ramanujan's sum for 1 'l/Jl (a; b; q,
137 137 138
z)
5.3 Bailey's sum of a very-wen-poised 6'l/J6 series
140
5.4 A general transformation formula for an r'l/Jr series
141
5.5 A general transformation formula for a very-wen-poised 2r'l/J2r series
143
5.6 Transformation formulas for very-wen-poised 8'l/J8 and
145
1O'l/J1O
series
Exercises Notes
146 152
6 The Askey-Wilson q- beta integral and some associated formulas
154
6.1 The Askey-Wilson q-extension of the beta integral
154
6.2 Proof of formula (6.1.1)
156
6.3 Integral representations for very-wen-poised 8¢7 series
157
6.4 Integral representations for very-wen-poised
159
1O¢9
series
6.5 A quadratic transformation formula for very-wen-poised balanced series 6.6 The Askey-Wilson integral when max
(Ial, Ibl, lei, Idl)
1O¢9
~ 1
Exercises Notes
7 Applications to orthogonal polynomials
162 163 168 173
175
7.1 Orthogonality
175
7.2 The finite discrete case: the q-Racah polynomials and some special cases
177
7.3 The infinite discrete case: the little and big q-Jacobi polynomials
181
7.4 An absolutely continuous measure: the continuous q-ultraspherical polynomials 7.5 The Askey-Wilson polynomials
184 188
Contents
x
7.6 Connection coefficients 7.7 A difference equation and a Rodrigues-type formula for the
195 197 199
Askey-Wilson polynomials Exercises Notes
213
8 Further applications 8.1 Introduction 8.2 A product formula for balanced
217 4
polynomials
8.3 Product formulas for q-Racah and Askey-Wilson polynomials
217 218 221
8.4 A product formula in integral form for the continuous q-ultraspherical 223
polynomials 8.5 Rogers' linearization formula for the continuous q-ultraspherical
226
polynomials 8.6 The Poisson kernel for
Cn(x; ,6lq)
227
8.7 Poisson kernels for the q-Racah polynomials
229
8.8 q-analogues of Clausen's formula
232
8.9 Nonnegative basic hypergeometric series
236
8.10 Applications in the theory of partitions of positive integers
239
8.11 Representations of positive integers as sums of squares
242
Exercises Notes
9 Linear and bilinear generating functions for basic orthogonal polynomials
245
257
259
9.1 Introduction 9.2 The little q-Jacobi polynomials
259
9.3 A generating function for Askey-Wilson polynomials
262
9.4 A bilinear sum for the Askey-Wilson polynomials I
265 269 270
9.5 A bilinear sum for the Askey-Wilson polynomials II 9.6 A bilinear sum for the Askey-Wilson polynomials III Exercises Notes
260
272 281
Contents
xi
10 q-series in two or more variables
282
10.1 Introduction 10.2 q-Appell and other basic double hypergeometric series
282 282
10.3 An integral representation for <[>(1) (qa; qb, qb'; qC; q; x, Y)
284
10.4 Formulas for <[>(2) (qa; qb, qb' ; qC, qC' ; q; x, Y)
286
10.5 Formulas for <[>(3) (qa, qa'; qb, qb'; qC; q; x, Y)
288
10.6 Formulas for a q-analogue of F4
290
10.7 An Askey-Wilson-type integral representation for a q-analogue of Fl
294
Exercises Notes
11 Elliptic, modular, and theta hypergeometric series
296 301
302
11.1 Introduction 11.2 Elliptic and theta hypergeometric series
302 303
11.3 Additive notations and modular series 11.4 Elliptic analogue of Jackson's S¢7 summation formula
312 321
11.5 Elliptic analogue of Bailey's transformation formula for a terminating 1O¢9
series
323
11.6 Multibasic summation and transformation formulas for theta hypergeometric series
325
11.7 Rosengren's elliptic extension of Milne's fundamental theorem
331
Exercises Notes
336 349
Appendix I Identities involving q-shifted factorials, q-gamma functions and q-binomial coefficients
351
Appendix II Selected summation formulas
354
Appendix III Selected transformation formulas
359
References
367
Symbol index
415
Author index
418
Subject index
423
Foreword
My education was not much different from that of most mathematicians of my generation. It included courses on modern algebra, real and complex variables, both point set and algebraic topology, some number theory and projective geometry, and some specialized courses such as one on Riemann surfaces. In none of these courses was a hypergeometric function mentioned, and I am not even sure if the gamma function was mentioned after an advanced calculus course. The only time Bessel functions were mentioned was in an undergraduate course on differential equations, and the only thing done with them was to find a power series solution for the general Bessel equation. It is small wonder that with a similar education almost all mathematicians think of special functions as a dead subject which might have been interesting once. They have no idea why anyone would care about it now. Fortunately there was one part of my education which was different. As a junior in college I read Widder's book The Laplace Transform and the manuscript of its very important sequel, Hirschman and Widder's The Convolution Transform. Then as a senior, 1. 1. Hirschman gave me a copy of a preprint of his on a multiplier theorem for Legendre series and suggested I extend it to ultraspherical series. This forced me to become acquainted with two other very important books, Gabor Szego's great book Orthogonal Polynomials, and the second volume of Higher Transcendental Functions, the monument to Harry Bateman which was written by Arthur Erdelyi and his co-workers W. Magnus, F. Oberhettinger and F. G. Tricomi. From this I began to realize that the many formulas that had been found, usually in the 18th or 19th century, but once in a while in the early 20th century, were useful, and started to learn about their structure. However, I had written my Ph.D. thesis and worked for three more years before I learned that not every fact about special functions I would need had already been found, and it was a couple of more years before I learned that it was essential to understand hypergeometric functions. Like others, I had been put off by all the parameters. If there were so many parameters that it was necessary to put subscripts on them, then there has to be a better way to solve a problem than this. That was my initial reaction to generalized hypergeometric functions, and a very common reaction to judge from the many conversations I have had on these functions in the last twenty years. After learning a little more about hypergeometric functions, I was very surprised to realize that they had occurred regularly in first year calculus. The reason for the subscripts on the parameters is that not all interesting polynomials are of degree one or two. For Xlll
xiv
Foreword
a generalized hypergeometric function has a series representation 00
(1) n=O
with cn+I/cn a rational function of n. These contain almost all the examples of infinite series introduced in calculus where the ratio test works easily. The ratio cn+I/cn can be factored, and it is usually written as
(n + b1 )
...
(n
(2)
+ bq ) (n + 1) .
Introduce the shifted factorial
(a)O = 1, (3)
(a)n = a(a + 1) ... (a + n - 1), Then if Co = 1, equation (2) can be solved for
n = 1,2, ... Cn
as
(al)n··· (ap)n xn (bdn ... (bq)n n!
Cn =
(4)
and
(5) is the usual notation. The first important result for a pFq with p sum 3F2
[
-n, a, b c, a
+b+ 1 -
C -
n
;
> 2,
q
> 1 is probably Pfaff's
(C - a)n(c - b)n (c)n(c - a - b)n'
]
1
n
=
0,1, ....
This result from 1797, see Pfaff [1797], contains as a limit when n ----; important result usually attributed to Gauss [1813],
F [a, b. 1] 2
c'
1
=
r(c)r(c-a-b) r(c-a)r(c-b)'
Re (c-a-b) > O.
00,
(6)
another
(7)
The next instance is a very important result of Clausen [1828]:
{ F [ 2
1
a, b ·x] a + b+ ~ ,
}2 =
3
F [2a, 2b, a + b ·x] 2 a + b + ~, 2a + 2b ' .
(8)
Some of the interest in Clausen's formula is that it changes the square of a class of 2Fl'S to a 3F2. In this direction it is also interesting because it was probably the first instance of anyone finding a differential equation satisfied by [Y(X)]2, y(x)z(x) and [Z(X)]2 when y(x) and z(x) satisfy
a(x)y" + b(x)y' + c(x)y
=
O.
(9)
Foreword
xv
This problem was considered for (9) by Appell, see Watson [1952]' but the essence of his general argument occurs in Clausen's paper. This is a common phenomenon, which is usually not mentioned when the general method is introduced to students, so they do not learn how often general methods come from specific problems or examples. See D. and G. Chudnovsky [1988] for an instance of the use of Clausen's formula, where a result for a 2Fl is carried to a 3F2 and from that to a very interesting set of expansions of 1[-1. Those identities were first discovered by Ramanujan. Here is Ramanujan's most impressive example:
There is another important reason why Clausen's formula is important. It leads to a large class of 3F2 's that are nonnegative for the power series variable between -1 and 1. The most famous use of this is in the final step of de Branges' solution of the Bieberbach conjecture, see de Branges [1985]. The integral of the 2Fl or Jacobi polynomial he had is a 3F2' and its positivity is an easy consequence of Clausen's formula, as Gasper had observed ten years earlier. There are other important results which follow from the positivity in Clausen's identity. Once Kummer [1836] wrote his long and important paper on 2Fl 's and IFI 's, this material became well-known. It has been reworked by others. Riemann redid the 2Fl using his idea that the singularities of a function go a long way toward determining the function. He showed that if the differential equation (9) has regular singularities at three points, and every other point in the extended complex plane is an ordinary point, then the equation is equivalent to the hypergeometric equation x(l - x)y"
+ [c -
(a
+ b + l)x]y' -
aby = 0,
(11)
which has regular singular points at x = 0,1,00. Riemann's work was very influential, so much so that much of the mathematical community that considered hypergeometric functions studied them almost exclusively from the point of view of differential equations. This is clear in Klein's book [1933], and in the work on multiple hypergeometric functions that starts with Appell in 1880 and is summarized in Appell and Kampe de Feriet [1926]. The integral representations associated with the differential equation point of view are similar to Euler's integral representation. This is
[a,
F 2
1
b. x]
c'
=
r(c) (1(1 _ xt)-at b- 1 (1 _ t)c-b- 1 dt r(b)r(c - b) io '
(12)
Ixl < 1, Re c > Re b > 0, and includes related integrals with different contours. The differential equation point of view is very powerful where it works, but it
xvi
Foreword
does not work well for p 2: 3 or q 2: 2 as Kummer discovered. Thus there is a need to develop other methods to study hypergeometric functions. In the late 19th and early 20th century a different type of integral representation was introduced. These two different types of integrals are best represented by Euler's beta integral
r1t
io
a-
1(1 _ t)b-1dt = r(a)r(b) , r(a+b)
Re (a, b)
>0
(13)
and Barnes' beta integral
-1
2n
1
00
r(a + it)r(b + it)r(c - it)r(d - it) dt
(14)
-00
r(a + c)r(a + d)r(b + c)r(b + d) ( ) ra+b+c+d
Re (a, b, c, d)
> o.
There is no direct connection with differential equations for integrals like (14), so it stands a better chance to work for larger values of p and q. While Euler, Gauss, and Riemann and many other great mathematicians wrote important and influential papers on hypergeometric functions, the development of basic hypergeometric functions was much slower. Euler and Gauss did important work on basic hypergeometric functions, but most of Gauss' work was unpublished until after his death and Euler's work was more influential on the development of number theory and elliptic functions. Basic hypergeometric series are series L Cn with cn+I/cn a rational function of qn for a fixed parameter q, which is usually taken to satisfy Iql < 1, but at other times is a power of a prime. In this Foreword Iql < 1 will be assumed. Euler summed three basic hypergeometric series. The one which had the largest impact was 00
2) _1)n q(3n -n)/2 = 2
(q; q)oo,
(15)
-00
where 00
(a; q)oo
=
II (1 -
aqn).
(16)
n=O If
(17) then Euler also showed that
1 xn ..,.....-....,...--2:-(x; q)oo - n=O (q; q)n' 00
Ixl
(18)
Foreword
xvii
and (19) Eventually all of these were contained in the q-binomial theorem
(20) While (18) is clearly the special case a = 0, and (19) follows easily on replacing x by xa- 1 and letting a ----t 00, it is not so clear how to obtain (15) from (20). The easiest way was discovered by Cauchy and many others. Take a = q-2N, shift n by N, rescale and let N ----t 00. The result is called the triple product, and can be written as 00
(x; q)00(qx- 1 ; q)oo(q; q)oo
=
L
(-1tq(~)xn.
(21)
n=-(XJ
Then q ----t q3 and x = q gives Euler's formula (15). Gauss used a basic hypergeometric series identity in his first proof of the determination of the sign of the Gauss sum, and Jacobi used some to determine the number of ways an integer can be written as the sum of two, four, six and eight squares. However, this particular aspect of Gauss' work on Gauss sums was not very influential, as his hypergeometric series work had been, and Jacobi's work appeared in his work on elliptic functions, so its hypergeometric character was lost in the great interest in the elliptic function work. Thus neither of these led to a serious treatment of basic hypergeometric series. The result that seems to have been the crucial one was a continued fraction of Eisenstein. This along with the one hundredth anniversary of Euler's first work on continued fractions seem to have been the motivating forces behind Heine's introduction of a basic hypergeometric extension of 2Fl (a, b; c; x). He considered
Ixl < 1.
(22)
Observe that
so
Heine followed the pattern of Gauss' published paper on hypergeometric series, and so obtained contiguous relations and from them continued fraction
xviii
Foreword
expansions. He also obtained some series transformations, and the sum a q. b '" [ q, c-a-b ] _ (c-a) q ; q 00 (c-b q ; q ) 00 c ' q, q ( c. ) (c-a-b. ) , q q,qooq ,qoo
2'/'1
(23)
This sum becomes (7) when q ---+ l. As often happens to path breaking work, this work of Heine was to a large extent ignored. When writing the second edition of K ugelfunctionen (Heine [1878]) Heine decided to include some of his work on basic hypergeometric series. This material was printed in smaller type, and it is clear that Heine included it because he thought it was important, and he wanted to call attention to it, rather than because he thought it was directly related to spherical harmonics, the subject of his book. Surprisingly, his inclusion of this material led to some later work, which showed there was a very close connection between Heine's work on basic hypergeometric series and spherical harmonics. The person Heine influenced was L. J. Rogers, who is still best known as the first discoverer of the Rogers-Ramanujan identities. Rogers tried to understand this aspect of Heine's work, and one transformation in particular. Thomae [1879] had observed this transformation of Heine could be written as an extension of Euler's integral representation (12), but Rogers was unaware of this explanation, and so discovered a second reason. He was able to modify the transformation so it became the permutation symmetry in a new series. While doing this he introduced a new set of polynomials which we now call the continuous q-Hermite polynomials. In a very important set of papers which were unjustly neglected for decades, Rogers discovered a more general set of polynomials and found some remarkable identities they satisfy, see Rogers [1893a,b, 1894, 1895]. For example, he found the linearization coefficients of these polynomials which we now call the continuous q-ultraspherical polynomials. These polynomials contain many of the spherical harmonics Heine studied. Contained within this product identity is the special case of the square of one of these polynomials as a double series. As Gasper and Rahman have observed, one of these series can be summed, and the resulting identity is an extension of Clausen's sum in the terminating case. Earlier, others had found a different extension of Clausen's identity to basic hypergeometric series, but the resulting identity was not satisfactory. The identity had the product of two functions, the same functions but one evaluated at x and the other at qx, and so was not a square. Thus the nonnegativity that is so useful in Clausen's formula was not true for the corresponding basic hypergeometric series. Rogers' result for his polynomials led directly to the better result which contains the appropriate nonnegativity. From this example and many others, one sees that orthogonal polynomials provide an alternative approach to the study of hypergeometric and basic hypergeometric functions. Both this approach and that of differential equations are most useful for small values of the degrees of the numerator and denominator polynomials in the ratio cn+I/cn , but orthogonal polynomials work for a larger class of series, and are much more useful for basic hypergeometric se-
Foreword
xix
ries. However, neither of these approaches is powerful enough to encompass all aspects of these functions. Direct series manipulations are surprisingly useful, when done by a master, or when a computer algebra system is used as an aid. Gasper and Rahman are both experts at symbolic calculations, and I regularly marvel at some of the formulas they have found. As quantum groups become better known, and as Baxter's work spreads to other parts of mathematics as it has started to do, there will be many people trying to learn how to deal with basic hypergeometric series. This book is where I would start. For many years people have asked me what is the best book on special functions. My response was George Gasper's copy of Bailey's book, which was heavily annotated with useful results and remarks. Now others can share the information contained in these margins, and many other very useful results. Richard Askey University of Wisconsin
Preface
The study of basic hypergeometric series (also called q- hypergeometric series or q-series) essentially started in 1748 when Euler considered the infinite product
(q; q)~,I
00
=
I1
(1 - qk+ 1)-1 as a generating function for p(n), the number of
k= 0
partitions of a positive integer n into positive integers (see §8.1O). But it was not until about a hundred years later that the subject acquired an independent status when Heine converted a simple observation that lim [(1- qa)j(1-q)] = a q---+I
into a systematic theory of 2 ¢I basic hypergeometric series parallel to the theory of Gauss' 2FI hypergeometric series. Heine's transformation formulas for 2¢1 series and his q-analogue of Gauss' 2FI (1) summation formula are derived in Chapter 1, along with a q-analogue of the binomial theorem, Jacobi's triple product identity, and some formulas for q-analogues of the exponential, gamma and beta functions. Apart from some important work by J. Thomae and L. J. Rogers the subject remained somewhat dormant during the latter part of the nineteenth century until F. H. Jackson embarked on a lifelong program of developing the theory of basic hypergeometric series in a systematic manner, studying q-differentiation and q-integration and deriving q-analogues of the hypergeometric summation and transformation formulas that were discovered by A. C. Dixon, J. Dougall, L. Saalschiitz, F. J. W. Whipple, and others. His work is so pervasive that it is impossible to cover all of his contributions in a single volume of this size, but we have tried to include many of his important formulas in the first three chapters. In particular, a derivation of his summation formula for an 8¢7 series is given in §2.6. During the 1930's and 1940's many important results on hypergeometric and basic hypergeometric series were derived by W. N. Bailey. Some mathematicians consider Bailey's greatest work to be the Bailey transform (an equivalent form of which is covered in Chapter 2), but equally significant are his nonterminating extensions of Jackson's 8¢7 summation formula and of Watson's transformation formula connecting very-well-poised 8¢7 series with balanced 4¢3 series. Much of the material on summation, transformation and expansion formulas for basic hypergeometric series in Chapter 2 is due to Bailey. D. B. Sears, L. Carlitz, W. Hahn, and L. J. Slater were among the prominent contributors during the 1950's. Sears derived several transformation formulas for 3 ¢2 series, balanced 4¢3 series, and very-well-poised n+ 1 ¢n series. Simple proofs of some of his 3 ¢2 transformation formulas are given in Chapter 3. Three of his very-well-poised transformation formulas are derived in Chapter 4, where we follow G. N. Watson and Slater to develop the theory of xxi
xxii
Preface
basic hypergeometric series from a contour integral point of view, an idea first introduced by Barnes in 1907. Chapter 5 is devoted to bilateral basic hypergeometric series, where the most fundamental formula is Ramanujan's ] 'l/J] summation formula. Substantial contributions were also made by Bailey, M. Jackson, Slater and others, whose works form the basis of this chapter. During the 1960's R. P. Agarwal and Slater each published a book partially devoted to the theory of basic hypergeometric series, and G. E. Andrews initiated his work in number theory, where he showed how useful the summation and transformation formulas for basic hypergeometric series are in the theory of partitions. Andrews gave simpler proofs of many old results, wrote review articles pointing out many important applications and, during the mid 1970's, started a period of very fruitful collaboration with R. Askey. Thanks to these two mathematicians, basic hypergeometric series is an active field of research today. Since Askey's primary area of interest is orthogonal polynomials, q-series suddenly provided him and his co-workers with a very rich environment for deriving q-extensions of beta integrals and of the classical orthogonal polynomials of Jacobi, Gegenbauer, Legendre, Laguerre and Hermite. Askey and his students and collaborators who include W. A. AI-Salam, M. E. H. Ismail, T. H. Koornwinder, W. G. Morris, D. Stanton, and J. A. Wilson have produced a substantial amount of interesting work over the past fifteen years. This flurry of activity has been so infectious that many researchers found themselves hopelessly trapped by this alluring "q-disease", as it is affectionately called. Our primary motivation for writing this book was to present in one modest volume the significant results of the past two hundred years so that they are readily available to students and researchers, to give a brief introduction to the applications to orthogonal polynomials that were discovered during the current renaissance period of basic hypergeometric series, and to point out important applications to other fields. Most of the material is elementary enough so that persons with a good background in analysis should be able to use this book as a textbook and a reference book. In order to assist the reader in developing a deeper understanding of the formulas and proof techniques and to include additional formulas, we have given a broad range of exercises at the end of each chapter. Additional information is provided in the Notes following the Exercises, particularly in relation to the results and relevant applications contained in the papers and books listed in the References. Although the References may have a bulky appearance, it is just an introduction to the vast literature available. Appendices I, II, and III are for quick reference, so that it is not necessary to page through the book in order to find the most frequently needed identities, summation formulas, and transformation formulas. It can be rather tedious to apply the summation and transformation formulas to the derivation of other formulas. But now that several symbolic computer algebraic systems are available, persons having access to such a system can let it do some of the symbolic manipulations, such as computing the form of Bailey's 1O¢9 transformation formula when its parameters are replaced by products of other parameters.
Preface
xxiii
Due to space limitations, we were unable to be as comprehensive in our coverage of basic hypergeometric series and their applications as we would have liked. In particular, we could not include a systematic treatment of basic hypergeometric series in two or more variables, covering F. H. Jackson's work on basic Appell series and the works of R. A. Gustafson and S. C. Milne on U(n) multiple series generalizations of basic hypergeometric series referred to in the References. But we do highlight Askey and Wilson's fundamental work on their beautiful q-analogue of the classical beta integral in Chapter 6 and develop its connection with very-well-poised 8(P7 series. Chapter 7 is devoted to applications to orthogonal polynomials, mostly developed by Askey and his collaborators. We conclude the book with some further applications in Chapter 8, where we present part of our work on product and linearization formulas, Poisson kernels, and nonnegativity, and we also manage to point out some elementary facts about applications to the theory of partitions and the representations of integers as sums of squares of integers. The interested reader is referred to the books and papers of Andrews and N. J. Fine for additional applications to partition theory, and recent references are pointed out for applications to affine root systems (Macdonald identities), association schemes, combinatorics, difference equations, Lie algebras and groups, physics (such as representations of quantum groups and R. J. Baxter's work on the hard hexagon model of phase transitions in statistical mechanics), statistics, etc. We use the common numbering system of letting (k.m.n) refer to the n-th numbered display in Section m of Chapter k, and letting (I.n), (Il.n) , and (1II.n) refer to the n-th numbered display in Appendices I, II, and III, respectively. To refer to the papers and books in the References, we place the year of publication in square brackets immediately after the author's name. Thus Bailey [1935] refers to Bailey's 1935 book. Suffixes a, b, ... are used after the years to distinguish different papers by an author that appeared in the same year. Papers that have not yet been published are referred to with the year 2004, even though they might be published later due to the backlogs of journals. Since there are three Agarwals, two Chiharas and three Jacksons listed in the References, to minimize the use of initials we drop the initials of the author whose works are referred to most often. Hence Agarwal, Chihara, and Jackson refer to R. P. Agarwal, T. S. Chihara, and F. H. Jackson, respectively. We would like to thank the publisher for their cooperation and patience during the preparation of this book. Thanks are also due to R. Askey, W. A. Ai-Salam, R. P. Boas, T. S. Chihara, B. Gasper, R. Holt, M. E. H. Ismail, T. Koornwinder, and B. Nassrallah for pointing out typos and suggesting improvements in earlier versions of the book. We also wish to express our sincere thanks and appreciation to our TgXtypist, Diane Berezowski, who suffered through many revisions of the book but never lost her patience or sense of humor.
Preface to the second edition
In 1990 it was beyond our wildest imagination that we would be working on a second edition of this book thirteen years later. In this day and age of rapid growth in almost every area of mathematics, in general, and in Orthogonal Polynomials and Special Functions, in particular, it would not be surprising if the book became obsolete by now and gathered dust on the bookshelves. All we hoped for is a second printing. Even that was only a dream since the main competitor of authors and publishers these days are not other books, but the ubiquitous copying machine. But here we are: bringing out a second edition with full support of our publisher. The main source of inspiration, of course, has been the readers and users of this book. The response has been absolutely fantastic right from the first weeks the book appeared in print. The kind of warm reception we enjoyed far exceeded our expectations. Years later many of the leading researchers in the field kept asking us if an updated version would soon be forthcoming. Yes, indeed, an updated and expanded version was becoming necessary during the latter part of the 1990's in view of all the explosive growth that the subject was experiencing in many different areas of applications of basic hypergeometric series (also called q-series). However, the most important and significant impetus came from an unexpected source - Statistical Mechanics. In trying to find elliptic (doubly periodic meromorphic) solutions of the so-called Yang-Baxter equation arising out of an eight-vertex model in Statistical Mechanics the researchers found that the solutions are, in fact, a form of hypergeometric series L anz n , where a n+ 1/ an is an elliptic function of n, with n regarded as a complex variable. In a span of only five years the study of elliptic hypergeometric series and integrals has become almost a separate area of research on its own, whose leading researchers include, in alphabetical order, R. J. Baxter, E. Date, J. F. van Diejen, G. Felder, P. J. Forrester, 1. B. Frenkel, M. Jimbo, Y. Kajihara, K. Kajiwara, H. T. Koelink, A. Kuniba, T. Masuda, T. Miwa, Y. van Norden, M. Noumi, Y. Ohta, M. Okado, E. Rains, H. Rosengren, S. N. M. Ruijsenaars, M. Schlosser, V. P. Spiridonov, L. Stevens, V. G. Turaev, A. Varchenko, S. O. Warnaar, Y. Yamada, and A. S. Zhedanov. Even though we had not had any research experience in this exciting new field, it became quite clear to us that a new edition could be justified only if we included a chapter on elliptic hypergeometric series (and modular and theta hypergeometric series), written in an expository manner so that it would be more accessible to non-experts and be consistent with the rest of the book. Chapter 11 is entirely devoted to that topic. Regrettably, we had to be ruthlessly selective about choosing one particular approach from many xxv
xxvi
Preface to the second edition
possible approaches, all of which are interesting and illuminating on their own. We were guided by the need for brevity and clarity at the same time, as well as consistency of notations. In addition to Chapter 11 we added Chapter 9 on generating and bilinear generating functions in view of their central importance in the study of orthogonal polynomials, as well as Chapter 10, covering briefly the huge topic of multivariable q-series, restricted mostly to F. H. Jackson's q-analogues of the four Appell functions Fl , F 2 , F 3 , F 4 , and some of their more recent extensions. In these chapters we have attempted to describe the basic methods and results, but left many important formulas as exercises. Some parts of Chapters 1-8 were updated by the addition of textual material and a number of exercises, which were added at the ends of the sections and exercises in order to retain the same numbering of the equations and exercises as in the first edition. We have corrected a number of minor typos in the first edition, some discovered by ourselves, but most kindly pointed out to us by researchers in the field, whose contributions are gratefully acknowledged. A list of errata, updates of the references, etc., for the first edition and its translation into Russian by N. M. Atakishiyev and S. K. Suslov may be downloaded at arxiv.org/ abs/math.CA/9705224 or at www.math.northwestern.edu/ ~george/ preprints/bhserrata, which is usually the most up-to-date. Analogous to the first edition, papers that have not been published by November of 2003 are referred to with the year 2003, even though they might be published later. The number of people to whom we would like to express our thanks and gratitude is just too large to acknowledge individually. However, we must mention the few whose help has been absolutely vital for the preparation of this edition. They are S. O. Warnaar (who proof-read our files with very detailed comments and suggestions for improvement, not just for Chapter 11, which is his specialty, but also the material in the other chapters), H. Rosengren (whose e-mails gave us the first clue as to how we should present the topic of Chapter 11), M. Schlosser (who led us in the right directions for Chapter 11), S. K. Suslov (who sent a long list of errata, additional references, and suggestions for new exercises), S. C. Milne (who suggested some improvements in Chapters 5,7,8, and 11), V. P. Spiridonov (who suggested some improvements in Chapter 11), and M. E. H. Ismail (whose comments have been very helpful). Thanks are also due to R. Askey for his support and useful comments. Finally, we need to mention the name of a behind-the-scene helper, Brigitta Gasper, who spent many hours proofreading the manuscript. The mention of our ever gracious 'IEXtypist, Diane Berezowski, is certainly a pleasure. She did not have to do the second edition, but she said she enjoys the work and wanted to be a part of it. Where do you find a more committed friend? We owe her immensely.
1 BASIC HYPERGEOMETRlC SERlES
1.1 Introduction
Our main objective in this chapter is to present the definitions and notations for hypergeometric and basic hypergeometric series, and to derive the elementary formulas that form the basis for most of the summation, transformation and expansion formulas, basic integrals, and applications to orthogonal polynomials and to other fields that follow in the subsequent chapters. We begin by defining Gauss' 2Fl hypergeometric series, the rFs (generalized) hypergeometric series, and pointing out some of their most important special cases. Next we define Heine's 2¢1 basic hypergeometric series which contains an additional parameter q, called the base, and then give the definition and notations for r¢s basic hypergeometric series. Basic hypergeometric series are called q-analogues (basic analogues or q-extensions) of hypergeometric series because an rFs series can be obtained as the q ---+ 1 limit case of an r¢s series. Since the binomial theorem is at the foundation of most of the summation formulas for hypergeometric series, we then derive a q-analogue of it, called the q-binomial theorem, and use it to derive Heine's q-analogues of Euler's transformation formulas, Jacobi's triple product identity, and summation formulas that are q-analogues of those for hypergeometric series due to Chu and Vandermonde, Gauss, Kummer, Pfaff and Saalschiitz, and to Karlsson and Minton. We also introduce q-analogues of the exponential, gamma and beta functions, as well as the concept of a q-integral that allows us to give a q-analogue of Euler's integral representation of a hypergeometric function. Many additional formulas and q-analogues are given in the exercises at the end of the chapter.
1.2 Hypergeometric and basic hypergeometric series
In 1812, Gauss presented to the Royal Society of Sciences at Gottingen his famous paper (Gauss [1813]) in which he considered the infinite series
ab a(a + l)b(b + 1) 2 a(a + 1)(a + 2)b(b + 1)(b + 2) 3 ( ) 1+-z+ z + z + ... 1.2.1 l·c 1.2·c(c+l) 1.2·3·c(c+l)(c+2) as a function of a, b, c, z, where it is assumed that c -I=- 0, -1, -2, ... , so that no zero factors appear in the denominators of the terms of the series. He showed that the series converges absolutely for Izi < 1, and for Izl = 1 when Re (c - a - b) > 0, gave its (contiguous) recurrence relations, and derived his famous formula (see (1.2.11) below) for the sum of this series when z = 1 and Re(c-a-b»O. 1
Basic hypergeometric series
2
Although Gauss used the notation F(a, b, c, z) for his series, it is now customary to use F(a, b; c; z) or either of the notations 2Fl
(a, b; c; z),
for this series (and for its sum when it converges), because these notations separate the numerator parameters a, b from the denominator parameter c and the variable z. In view of Gauss' paper, his series is frequently called Gauss'series. However, since the special case a = 1, b = c yields the geometric senes 1+z+z2 +z3 + ... , Gauss' series is also called the (ordinary) hypergeometric series or the Gauss hypergeometric series. Some important functions which can be expressed by means of Gauss' series are (1 + z)a = F( -a, b; b; -z), 10g(1
+ z) = zF(l, 1; 2; -z),
sin- 1 z = zF(1/2, 1/2; 3/2; z2), tan- 1
z
e
Z
=
(1.2.2)
zF(1/2, 1;3/2; _z2),
= a-HXl lim F(a, b; b; z/a),
where Izl < 1 in the first four formulas. Also expressible by means of Gauss' series are the classical orthogonal polynomials, such as the Tchebichef polynomials of the first and second kinds
= F( -n, n; 1/2; (1 - x)/2), = (n + l)F( -n, n + 2; 3/2; (1 - x)/2),
Tn(x) Un(x)
(1.2.3) (1.2.4)
the Legendre polynomials
Pn(x)
=
F( -n, n
+ 1; 1; (1
- x)/2),
(1.2.5)
the Gegenbauer (ultraspherical) polynomials
(2'\)n F( -n, n + 2'\;'\ + 1/2; (1 - x)/2), n. and the more general Jacobi polynomials >.
en (x)
= -,-
p~a,(3)(x)= where n =
(a+,l)n F(-n,n+a+,B+1; a+1;(1-x)/2), n. 0,1, ... , and (a)n denotes the shifted factorial defined by
(a)o=l, (a)n=a(a+1)···(a+n-1)=
r(a+n) r(a) ,
(1.2.6)
(1.2.7)
n=1,2, .... (1.2.8)
Before Gauss, Chu [1303] (see Needham [1959, p. 138], Takacs [1973] and Askey [1975, p. 59]) and Vandermonde [1772] had proved the summation formula (c- b)n (1.2.9) F( -n, b; c; 1) = (c)n ' n = 0,1, ... ,
1.2 Hypergeometric series
3
which is now called Vandermonde's formula or the Chu- Vandermonde formula, and Euler [1748] had derived several results for hypergeometric series, including his transformation formula
F(a, b; c; z)
=
(1 - z)c-a-b F(c - a, c - b; c; z),
Formula (1.2.9) is the terminating case a
F(a, b; c; 1)
=
r(c)r(c - a - b) r(c _ a)r(c _ b)'
Izl < 1.
(1.2.10)
= -n of the summation formula Re(c - a - b) > 0,
(1.2.11)
which Gauss proved in his paper. Thirty-three years after Gauss' paper, Heine [1846, 1847, 1878] introduced the series (1 - qa)(l - qb) (1 - qa)(l - qa+ 1)(1 - qb)(l - qb+ I) 2 1+ (1 _ q)(l _ qc) z+ (1- q)(l _ q2)(1 _ qC)(l _ qC+ I) Z +- .. , (1.2.12) where it is assumed that q i- 1, c i- 0, -1, -2, ... and the principal value of each power of q is taken. This series converges absolutely for Izl < 1 when Iql < 1 and it tends (at least termwise) to Gauss' series as q ----> 1, because 1 _ qa lim - - = a. (1.2.13) q--+I 1 - q The series in (1.2.12) is usually called Heine's series or, in view of the base q, the basic hypergeometric series or q-hypergeometric series. Analogous to Gauss' notation, Heine used the notation ¢(a, b, c, q, z) for his series. However, since one would like to also be able to consider the case when q to the power a, b, or c is replaced by zero, it is now customary to define the basic hypergeometric series by
¢(a,b;c;q,z) == 2¢I(a,b;c;q,z) ==
=
f
~a; q~n~b; q~n
2¢1
[a~b;q,z]
zn,
(1.2.14)
n= 0 q; q n c; q n
where
(a; q)n
= {
~i _ a)(l -
aq) ... (1 _ aqn-I),
n=O, n= 1,2, ... ,
(1.2.15)
is the q-shifted factorial and it is assumed that c i- q-rn for m = 0,1, .... Some other notations that have been used in the literature for the product (a; q)n are (a)q,n' [a]n, and even (a)n when (1.2.8) is not used and the base is not displayed. Another generalization of Gauss' series is the (generalized) hypergeometric series with r numerator parameters aI, ... ,ar and s denominator parameters bl , ... ,bs defined by
rFs(al' a2, ... , ar; bl , ... , bs ;
z) == rFs [alb' a2,·· ·b' ar ; z] I, ... , s
(1.2.16)
Basic hypergeometric series
4
Some well-known special cases are the exponential function (1.2.17)
e Z = oFo(-;-;z),
the trigonometric functions sin z
=
cosz
=
z oFI (-; 3/2; -z2/4), oFI (-; 1/2;
_Z2
/4),
(1.2.18)
the Bessel function Ja(z) = (z/2t oFI (-; a
+ 1; _z2 /4)/r(a + 1),
(1.2.19)
where a dash is used to indicate the absence of either numerator (when r = 0) or denominator (when s = 0) parameters. Some other well-known special cases are the Hermite polynomials Hn(x) = (2x)n 2Fo( -n/2, (1 - n)/2; -; _x- 2 ),
(1.2.20)
and the Laguerre polynomials
= (a +,l)n
(1.2.21 ) IFI ( -n; a + 1; x ) . n. Generalizing Heine's series, we shall define an r¢s basic hypergeometric series by Lna (x)
A. ( A. [aI, a2, ... , a r r'f's al,a2,···,ar ; bI,···, bs;q,z ) -= r'f's b b ;q,z ]
=
f n= 0
1, ... , s
(al;q)n(a2;q)n···(ar ;q)n (q; q)n(bl ; q)n··· (b s; q)n
[(_l t qG)]I+s-r zn
(1.2.22)
with (~) = n(n - 1)/2, where q -=I 0 when r > s + 1. In (1.2.16) and (1.2.22) it is assumed that the parameters bl , ... , bs are such that the denominator factors in the terms of the series are never zero. Since (1.2.23) (-m)n = (q-m; q)n = 0, n = m + 1, m + 2, ... , an rFs series terminates if one of its numerator parameters is zero or a negative integer, and an r¢s series terminates if one of its numerator parameters is of the form q-m with m = 0,1,2, ... , and q -=I O. Basic analogues of the classical orthogonal polynomials will be considered in Chapter 7 as well as in the exercises at the ends of the chapters. Unless stated otherwise, when dealing with nonterminating basic hypergeometric series we shall assume that Iql < 1 and that the parameters and variables are such that the series converges absolutely. Note that if Iql > 1, then we can perform an inversion with respect to the base by setting p = q-I and using the identity (1.2.24) to convert the series (1.2.22) to a similar series in base p with Ipi < 1 (see Ex. 1.4(i)). The inverted series will have a finite radius of convergence if the original series does.
1.2 Hypergeometric series
5
Observe that if we denote the terms of the series (1.2.16) and (1.2.22) which contain zn by Un and Vn , respectively, then U n+ 1
(al
+ n)(a2 + n) ... (a r + n)
--= Un (l+n)(b 1 +n)···(bs+n)
z
(1.2.25)
is a rational function of n, and
(1.2.26)
L::
L::
is a rational function of qn. Conversely, if 0 Un and 0 Vn are power series with Uo = Vo = 1 such that u n+ dUn is a rational function of nand vn+ l/V n is a rational function of qn, then these series are of the forms (1.2.16) and (1.2.22), respectively. By the ratio test, the rFs series converges absolutely for all z if r ::; s, and for Izl < 1 if r = s + 1. By an extension of the ratio test (Bromwich [1959, p. 241]), it converges absolutely for Izl = 1 if r = s + 1 and Re [b 1 + ... + bs (a 1 + ... + ar )] > o. If r > s + 1 and z i=- 0 or r = s + 1 and Iz I > 1, then this series diverges, unless it terminates. If 0 < Iql < 1, the r¢s series converges absolutely for all z if r ::; sand for Izl < 1 if r = s + 1. This series also converges absolutely if Iql > 1 and Izl < Ib 1b2·· ·bsql/lala2·· ·arl· It diverges for z i=- 0 if 0 < Iql < 1 and r > s + 1, and if Iql > 1 and Izl > Ib1 b2·· ·bsql/lala2· ··arl, unless it terminates. As is customary, the rFs and r¢s notations are also used for the sums of these series inside the circle of convergence and for their analytic continuations (called hypergeometric functions and basic hypergeometric functions, respectively) outside the circle of convergence. Observe that the series (1.2.22) has the property that if we replace z by z/a r and let ar ---+ 00, then the resulting series is again of the form (1.2.22) with r replaced by r -1. Because this is not the case for the r¢s series defined n)]l+s-r without the factors [ (_I)nq ( 2 in the books of Bailey [1935] and Slater
[1966] and we wish to be able to handle such limit cases, we have chosen to use the series defined in (1.2.22). There is no loss in generality since the Bailey and Slater series can be obtained from the r = s + 1 case of (1.2.22) by choosing s sufficiently large and setting some of the parameters equal to zero. An r+ 1 Fr series is called k-balanced if b1 + b2 + ... + br = k + al + a2 + ... + ar+ 1 and z = 1; a I-balanced series is called balanced (or Saalschutzian). Analogously, an r+ 1 ¢r series is called k- balanced if b1 b2 ... br = qk al a2 ... ar+ 1 and z = q, and a I-balanced series is called balanced (or Saalschutzian). We will first encounter balanced series in §1.7, where we derive a summation formula for such a series. For negative subscripts, the shifted factorial and the q-shifted factorials are defined by ( a ) -n
1 (a-l)(a-2)···(a-n)
= -:------:--:------:---:------:-
1
(a-n)n
(1.2.27)
6
Basic hypergeometric series
1
1
(a;q)-n= (1-aq-l)(1-aq-2) ... (1-aq-n)
(aq-n;q)n
( -qja)nqG) (qja; q)n ' (1.2.28)
where n
= 0,1, .... We also define 00
(a; q)oo =
II (1- aqk)
(1.2.29)
k= 0
for Iql < 1. Since the infinite product in (1.2.29) diverges when a -=I- 0 and Iql 2:: 1, whenever (a; q)oo appears in a formula, we shall assume that Iql < 1. The following easily verified identities will be frequently used in this book:
() a; q
n =
(
(a; q)oo aq n ; q) 00 ,
(a- 1 ql-n; q)n = (a; q)n( _a- 1 )nq-G), (a;q)n-k =
(a;q)n (_qa-l)kq(~)-nk, (a- 1 ql-n; q)k
(a; q)n+ k = (a; q)n(aqn; q)k,
(1.2.30) (1.2.31 ) (1.2.32) (1.2.33) (1.2.34)
(a;q)n (aq k) ;q n-k = -(-)-, a;q k
(1.2.35)
_ (a; q)n(aqn; q)k (aq 2k.) ,q n-k () , a; q 2k
(1.2.36)
) = (q; q)n (_1)k (~)-nk (q-no,qk () q, q; q n-k
(1.2.37)
q)k(qa- 1 ; q)n -nk (aq -no,q )k -_ (a;(-1 , a q l-k.) ,q n q
(1.2.38)
(a; q)zn
(a; q2)n(aq; q2)n,
(1.2.39)
(a 2; q2)n = (a; q)n( -a; q)n,
(1.2.40)
=
where nand k are integers. A more complete list of useful identities is given in Appendix I at the end of the book. Since products of q-shifted factorials occur so often, to simplify them we shall frequently use the more compact notations
(ai, a2,···, am; q)n = (al; q)n(a2; q)n··· (am; q)n,
(1.2.41 )
(ai, a2, ... , am; q)oo = (al; q)oo (a2; q)oo ... (am; q)oo.
(1.2.42)
1.2 Hypergeometric series
7
The ratio (1 - qa)/(1 - q) considered in (1.2.13) is called a q-number (or basic number) and it is denoted by 1 _ qa (1.2.43) [a]q = - 1 - ' q -I- 1. -q It is also called a q-analogue, q-deformation, q-extension, or a q-generalization of the complex number a. In terms of q-numbers the q-number factorial [n]q! is defined for a nonnegative integer n by n
[n]q! =
II [k]q,
(1.2.44)
k= I
and the corresponding q-number shifted factorial is defined by
n-I [a]q;n
=
II [a + k]q.
(1.2.45)
k=O
Clearly,
· []' 11m n q. -- n.,,
q-tl
lim [a]q q-tl
=
a,
(1.2.46)
and lim [a]q;n = (a)n. q-tl
(1.2.4 7)
Corresponding to (1.2.41) we can use the compact notation
[aI, a2,···, am]q;n = [adq;n[a2]q;n'" [am]q;n. Since
(1.2.48)
f
[al,a2, ... ,ar]q;n [(_I)n qG)]I+s-r zn n= 0 [n]q![b l , ... , bs]q;n
(a1
A.. -_ r'f/s q , qa2 , ... , qar ,• qb, , ... , qb s ,• q, z (1 - q)I+s-r) ,
(1.2.49)
anyone working with q-numbers and the q-number hypergeometric series on the left-hand side of (1.2.49) can use the formulas for reps series in this book that have no zero parameters by replacing the parameters by q"lj powers and applying (1.2.49). As in Frenkel and Turaev [1995] one can define a trigonometric number [a; u] by [a'u] = sin(7rua) (1.2.50) , sin( 7ru) for noninteger values of u and view [a; u] as a trigonometric deformation of a since lima-to [a; u] = a. The corresponding rts trigonometric hypergeometric series can be defined by
rts(al, a2, ... , ar; bl , ... , bs ; u, z)
=
f
[al,a2, ... ,ar ;u]n n= 0 [n; u]![bl , ... , bs; u]n
[(_lt e11"ia G)]I+s-r zn ,
(1.2.51 )
Basic hypergeometric series
8 where
n
[n; u]! =
II [k; u],
n-I
[a; u]n =
k= I
II [a + k; u],
(1.2.52)
k= 0
and (1.2.53) From eniaa _ e- niaa
[a·, u] = -:-----,---e7ria _ e- 7ricr
qa/2 _ q-a/2 _ 1 - qa (l-a)/2 ql/2 _ q-I/2 - 1 - q q ,
(1.2.54)
where q = e 2nia , it follows that
(qa;q)n n(l-a)/2-n(n-I)/4 [a,. u ]n -_ (1 _ q)n q ,
(1.2.55)
rts(al' a2,···, ar ; bl ,···, bs; u, z) rI-. (a1 -_ r'l's q ,qa2 , ... , qar •,q b1 , ... , qbs •,q, cz )
(1.2.56)
and hence
with (1.2.57) which shows that the rts series is equivalent to the reps series in (1.2.49). Elliptic numbers [a; u, T], which are a one-parameter generalization (deformation) of trigonometric numbers, are considered in §1.6, and the corresponding elliptic (and theta) hypergeometric series and their summation and transformation formulas are considered in Chapter 11. We close this section with two identities involving ordinary binomial coefficients, which are particularly useful in handling some powers of q that arise in the derivations of many formulas containing q-series: (1.2.58) (1.2.59)
1.3 The q-binomial theorem One of the most important summation formulas for hypergeometric series is given by the binomial theorem:
2FI (a, C; C; z) where Izl
=
IFo(a;-; z)
=
f
(a),n zn n=O n.
= (1- z)-a,
(1.3.1)
< 1. We shall show that this formula has the following q-analogue
rI-. ( . _ . 1'l'0 a, ,q,
) _
z -
~ (a;q)n n _ (az;q)oo ~ ( ) z - ( ) , n= 0 q; q n Z; q 00
Izl
< 1, Iql < 1,
(1.3.2)
1.3 The q-binomial theorem
9
which was derived by Cauchy [1843]' Heine [1847] and by other mathematicians. See Askey [1980a], which also cites the books by Rothe [1811] and Schweins [1820]' and the remark on p. 491 of Andrews, Askey, and Roy [1999] concerning the terminating form of the q-binomial theorem in Rothe [1811]. Heine's proof of (1.3.2), which can also be found in the books Heine [1878], Bailey [1935, p. 66] and Slater [1966, p. 92]' is better understood if one first follows Askey's [1980a] approach of evaluating the sum of the binomial series in (1.3.1), and then carries out the analogous steps for the series in (1.3.2). Let us set .. ( ) Ja Z
=
~ (a)n n ~ ,z. n=O n.
(1.3.3)
Since this series is uniformly convergent in Izl ::; differentiate it termwise to get
f =f
f~(z) =
E
when 0
<
E
< 1, we may
n(a/ n zn-l n= 1 n. n= 0
(a)~+ 1 zn = afa+ 1 (z). n.
(1.3.4)
Also
.. ( ) _.. () _ ~ (a)n - (a + l)n n Ja Z Ja+ 1 Z - ~ , Z n= 1 n.
=
f
n= 1
f
(a + l?n-l [a _ (a + n)] zn = _ n(a + ~)n-l zn n. n= 1 n.
-_ - ~(a+1)nn+l_ ,z - n. ~
..a+ 1 () Z .
(1.3.5)
ZJ
n= 0 Eliminating fa+ 1 (z) from (1.3.4) and (1.3.5), we obtain the first order differential equation (1.3.6) subject to the initial condition fa(O) = 1, which follows from the definition (1.3.3) of fa(z). Solving (1.3.6) under this condition immediately gives that fa(z) = (1 - z)-a for Izl < 1. Analogously, let us now set
h ( ) - ~ (a; q)n n aZ-~( )z' n= 0 q; q n
Izl
< 1, Iql < 1.
(1.3.7)
Clearly, hqa (z) ----+ fa(z) as q ----+ 1. Since haq(z) is a q-analogue of fa+ 1 (z), we first compute the difference
= ~ (a; q)n - (aq; q)n n h a(Z ) _ h aq () Z ~ () Z n= 1 q; q n
10
Basic hypergeometric series
=
f
n= 1 =
-a
(aq; q)n-l [1 - a - (1 - aqn)] zn (q; q)n
f
n= 1
(1 - qn)(aq; q)n-l zn (q; q)n
__ ~ (aq; q)n-l n - _ h () - a~ ( ) z - az aq Z , n= 1 q; q n-l
(1.3.8)
giving an analogue of (1.3.5). Observing that
!'(z) = lim J(z) - J(qz) q-+l (1 - q)z for a differentiable function
J,
we next compute the difference
ha(z) - ha(qz) =
f
f
(a; q)n (zn _ qnzn) n= 1 (q; q)n
(a;q)n zn n=l (q;q)n-l = (1 - a)zhaq(z).
=
(1.3.9)
=
f
(a;q)n+l zn+l n=O (q;q)n (1.3.10)
Eliminating haq(z) from (1.3.8) and (1.3.10) gives
ha(z)
1- az --ha(qz). 1-z
=
(1.3.11)
Iterating this relation n - 1 times and then letting n -----
00
we obtain
(n) h aZ () = (az;q)nh (z;q )n aqz
= (az; q)oo ha(O) = (az; q)oo,
(1.3.12)
(z; q)oo (z; q)oo and ha(O) = 1 by (1.3.7), which completes the proof
since qn _____ 0 as n _____ 00 of (1.3.2). One consequence of (1.3.2) is the product formula
l¢o(a;-;q,z) l¢o(b;-;q,az) = l¢o(ab;-;q,z),
(1.3.13)
which is a q-analogue of (1 - z)-a(1- z)-b = (1 - z)-a-b. In the special case a = q-n, n = 0,1,2, ... , (1.3.2) gives
l¢O(q-n;_;q,z) = (zq-n;q)n = (-ztq-n(n+l)/2(q/Z;q)n,
(1.3.14)
where, by analytic continuation, z can be any complex number. From now on, unless stated othewise, whenever q- j , q-k, q-m, q-n appear as numerator parameters in basic series it will be assumed that j, k, m, n, respectively, are nonnegative integers. If we set a = 0 in (1.3.2), we get
zn
1
L -(-.-) = (Z,. q) n= q, q n 00
l¢O(O;-;q,z) =
0
00
,Izl < 1,
(1.3.15)
1.3 The q-binomial theorem
11
which is a q-analogue of the exponential function e Z • Another q-analogue of e Z can be obtained from (1.3.2) by replacing z by -z/a and then letting a ----700 to get 00
ocPo(-; -; q, -z) =
qn(n-l)/2 n z = (-z; q)oo.
L (.) n=O q,q n
(1.3.16)
Observe that if we denote the q-exponential functions in (1.3.15) and (1.3.16) by eq(z) and Eq(z), respectively, then eq(z)Eq( -z) = 1, eq-" (z) = Eq( -qz) by (1.2.24), and lim eq(z(l - q))
q---+l-
=
lim Eq(z(l - q))
q---+l-
=e
Z •
(1.3.17)
In deriving q-analogues of various formulas we shall sometimes use the observation that
Thus lim (qa z ; q)oo = (1 _ z)-a, q---+l- (z; q)oo
Izl < 1,
a real.
(1.3.19)
By analytic continuation this holds for z in the complex plane cut along the positive real axis from 1 to 00, with (1 - z)-a positive when z is real and less than 1. Let Do and \7 be the forward and backward q-difference operators, respectively, defined by
Dof(z)
f(qz) - f(z),
=
where we take 0 < q analytic solutions of
Do~~)
=
f(z),
\7 f(z) = f(q-l z) - f(z),
(1.3.20)
< 1, without any loss of generality. Then the unique f( o) = 1 an d
\7g(z) - () \7z - g z ,
g(O) = 1,
(1.3.21 )
are
f(z) = eq(z(l - q))
and
g(z) = Eq(z(l - q)).
(1.3.22)
The symmetric q-difference operator 6q is defined by
(1.3.23) If we seek an analytic solution of the initial-value problem 6q!(Z) uqz in the form
2::
0
= f(z),
f(O) = 1,
(1.3.24)
anz n , then we find that 1 - q nl2 an+1 = 1 -q n+ 1 q an,
aO = 1,
(1.3.25)
12
Basic hypergeometric series
n = 0,1,2,... Hence, an q-exponential function 00
eXPq(z) = with q
=
L n=O
(1 - q)n q(n 2-n)/4 /(q; q)n, and we have a third
=
(1 _
)n (n 2-n)/4 00 1 q( .q) zn = zn q,q n n=O n,u.
L -[-.-]'
(1.3.26)
e21ria . This q-exponential function has the properties
(1.3.27) and it is an entire function of z of order zero with an infinite product representation in terms of its zeros. See Nelson and Gartley [1994]' and Atakishiyev and Suslov [1992a]. The multi-sheet Riemann surface associated with the qlogarithm inverse function z = lnq(w) of W = eXPq(z) is considered in Nelson and Gartley [1996]. Ismail and Zhang [1994] found an extension of eXPq(z) in the form m
2
/4
L -q-- (a q1 -;=+z,aq1 -;=-Z;q) m=O (q;q)m 00
f(z)
=
bm ,
(1.3.28)
1/2),
(1.3.29)
m
which has the property
8f(z) 8x(z)
=
f( ) z ,
8f(z)
=
f(z
+ 1/2) - f(z -
where (1.3.30) with C = -abql/4/(1 - q) is the so-called q-quadmtic lattice, and a and b are arbitrary complex parameters such that labl < 1. In the particular case qZ = e-if}, 0 ~ () ~ Jr, X = cos (), the q-exponential function in (1.3.28) becomes the function 00 m /4 q (1-=.f) 1-= Of}) bm . =" ~ -(-.-)- q-2-ae' ,q-2-ae-';q 2
Eq(x;a,b)
m=O q,q
m
(1.3.31 )
m
Ismail and Zhang showed that lim Eq(x; a, b(l - q)) q--->l
= exp[(l + a2
-
2ax)b],
(1.3.32)
and that Eq(x; a, b) is an entire function of x when labl < 1. From (1.3.32) they observed that Eq(x; -i, -it/2) is a q-analogue of ext. It is now standard to use the notation in Suslov [2003] for the slightly modified q-exponential function
Eq(X; a) =
(a 2. q2) 00 " 00 qm 2 /4 (1-= of) 1-= of} ) ; 2 ~ - - ( -iar -iq-2-e', -iq-2-e-' ; q (qa;q)oom=O(q;q)m
, m
(1.3.33) which, because of the normalizing factor that he introduced, has the nice property that Eq(O;a) = 1 (see Suslov [2003, p.17]).
1.4 Heine's transformation formulas for 2
13
1.4 Heine's transformation formulas for 2
(b, az; q)oo 2
(1.4.1)
00
where Izl < 1 and Ibl < 1. To prove this transformation formula, first observe from the q-binomial theorem (1.3.2) that
(cqn; q)oo (bqn; q)oo Hence, for
f
=
m= 0
(clb; q)m (bqn)m. (q; q)m
Izl < 1 and Ibl < 1, '" ( b.. ) _ (b; q)oo ~ (a; q)n(cqn; q)oo n a, ,c, q, z - ( ) ~ ( ) ( ) z c; q 00 n= 0 q; q n bqn; q 00
2 '/'1
f f f
= (b; q)oo
f
(a; q)n zn (clb; q)m (bqn)m (c;q)oo n=O (q;q)n m=O (q;q)m
= (b; q)oo (c; q)oo
= (b; q)oo (c; q)oo
(clb; q)m bm (a; q)n (zqmt (q; q)m n= 0 (q; q)n
m= 0
(clb; q)m bm (azqm; q)oo (q; q)m (zqm; q)oo
(b, az; q)oo
= (
c, z; q
f
m= 0
)
2
(
I b,z;az;q,b )
C
00
by (1.3.2), which gives (1.4.1). Heine also showed that Euler's transformation formula
2FI (a, b; c; z) = (1 - z)c-a-b 2FI (c - a, c - b; c; z)
(1.4.2)
has a q-analogue of the form
2
(abz I c; q)oo ( ) 2
(1.4.3)
A short way to prove this formula is just to iterate (1.4.1) as follows
(b, az; q)oo 2
I I) = (clb, ( bz; )q)oo 2
=
(1.4.4)
00
(1.4.5)
00
(abzlc; q)oo ( ) 2
(1.4.6)
14
Basic hypergeometric series
1.5 Heine's q-analogue of Gauss' summation formula In order to derive Heine's [1847] q-analogue of Gauss' summation formula (1.2.11) it suffices to set z = e/ab in (1.4.1), assume that Ibl < 1, le/abl < 1, and observe that the series on the right side of
(a,b;e;q,e /ab ) = ((b, e/b; / b q)oo )
2¢1
e, e a ; q
I¢O
( /
e ab;-;q,b
)
00
can be summed by (1.3.2) to give
2¢1
(a,b;c;q,e /ab ) = (e/a,e/b;q)oo ( / b ) . c, e a ; q
(1.5.1)
00
By analytic continuation, we may drop the assumption that Ibl < 1 and require only that le/abl < 1 for (1.5.1) to be valid. For the terminating case when a = q-n, (1.5.1) reduces to A. (
2 '/'1
b c; q, eq nib) = (e/b;q)n q-n ,; ( ) . c;q n
(1.5.2)
By inversion or by changing the order of summation it follows from (1.5.2) that A. (
2 '/'1
q
-n b
, ; e; q, q
)
= (e/b;q)nbn () . e;q
(1.5.3)
n
Both (1.5.2) and (1.5.3) are q-analogues of Vandermonde's formula (1.2.9). These formulas can be used to derive other important formulas such as, for example, Jackson's [1910a] transformation formula A. (
2 '/'1
b.. ) = (az; q)oo ~ (a, c/b; q)k (-b )k (~) a, ,e, q, z ( ) ~ ( ) z q z; q 00 k= 0 q, e, az; q k
= (az; ( q)oo ) z;q
2¢2
( a,e/ b;c,az;q,bz ) .
(1.5.4)
00
This formula is a q-analogue of the Pfaff-Kummer transformation formula 2FI
(a, b; e; z) = (1 - z)-a
2FI
To prove (1.5.4), we use (1.5.2) to write
(a, e - b; e; z/(z - 1)).
(1.5.5)
1.6 Jacobi's triple product identity
15
and hence
(a, b; c; q, z)
2¢1
f t =f f ff (a;q)k+n(cjb;q)n(_bz)nZkq(~) f f f =
(a; q)k zk (q-k, cjb; q)n (bqk) n k= 0 (q; qh n= 0 (q, c; q)n
(a; qh(cjb; q)n zk( -btqG) n= 0 k= n (q; q)k-n(q, c; q)n
=
n= 0 k= 0 (q; qh(q, c; q)n
=
(a, cjb; q)n (-bz)nqG) (aqn; q)k zk n=O (q,c;q)n k=O (q;qh
(a, cjb; q)n (-bztqG), = (az; q)oo (z;q)oo n=O (q,c,az;q)n
by (1.3.2). Also see Andrews [1973]. If a = q-n, then the series on the right side of (1.5.4) can be reversed (by replacing k by n - k) to yield Sears' [1951c] transformation formula 2¢1
=
(q-n, b; c; q, z)
(~jb; r)n c;q n
(bZ)n 3¢2(q-n, qjz, c-1ql-n; bc-1ql-n, 0; q, q). q
(1.5.6)
1.6 Jacobi's triple product identity, theta functions, and elliptic numbers Jacobi's [1829] well-known triple product identity (see Andrews [1971]) 00
L
(zq!,q!jz,q;q)oo=
n=
(_lt qn2/2 zn,
zyfO,
(1.6.1)
-00
can be easily derived by using Heine's summation formula (1.5.1). First, set c = bzq! in (1.5.1) and then let b ---40 and a ---4 00 to obtain
(_1)nqn2/2
L (q; q)n n=O 00
1
zn
=
Similarly, setting c = zq in (1.5.1) and letting a ---4 00
qn 2 zn
~ (q,zq;q)n
1
(zq; q)oo .
Now use (1.6.2) to find that
(zq!, q! jz; q)oo 00 00 (_l)m+ nqCm2+ n2)/2 = zm-n m= 0 n= 0 (q; q)m(q; q)n
LL
(1.6.2)
(Zq2; q)oo. 00
and b ---400 we get (1.6.3)
16
Basic hypergeometric series 00
_ '" - ~
(_l)n n2/2 00 k2 q zn ' " q qnk (q;q)n ~ (q,qn+ l;qh
(-l)n n 2/2 00 k2 q -n ' " q nk ~ (q;q)n Z ~(q,qn+1;qhq· 00
+ '"
(1.6.4)
Formula (1.6.1) then follows from (1.6.3) by observing that 1
An important application of (1.6.1) is that it can be used to express the theta functions (Whittaker and Watson [1965, Chapter 21]) 00
19 1(x, q)
=
2 L( _1)nq(n+ 1/2)2 sin(2n + l)x,
(1.6.5)
n= 0 00
19 2 (x, q)
=
2L
q(n+
1/2)2
cos(2n
+ l)x,
(1.6.6)
n= 0 00
193 (x,q)
= 1+ 2L
qn2 cos2nx,
(1.6.7)
n= 1 00
194 (x,q)
=
1 + 2 L ( - l tqn2 cos2nx
(1.6.8)
n= 1
in terms of infinite products. Just replace q by q2 in (1.6.1) and then set z equal to qe2ix , _qe2ix , _e 2ix , e2ix , respectively, to obtain 00
19 1(x, q)
= 2q1/4 sinx II (1 -
+ q4n),
(1.6.9)
II (1- q2n)(1 + 2q2n cos2x + q4n),
(1.6.10)
q2n)(1 - 2l n cos 2x
n= 1 00
192 (x, q)
=
2q1/4 cos x
n= 1 00
193 (X, q)
=
II (1- q2n)(1 + 2q2n-1 cos2x + q4n-2),
(1.6.11)
n= 1
and 00
194 (X, q)
=
II (1- q2n)(1_ 2q2n-1 cos2x + q4n-2).
(1.6.12)
n= 1
It is common to write 19 k (x) for 19 k (x, q), k = 1, ... ,4. Since, from (1.6.9) and (1.6.10), (1.6.13) one can think of the theta functions 19 1 (x, q) and 192 (x, q) as one-parameter deformations (generalizations) of the trigonometric functions sin x and cos x,
1. 7 q-Saalschutz formula
17
respectively. This led Frenkel and Turaev [1995] to define an elliptic number [a; CT, T] by (1.6.14) where a is a complex number and the modular parameters CT and T are fixed and CT -I- m + nT for integer valcomplex numbers such that 1m (T) > ues of m and n, so that the denominator f)l (1[CT, e7riT) in (1.6.14) is never zero. Then, from (1.6.9) it is clear that [a; CT, T] is well-defined, [-a; CT, T] = -[a;CT,TJ, [1;CT,T] = 1, and . sin( 1[CTa) (1.6.15) hm [a; CT, T] = . ( ) = [a;CT]. III T-tOO sIn 1[CT
°
Hence, the elliptic number [a; CT, T] is a one-parameter deformation of the trigonometric number [a; CT] and a two-parameter deformation of the number a. Notice that [a; CT, T] is called an "elliptic number" even though it is not an elliptic (doubly periodic and meromorphic) function of a. However, [a;CT,T] is a quotient of f)l functions and, as is well-known (see Whittaker and Watson [1965, §21.5]), any (doubly periodic meromorphic) elliptic function can be written as a constant multiple of a quotient of products of f)l functions. The corresponding elliptic hypergeometric series are considered in Chapter 11.
1. 7 A q-analogue of Saalschiitz's summation formula
Pfaff [1797] discovered the summation formula 3 F 2(a,b,-n;c,l+a+b-c-n;I)=
(c - a)n(c - b)n (c)n(c-a-b)n'
n=O,I, ... , (1.7.1)
which sums a terminating balanced 3F2(1) series with argument 1. It was rediscovered by Saalschutz [1890] and is usually called Saalschiitz formula or the Pfaff-Saalschiitz formula; see Askey [1975]. To derive a q-analogue of (1.7.1), observe that since, by (1.3.2),
(abzjc; q)oo (z;q)oo
=
f
(abjc; q)k zk k=O (q;q)k
the right side of (1.4.3) equals
ff
k= 0 m= 0
(abjc; q)k(cja, cjb; q)m (ab)m zk+ m, (q; qh(q, c; q)m C
and hence, equating the coefficients of zn on both sides of (1.4.3) we get
t .
J= 0
(q-n, cja,cjb; q)j qj (q,c,cql-njab;q)j
= (a,b;q)n . (c,abjc;q)n
Replacing a, b by cj a, cjb, respectively, this gives the following sum of a terminating balanced 3 (P2 series
3(P2 (a,b,q -n ;c,abc-I qI-n ;q,q ) =
(cja, cjb; q)n ( j b ) , c, c a ; q n
n=O,I, ... ,
(1. 7.2)
18
Basic hypergeometric series
which was first derived by Jackson [1910a]. It is easy to see that (1.7.1) follows from (1.7.2) by replacing a, b, c in (1. 7.2) by qa, qb, qC, respectively, and letting q -+ 1. Note that letting a -+ 00 in (1.7.2) gives (1.5.2), while letting a -+ 0 gives (1.5.3). 1.8 The Bailey-Daum summation formula
Bailey [1941] and Daum [1942] independently discovered the summation formula ,/, ( b' jb' _ jb) = (-q; q)oc,(aq, aq2 jb2; q2)00 (1.8.1) 2'f'1 a, ,aq ,q, q (aq jb , _ q jb',q) 00 , which is a q-analogue of Kummer's formula
2FI(a,b;l+a-b;-I)=
r(1 (
r
+ a - b)r(1 + ta) ) ( I ). r 1 + 'ia - b
1+a
(1.8.2)
Formula (1.8.1) can be easily obtained from (1.4.1) by using the identity (1.2.40) and a limiting form of (1.2.39), namely, (a; q)oo = (a, aq; q2)00, to see that 2¢1 (a, b; aqjb; q, -qjb)
(a, -q; q)oo ,/, (jb jb ) (aq,q,qoo j b _ jb') 2'f'1 q ,-q ;-q;q,a (a,-q;q)oo ~ (q2jb 2;q2)n n = (aqjb, -qjb; q)oo ~ (q2; q2)n a (a, -q; q)oo (aq2 jb2; q2)00 (aqjb, -qjb; q)oo (a; q2)00 2 (-q; q)oo (aq, aq2 jb ; q2 )00 (aqjb, -qjb; q)oo
by (1.3.2)
1.9 q-analogues of the Karlsson-Minton summation formulas
Minton [1970] showed that if a is a negative integer and ml, m2, ... , mr are nonnegative integers such that -a 2: ml + ... + mT) then
F r+ 2 r+
[a, b, I
bb ++ 1, b l
+ mr . 1]
ml , ... , br l , ... , br
'
r(b + 1)r(1 - a) (b l - b)ml ... (b r - b)m r (1.9.1) r(1 + b - a) (bdml ... (br)m r where, as usual, it is assumed that none of the factors in the denominators of the terms of the series is zero. Karlsson [1971] showed that (1.9.1) also holds when a is not a negative integer provided that the series converges, i.e., if Re( -a) > ml + ... + mr - 1, and he deduced from (1.9.1) that F [a, r+1 r
b+b l
ml , ... , br b 1, ... , r
+ mr , . 1] - 0 - ,
Re (-a)
> ml + ... + mT) (1.9.2)
1.9 q-Karlsson-Minton formulas
19
(1.9.3) These formulas are particularly useful for evaluating sums that appear as solutions to some problems in theoretical physics such as the Racah coefficients. They were also used by Gasper [1981b] to prove the orthogonality on (0,27f) of certain functions that arose in Greiner's [1980] work on spherical harmonics on the Heisenberg group. Here we shall present Gasper's [1981a] derivation of q-analogues of the above formulas. Some of the formulas derived below will be used in Chapter 7 to prove the orthogonality relation for the continuous q- ultraspherical polynomials. Observe that if m and n are nonnegative integers with m ~ n, then
'" (-n -m b ) 2'1-'1 q , q ; r; q, q
=
(brqm; q)n -mn (b') q r,qn
by (1.5.3), and hence
'" [al, ... ,ar,brqm. ] r+ 1 'l-'r b q, Z I,···, br-I, b' r
=
~ ~
n= 0 =
(al, ... ,ar;q)n n~ (q-n,q-m;q)k mn+k Z ~ q (q,bl , ... ,br-I;q)n (q,br;q)k k= 0
ff
n= 0 k= 0
(al, ... ,ar;q)n(q-m;qh zn(_l)kqmn+k-nk+(~) (bl, ... ,br-I;q)n(q;q)n-k(q,br;q)k
~ (q-m,al, ... ,ar;qh( m)k _(k) -zq q 2 k=O (q,bl, ... ,br;q)k
= ~
X
alqk, ... ,arqk m-k] r¢r-I [ b k b k;q,zq , Iq , ... , r-Iq
Izl < 1. (1.9.4)
This expansion formula is a q-analogue of a formula in Fox [1927, (1.11)] and independently derived by Minton [1970, (4)]. When r = 2, formulas (1.9.4), (1.5.1) and (1.5.3) yield
¢ [a,b,b1qm.
3 2
bq,b l
,q,a
-I I-m] _ q
-m
(q,bq/a;q)oo ¢ ( - (bq,q/a;q)oo 2 1 q
b'b' ) "I,q,q
= (q,bq/a;q)oo(bl/b;q)mb m , ( ) ( ) / bq, q a; q
00
bl ; q
(
1.9.5
)
m
provided that la-1ql-ml < 1. By induction it follows from (1.9.4) and (1.9.5) that if ml,"" mr are nonnegative integers and la-1ql-(m 1 +... + mr ) I < 1, then
'" [a,b,blqml, ... ,brqmr. -I 1-(m1+ ... +m r )] r+ 2 'l-'r+ 1 b b b ' q, a q q, 1,···, r = (q, bq/a; q)oo (bl /b; q)ml ... (br/b; q)mr bm1+ ... + mr (bq, q/a; q)oo (b l ; q)ml ... (b r ; q)mr
(1.9.6)
20
Basic hypergeometric series
which is a q-analogue of (1.9.1). Formula (1.9.1) can be derived from (1.9.6) by replacing a, b, bl , ... , br by qa, qb, qb 1, ... , qb r , respectively, and letting q ----7 1. Setting br = b, mr = 1 and then replacing r by r + 1 in (1.9.6) gives '" [a,blqml, ... ,brqmr. -I -cm1+ ... +mr)] _ 0 I -I -Cm1+···+mr)1 r+ I '/'r bI, ... , br ' q, a q - , a q
< 1,
(1.9.7) while letting b ----700 in the case a '"
r+I'/'r
=
q-Cml+ ... + m r ) of (1.9.6) gives
] [q -Cm1+ ...+mr) ' bI qml , ... , br qmr 1 bI, ... , br ;q,
(_I)ml+···+mr(. ) ( ) q(q m)+"'+mrq-cml+"'+mrHml+"'+mr+I)/2, (1.9.8) bl ; q ml ... br ; q mr
=
which are q-analogues of (1.9.2) and (1.9.3). Another q-analogue of (1.9.3) can be derived by letting b ----7 0 in (1.9.6) to obtain
(1.9.9) when la-Iql-Cm1+···+mr)1 < 1. In addition, if a = q-n and n is a nonnegative integer then we can reverse the order of summation of the series in (1.9.6), (1.9.7) and (1.9.9) to obtain
n 2 ml '" r+ I '/'r
[
+ ... + mn
mr. ] - 0 q -n , bI qml , ... , br q bI, ... , br ' q, q - ,
(1.9.10)
(1.9.11)
and the following generalization of (1.9.8) '" [q-n,blqml, ... ,brqmr. r+ I '/'r bI,···, br ' q,
1] -_ (_I)n(q;q)nq-nCn+I)/2 (b) (b) , I;qml'"
r;qm r
(1.9.12)
where n 2 ml + ... + mn which also follows by letting b ----7 00 in (1.9.10). Note that the b ----7 0 limit case of (1.9.10) is (1.9.11) when n > ml + ... + mn and it is the a = q-Cml+ ... + m r ) special case of (1.9.9) when n = ml + ... +mr .
1.10 The q-gamma and q-beta functions The q-gamma function
r q (x ) = (x.) (q; q)oo q ,q
00
(1 - q )I-x , 0
< q < 1,
(1.10.1)
1.10 The q-gamma and q- beta functions
21
was introduced by Thomae [1869] and later by Jackson [1904e]. Heine [1847] gave an equivalent definition, but without the factor (l-q)l-x. When x = n+ 1 with n a nonnegative integer, this definition reduces to (1.10.2) which clearly approaches n! as q ---+ 1-. Hence f q(n + 1) tends to f( n + 1) = n! as q ---+ 1-. The definition of f q (x) can be extended to Iq I < 1 by using the principal values of qX and (1 - q)l-x in (1.10.1). To show that lim fq(x) = f(x) (1.10.3) q---+I we shall give a simple, formal proof due to Gosper; see Andrews [1986]. From (1.10.1),
Hence lim fq(x q---+I-
+ 1)
=
(+I)X II _n_ _n_ n+x n 00
n= I
=
xr(x)
=
r(x
+ 1)
by Euler's product formula (see Whittaker and Watson [1965, §12.11]) and the well-known functional equation for the gamma function r(x
+ 1) =
xr(x),
r(1) = 1.
(1.10.4)
For a rigorous justification of the above steps see Koornwinder [1990]. From (1.10.1) it is easily seen that, analogous to (1.10.4), fq(x) satisfies the functional equation 1- qX (1.10.5) f(x + 1) = - 1 - f (x), f(l) = 1. -q Askey [1978] derived analogues of many of the well-known properties of the gamma function, including its log-convexity (see the exercises at the end of this chapter), which show that (1.10.1) is a natural q-analogue of f(x). It is obvious from (1.10.1) that fq(x) has poles at x = 0, -1, -2, .... The residue at x = -n is lim (x x---+-n
+ n) f q(x)
(1 _q)n+1 x+n lim (1 - q-n)(1 - ql-n) ... (1 - q-I) x---+-n 1 _ qX+ n (1 _ q)n+ I (1.10.6)
= --;--------:-----:----'---:--'----,-----------,----:-:-
22
Basic hypergeometric series
The q-gamma function has no zeros, so its reciprocal is an entire function with zeros at x = 0, -1, -2, .... Since 1
00
r (x) =
(1- q)x-l
1- qn+ x
II 1 _ qn+
1'
(1.10.7)
n= 0
q
the function 1/r q (x) has zeros at x = -n ± 27rik / log q, where k and n are nonnegative integers. A q-analogue of Legendre's duplication formula r(2x)r
(~)
= 22x - 1r(x)r
(x +~)
(1.10.8)
can be easily derived by observing that
(q,q2;q2)00 (1_ q2)1-2x (q2x , q2x+ 1 ; q2 ) 00
rq2(X)rq2(X+t) r q2 (t) =
(q; q)oo (1 _ q2)1-2x (q2x;q)00
=
(1
+ q)I- 2x r
(2x) q
and hence (1.10.9) Similarly, it can be shown that the Gauss multiplication formula r(nx)(27r)(n-l)/2
=
nnx-!r(x)r (x
+ ~) ... r
(x
+ n: 1 )
(1.10.10)
has a q-analogue of the form
rq(nx)rr
(~) rr (~) ... rr (n: 1)
= (1 + q + ... + qn-l tx-1rr(x)rr (x +
~) ... rr (x + n: 1)
(1.10.11)
with r = qn; see Jackson [1904e, 1905d]. The q-gamma function for q > 1 is considered in Exercise 1.23. For other interesting properties of the q-gamma function see Askey [1978] and Moak [1980a,b] and Ismail, Lorch and Muldoon [1986]. Since the beta function is defined by
B(
X,y
)
=
r(x)r(y)
r(x+y)'
(1.10.12)
it is natural to define the q-beta function by (1.10.13)
1.11 The q-integral which tends to B(x, y) as q
23
1-. By (1.10.1) and (1.3.2),
----+
B (x, y) = (1 _ q) (q, qX+ Y; q)oo
q
(qX,qY;q)oo
= (1 - q) (q; q)oo
f
(qY; q)n qnx (qY;q)oo n=O (q;q)n n+ 1
)
= (1 - q) L (q n+ Y: q) qnx, 00
(
Re x, Re y
00
n=O q
,qoo
> O. (1.10.14)
This series expansion will be used in the next section to derive a q-integral representation for Bq(x, y).
1.11 The q-integral Thomae [1869, 1870] and Jackson [1910c, 1951] introduced the q-integral
11
f(t) dqt = (1 - q) ~ f(qn)qn
(1.11.1)
and Jackson gave the more general definition
lb
f(t) dqt
=
1b
f(t) dqt -
1 a
f(t) dqt,
(1.11.2)
where (1.11.3) Jackson also defined an integral on (0, (0) by (1.11.4) The bilateral q-integral is defined by
1
00
f(t) dqt = (1 - q)
f n=
-00
[f(qn) + f( _qn)] qn.
(1.11.5)
-00
If f is continuous on [0, a], then it is easily seen that
lim q---+l
r f(t) dqt = iofa f(t) dt
(1.11.6)
io
and that a similar limit holds for (1.11.4) and (1.11.5) when restricted. By (1.11.1), it follows from (1.10.14) that
1
f is suitably
t x-I ((tq; q)oo ) dqt, R ex> 0 , y r--I- 0 ,- 1,- 2, ... , tqY;qoo which clearly approaches the beta function integral B q(x, y ) =
1
o
B(x, y)
=
11
t x - 1 (1 - t)y-l dt,
Re x, Re y
> 0,
(1117) ..
(1.11.8)
24
Basic hypergeometric series
as q ----+ 1-. Thomae [1869] rewrote Heine's formula (1.4.1) in the q-integral form 2¢1
) _ fq(c) ( a b. c. q ,q ,q ,q,z - fq(b)fq(c-b)
t t b-I (tz,tqc-b;q)cx; (tzqa,tq;q)cx; dqt,
io
(1.11.9)
which is a q-analogue of Euler's integral representation 2FI
(a, b; c; z)
=
f(b)~~~ _ b)
11
t b- I (1 -
W- b- I (1 -
tz)-a dt,
(1.11.10)
where Iarg(l- z)1 < 7r and Re c> Re b > O. The q-integral notation is, as we shall see later, quite useful in simplifying and manipulating various formulas involving sums of series.
Exercises
1.1 Verify the identities (1.2.30)-(1.2.40), and show that
(aq-n; q)n
(i) (ii)
(iii)
(vi)
(qja; q)n ( _~) n q-G),
(_~)n -nk-(~) (aq -k-n.,q)n = (qja;q)n+k ( , qk q q q ja;) (qa!, -qa!; q)n 1 - aq2n (a!, -a!; q)n 1- a ' (a;q)zn
(iv)
(v)
=
=
(a!,-a!,(aq)!,-(aq)!;q)n'
(a; q)n(qja; q)-n = (-atq(~), (q, -q, _q2; q2)cx; = 1.
1.2 The q-binomial coefficient is defined by
[n]
(q;q)n
k q - (q; q)k(q; q)n-k
for k
= 0,1, ... , n,
and by
(q!3+ I, q<>-!3+ I; q)cx; (q, q<>+ I; q)cx; for complex a and f3 when Iql
(i) (ii) (iii)
(q<>+ I; q)k (q;q)k '
< 1. Verify that
25
Exercises (iv)
(v)
[ -kG
[G:
L[
1
G
+~- 1
L= [~L
qk
L
(_q-a)kq-(~),
lL = [~L + [k ~ lL
+ [k ~
q<>+ I-k,
k
n
(Z;q)n=2::[~] (-z)k q(2),
(vi)
k= 0 q when k and n are nonnegative integers. 1.3 (i) Show that the binomial theorem
(a+bt
=
t (~)akbn-k k= 0
where n
= 0,1, ... , has a q-analogue of the form (ab; q)n
=
t [~] t [~] k= 0
=
bk(a; q)k(b; q)n-k q
an-k(a; qh(b; q)n-k. k= 0 q (ii) Extend the above formula to the q-multinomial theorem (ala2 ... am+I; q)n
2:: 0:Sklo···,O::;km,
[k
n l , ...
,km
a~l a~l + k2 ... a~: ~2+ ... + krn
] q
kl +···+krn:Sn
x (al; q)k 1 (a2; q)k 2 ... (am; q)krn (a m+I; q)n-(k 1 +... + krn ), where m = 1,2, ... , n = 0,1, ... , and
is the q-multinomial coefficient. 1.4 (i) Prove the inversion formula
r¢s [a l ' ... ' ar ; q, z] bl
, ...
,bs
(-I -I -I) aI' ... ,ar ; q n I ~ ( -I bb- I -I ) n n=oq 'I,···,s;q
= '"""' (Xl
(ii) By reversing the order of summation, show that A.
r+I'Ps
[al' ... ' an q-n . ] b1, ... , bs ,q,z
26
Basic hypergeometric series when n = 0,1, .... (iii) Show that
r+ 1'. q, qz] " [a1'b ... ' ab
r+ 1 'Pr
1, ... , r
t
(al, ... , ar+ 1; q)oo
z-1
= (l-q)(q,b 1 , •.• ,br ;q)00}o t
°
when < q terminate.
(qt, b1t, ... , brt; q)oo d (al t , ... ,ar+l t ;q)oo qt,
< 1, Re z > 0, and the series on the left side does not
1.5 Show that
(e,bqn;q)rn _ (b/e;q)n ~ (q-n,e;q)kqk ( k. ) ~0 ( q, eq 1 -n / b; q) k cq, q rn· ( b; ) q rn - ( b; q) n k= 1.6 Prove the summation formulas
(~:;.q2inq-(~),
(i) 2¢I(q-n,ql-n;qb2;q2,q2) =
,q
n
" (a; e; q, e /) (e/a; )q)oo , (11.. ) 1 'PI a = ( e; q 00 (iii) 2¢o(a, q-n; -; q, qn fa) = a- n , (iv)
2
n 2 -n
00
L - ( q)2 n= q; q n
=
(
)
,
q; q 00 (zp-l ;p-l)oo -1 (v) l¢o(a;-;p,z) = (azp -1 ;p-1) 00 , Ipl > 1, lazp 1<1, (a/e, b/e;p-l )00 I I (VI.) 2¢1 (a, b; e;p,p ) = (1/ b/. -1) 00 , p > 1. e, a e,p 0
1.7 Show that, for
Izl < 1, 2 ) ( ) 2¢1 ( a ,aq;a;q,z = l+az
1.8 Show that, when 2¢1
=
lal < 1 and
(a 2qz; q)oo ( ) . z;q 00
Ibq/a 2 < 1, 1
(a 2, a2/b; b; l, bq/a2)
(a2,q;f)~
[(b/~;q)oo + (-b/~;q)oo] .
2(b, bq/ a ,q)oo
(a, q)oo
(-a, q)oo
(Andrews and Askey [1977]) 1.9 Let ¢(a, b, c) denote the series q-contiguous relations: .
(1) ¢(a, b, eq
-I
2¢1
(a, b; e; q, z). Verify Heine's [1847]
(l-a)(I-b) ) - ¢(a, b, c) = ez ( )( ) ¢(aq, bq, cq), q-e l-e
(ii) ¢(aq, b, c) - ¢(a, b, c) = az ~
=~ ¢(aq, bq, eq),
Exercises
27
... (l-b)(l-c/a) 2 (m) ¢(aq, b, cq) - ¢(a, b, c) = az ( )( ) ¢(aq, bq, cq ), 1-c 1-cq . 1 (1- b/aq) (IV) ¢(aq, bq- ,c) - ¢(a, b, c) = az 1 _ c ¢(aq, b, cq). 1.10 Denoting 2¢1 (a, b; c; q, z), 2¢1 (aq±l, b; c, q, z), 2¢1 (a, bq±l; c; q, z) and 2¢1 (a, b; cq±l; q, z) by ¢, ¢(aq±I), ¢(bq±l) and ¢(cq±I), respectively, show that
(i) b(l - a)¢(aq) - a(l - b)¢(bq)
=
(b - a)¢,
(ii) a(l - b/c)¢(bq-l) - b(l - a/c)¢(aq-l) = (a - b)(l - abz/cq)¢, (iii) q(l - a/c)¢(aq-l) + (1 - a)(l - abz/c)¢(aq) = [1 + q - a - aq/c + a2z(l - b/a)/c]¢, (iv) (1 - c)(q - c) (abz - c)¢(cq-l) + (c - a)(c - b)z¢(cq)
= (c - l)[c(q - c) + (ca + cb - ab - abq)z]¢. (Heine [1847]) 1.11 Let g(e;A,/-l,v) = (AeiO,/-lV;q)oo 2¢1(/-le- iO ,ve- iO ;/-lv;q, Ae iO ). Prove that g(e; A, /-l, v) is symmetric in A, /-l, v and is even in e. 1.12 Let Vq be the q-derivative operator defined for fixed q by
V J(z) = J(z) - J(qz) q (1 - q)z ' and let V~u = Vq(D~-IU) for n = 1,2, .... Show that (i) lim VqJ(z) = dd J(z) if J is differentiable at z,
q---+l (11.. ) 'T"In Vq
z
rI-.
2'/-'1
(b a, ;c;q,z ) =
... ) 'T"In { (z; q)oo (m Vq (abz/c;q)oo
rI-.
2'/-'1
(
(a, b; q)n ) (
c; q
)
1- q
n
rI-. (
n
2 '/-'1
aq n , bqn ;cqn ;q,z ),
(b )} a, ;c;q,z
_ (c/a,c/b;q)n (ab)n (zqn;q)oo ¢ ( b' n. n) - (c; q)n(1- q)n C (abz/c; q)oo 2 1 a, ,cq ,q, zq . (iv) Prove the q-Leibniz formula
V;[J(z)g(z)]
=
~ [~] qV;-k J(zqk)V;g(z).
1.13 Show that u(z) = 2¢1 (a, b; c; q, z) satisfies (for Izl < 1 and in the formal power series sense) the second order q-differential equation
z(c _ abqz)V2 u + [1 - c q 1-q (l-a)(l-b) (1 _ q)2 u = 0,
+ (1 -
a)(l - b) - (1 - abq) z] V u 1-q q
where Vq is defined as in Ex. 1.12. By replacing a, b, c, respectively, by qa, qb, qC and then letting q ----+ 1- show that the above equation tends to the second order differential equation
z(l - z)v" + [c - (a + b + l)z]v' - abv = 0
28
Basic hypergeometric series for the hypergeometric function v(z) (Heine [1847])
1.14 Let Ixl
=
2F1 (a, b; c; z), where Izl < 1.
< 1 and let eq(x) and Eq(x) be as defined in §1.3. Define . Slllq (x)
COS q
=
eq(ix) - eq( -ix) 2. ~
(_I)nx2n+ 1
L (.q, q)2n+ , n= 00
=
0
1
( ) _ eq(ix) +eq(-ix) _ ~ (_I)nx2n x 2 - ~ (.) . n= 0 q, q 2n
Also define
S.
()
lllq
x =
Eq(ix) - Eq( -ix) 2i
'
Show that
eq(ix) = cosq(x) + i sinq(x), Eq(ix) = Cosq(x) + iSinq(x), sinq(x)Sinq(x) + cosq(x)Cosq(x) = 1, sinq(x)Cosq(x) - Sinq(x) cosq(x) = 0.
(i) (ii) (iii) (iv)
For these identities and other identities involving q-analogues of sin x and cosx, see Jackson [1904a] and Hahn [1949c]. 1.15 Prove the transformation formulas A.
(i)
2'/'1
A.
(ii)
2'/'1
[q-n,b. C'
q, Z
] _ (bzq-n/c;q)oo (bz/c; q)oo
[q-n,c/b,o. ] c, cq/bz ,q, q ,
[q-n,b. ] _ (c/b;q)nbn A. [q-n,b,q/z. /] C ,q,z - (C;q)n 3'/'1 bq1-n/C ,q,z C
q-n,b. ] _ (c/b;q)n 21;1 [ C ,q,z - (.) C, q n
(iii)
A.
3 '/'2
31;2
,
°
[q-n,b,bzq-n/C. ] bq 1-n/ C, ,q,q .
(See Jackson [1905a, 1927]) 1.16 Show that
1.17 Show that
t
(a, b; q)k (_ab)n-kq(n-k)(n+ k-1)/2 k=O (q;q)k n
=
(-b)kq(~)
(a; q)n+ 1 ~ (q; qh(q; q)n-k(1 _ aqn-k)·
(Carlitz [1974])
Exercises
29
1.18 Show that
(i) (ii) (iii)
(c;q)oo 1¢I(a;c;q,z) = (z;q)oo 1¢I(az/c;z;q,c), and deduce that 1¢1(-bq;O;q,-q) = (-bq2;q2)00/(q;q2)00, (z;q)oo 2¢I(a,0;c;q,z) = (az;q)oo 1¢2(a;c,az;q,cz),
f
n= 0
((a;2~)n) q, a ,q
qG)(at/z)n 2¢I(q-n,a;ql-n/a;q, qz2/a) n
Iztl < 1.
= (-zt;q)oo 2¢I(a,a/z2;a2;q,-zt), 1.19 Using (1.5.4) show that
(ii)
[a, q/a. -b] = (ab, bq/a; q2)00 -q, b ,q, (b; q)oo ' a2,b2 ] (a 2q,b2q;q2)00 [ b 1 b 1 ; q, -q = a q2,-a q2 (q,a 2b2q;q2)00 . A..
(i)
2'1-'2
2¢2
(Andrews [1973]) 1.20 Prove that if Re x
> 0 and 0 < q < 1, then
(i) (ii)
1.21 For 0
< q < 1 and x > 0, show that d2 2 00 qn+ x dx2 logfq(x) = (logq) ~ (1- qn+ x)2'
which proves that log f q (x) is convex for x
> 0 when 0 < q < 1.
1.22 Conversely, prove that if f(x) is a positive function defined on (0, oo)which satisfies 1- qX f(x + 1) = --f(x) for some q, 0 < q < 1, l-q
f(l) = 1, and logf(x) is convex for x > 0, then f(x) = fq(x). This is Askey's [1978] q-analogue of the Bohr-Mollerup [1922] theorem for r(x). For two extensions to the q > 1 case (with f q (x) defined as in the next exercise), see Moak [1980b]. 1.23 For q > 1 the q-gamma function is defined by f ()= (q-I;q-I)oo( _1)I-X x(x-I)/2
qX
(q-x;q-I)oo q q .
Basic hypergeometric series
30
Show that this function also satisfies the functional equation (1.10.5) and that rq(x) ----t r(x) as q ----t 1+. Show that for q > 1 the residue of rq(x) at x = -n is
n+ 1) (q _ l)n+ 1q ( 2 (q; q)n logq
1.24 Jackson [1905a,b,el gave the following q-analogues of Bessel functions:
(qv+ 1. q) ( ') 00 (x/2)V 2¢1 (0, 0; qV+ 1; q, -x2/4), q;q 00 (qv+ 1. q) ( x2qv+ J~2) (x; q) = (q; ;)0000 (x/2t 0¢1 - ; qV+ 1; q, - - 4 - , J~l) (x; q)
=
1)
J~3)(x;q)
=
(qv+ 1. q) ( \ 00 (x/2t 1¢1(0;qv+1;q,qx2/4), q;q 00
where 0 < q < 1. The above notations for the q-Bessel functions are due to Ismail [1981, 1982, 2003cl. Show that J~2)(x;q)
= (-x2/4;q)ooJ~1)(x;q),
Ixl < 2,
(Hahn [1949c])
and lim J~k)(x(l- q);q)
q--+1
= Jv(x),
k
= 1,2,3.
1.25 For the q-Bessel functions defined as in Exercise 1.24 prove that (i)
qV ik) (x·, q) = 2(1 -X qV) fk) (x· q) _ ik) (x· q) k = 1 , 2·, v+ 1 v' v-I"
(ii)
J~l) (xq!; q) = qv/2 (J~l) (x; q) + ~J~~ 1 (x; q)) ;
(iii)
J~l)(xq!;q)
(iv) qV+
1
=
J~~ 1 (xq1/2; q) =
q-v/2 (J~l)(x;q) - ~J~~l(X;q)). 2(1 - qV) J~3) (x; q) -
J~~l (x; q).
X
1.26 (i) Following Ismail [1982]' let
fv(x) = J~l) (x; q)J£12 (xq!; q) - J£12 (x; q)J~l) (xq!; q). Show that
and deduce that, for non-integral v,
fv(x)
=
q-v/2(qv, q1-v; q)oo/(q, q, _x 2/4; q)oo.
(ii) Show that
9v(qX)
+ (x2 /4 -
qV - q-v)9v (X)
+ 9v(xq-1) = 0
Exercises
31
with gl/(x) = J~3)(xql//2;q2) and deduce that gl/ (X) g-I/ ( xq
-1 )
- g-I/
() X
(
gl/ xq
-1) _
-
( 21/
q ,q
1-21/. 2)
,q (2 2. 2) q ,q ,q
00
q
1/( 1/-1)
.
00
(Ismail [2003c]) 1.27 Show that 00
n=
-CX)
00
Both of these are q-analogues of the generating function 00
L n=
t n In(x)
= ex (t-C
1
)/2.
-CX)
1.28 The continuous q-Hermite polynomials are defined in Askey and Ismail [1983] by H
(q; q)n i(n-2k)IJ ( I ) - ~ nxq-L..J( )( ) e , k= 0 q; q k q; q n-k
where x = cos B; see Szego [1926]' Carlitz [1955, 1957a, 1958, 1960] and Rogers [1894, 1917]. Derive the generating function n f:'~o Hn(xlq) (q; q)n t
1 =
(teiIJ, te-iIJ; q)oo'
It I < 1.
(Rogers [1894])
1.29 The continuous q-ultraspherical polynomials are defined in Askey and Ismail [1983] by
C ( . (31 ) - ~ ((3; q)k((3; q)n-k i(n-2k)IJ n x, q - L..J ( ) ( ) e , k= 0
q; q k q; q n-k
and 00
' " Cn(x; (3lq)t L..J n=O
n =
((3te iIJ , (3te- iIJ ; q)oo ( ·IJ -"IJ) te' te ' . q ' ,00
,It I < 1.
(Rogers [1895])
32
Basic hypergeometric series
1.30 Show that if
1.31 Let
~b
ml , ...
,mr are nonnegative integers, then
denote the q-difference operator defined for a fixed q by ~bJ(Z)
Then
~1
is the
~
= bJ(qz) - J(z).
operator defined in (1.3.20). Show that ~bXn
= (bqn - 1)xn
and, if
vn () z then
(al,"" ar; q)n (_1)(1+ s-r)n q(l+ s-r)n(n-l)/2 z n,
= (q,b 1 ,···,bs;q )n
(~~bdq~b2/q'" ~bs/q)Vn(z) = z(~al ~a2
... ~aJVn-l (zql+ s-r),
n
=
1,2, ...
Use this to show that the basic hypergeometric series
v(z)
= r¢s(al, ... , ar; b1 ,···, bs; q, z)
satisfies (in the sense of formal power series) the q-difference equation (~~bl/q~b2/q'" ~bs/q)V(z)
= Z(~al ... ~ar)v(zql+ s-r).
This is a q-analogue of the formal differential equation for generalized hypergeometric series given, e.g. in Henrici [1974, Theorem (1.5)] and Slater [1966, (2.1.2.1)]. Also see Jackson [191Od, (15)]. 1.32 The little q-Jacobi polynomials are defined by
Pn(x; a, b; q) = 2¢1 (q-n, abqn+ 1; aq; q, qx). Show that these polynomials satisfy the orthogonality relation
f
j= 0
~b~; ql j (aq)j Pn (qj; a, b; q)Pm( qj; a, b; q) q, q
J
0,
= { (q, bq; q)n(1- abq) (aqt (abl; q)= (aq,abq;q)n(l-abq2n+l) (aq;q)= '
if m =I- n, if m = n.
Exercises
33
(Andrews and Askey [1977 ]) 1.33 Show for the above little q-Jacobi polynomials that the formula n
Pn(x; c, d; q)
=
L ak,nPk(X; a, b; q) k= 0
holds with
_ _ k (k~l)(q-n,aq,cdqn+l;q)k ak,n - ( 1) q (q,cq,a bqk+I.) ,q k
3(Pl
[qk-n,Cdqn+k+l,aqk+l. ] cqk+1 , abq2k+2 ,q,q.
(Andrews and Askey [1977]) 1.34 (i) If m, ml, m2, ... , mr are arbitrary nonnegative integers and la- I qm+l-(m 1+···+m r)1 < 1, show that A..
r+ 2 'f'r+
I
[
a ' b,I b qml , ... , br qmr . a-I m+I-(ml+···+mr) ] bq 1+ m ,I, b ... , b ' q, q r
= (q, bq/a; q)oo(bq; q)m(bl /b; q)ml ... (br/b; q)mr bm1+ ... + mr-m (bq, q/a; q)oo(q; q)m(b l ; q)ml ... (b r ; q)mr X
q-m, b, bq/bl , ... ,bq/br ] [ r+ 2 ¢r+ I bq/ a, bqI -ml /b I,···, bqI-mr /b r ; q, q ;
(ii) if ml, m2, ... , mr are nonnegative integers and la-Iql-(m 1+ ... + mr ) 1< 1, Icql < 1, show that A..
r+ 2 'f'r+
=
I
[
a ' b,I b qml , ... , br qmr . a-II-(m1+···+mr) ] b b b ' q, q cq, I,···, r
(bq/a, cq; q)oo (b l /b; q)ml ... (br/b; q)mr bm1+ ... + mr (bcq, q/ a; q)oo (b l ; q)ml ... (b r ; q)mr X
c- I , b, bq/bl , ... ,bq/br ] r+ 2 ¢r+ I [ bq/ a, bqI -ml /b I , ... , bqI -m r /b r ; q, cq .
(Gasper [1981a]) 1.35 Use Ex. 1.2(v) to prove that if x and yare indeterminates such that xy = qyx, q commutes with x and y, and the associative law holds, then
(See Cigler [1979], Feinsilver [1982]' Koornwinder [1989], Potter[1950]' Schiitzenberger [1953], and Yang [1991]). 1.36 Verify that if x and yare indeterminates satisfying the conditions in Ex. 1.35, then
(i) (ii)
eq(y)eq(x) = eq(x + y), eq(x)eq(y) = eq(x + y - yx); Eq(x)Eq(y) = Eq(x + y), Eq(y)Eq(x) = Eq(x + y + yx).
34
Basic hypergeometric series (Fairlie and Wu [1997]; Koornwinder [1997], where q-exponentials with q- Heisenberg relations and other relations are also considered.)
1.37 Show that Eq(z;a)
(a 2;q2)= {
= (qa2.,q2) =
2
2 2]
[_qe2iO,_qe-2iO ;q ,a q
2q!/4 + --acose 2
with
Z
= cose.
1.38 Extend Jacobi's triple product identity to the transformation formula 1+
f)-ltq(~)(an+bn) = (q,a,b;q)=f
(ab/q;q)znqn. n=O (q,a,b,ab;q)n
n=!
Deduce that
=
( ) n 2n 2 1+ 2~ ~a q = q; q n= !
= (.) ~ -a, q 2nq n = aq; q2) = n= ~ ( . ) ( . 2) . q, -aq, q n aq, q n (
0
(Warnaar [2003a]) Notes §§1.1 and 1.2 For additional material on hypergeometric series and orthogonal polynomials see, e.g., the books by Erdelyi [1953], Rainville [1960], Szego [1975], Whittaker and Watson [1965], Agarwal [1963], Carlson [1977], T.S. Chihara [1978], Henrici [1974], Luke [1969], Miller [1968], Nikiforov and Uvarov [1988], Vilenkin [1968], and Watson [1952]. Some techniques for using symbolic computer algebraic systems such as Mathematica, Maple, and Macsyma to derive formulas containing hypergeometric and basic hypergeometric series are discussed in Gasper [1990]. Also see Andrews [1984d, 1986, 1987b], Andrews, Crippa and Simon [1997], Andrews and Knopfmacher [2001], Andrews, Knopfmacher, Paule and Zimmermann [2001]' Andrews, Paule and Riese [2001a,b], Askey [1989f, 1990], Askey, Koepf and Koornwinder [1999], Baing and Koepf [1999], Garoufalidis [2003], Garoufalidis, Le and Zeilberger [2003], Garvan [1999], Garvan and Gonnet [1992]' Gosper [2001], Gosper and Suslov [2000], Koepf [1998], Koornwinder [1991b, 1993a, 1998], Krattenthaler [1995b], Paule and Riese [1997], Petkovsek, Wilf and Zeilberger [1996], Riese [2003], Sills [2003c], Wilf and Zeilberger [1990], and Zeilberger [1990b]. §§1.3-1.5 The q-binomial theorem was also derived in Jacobi [1846]' along with the q-Vandermonde formula. Bijective proofs of the q-binomial theorem, Heine's 2
Notes
35
sum in the form of a generalized q-binomial Vandermonde convolution. The even and odd parts of the infinite series on the right side of (1.3.33) appeared in Atakishiyev and Suslov [1992a], but without any explicit reference to the q-exponential function. Also see Suslov [1998-2003] and the q-convolutions in Carnovale [2002], Carnovale and Koornwinder [2000], and Rogov [2000]. §1.6 Other proofs of Jacobi's triple product identity and/or applications of it are presented in Adiga et al. [1985], Alladi and Berkovich [2003], Andrews [1965], Cheema [1964], Ewell [1981]' Gustafson [1989], Joichi and Stanton [1989], Kac [1978, 1985], Lepowsky and Milne [1978], Lewis [1984]' Macdonald [1972], Menon [1965], Milne [1985a], Sudler [1966], Sylvester [1882]' and Wright [1965]. Concerning theta functions, see Adiga et al. [1985], Askey [1989c], Bellman [1961]' and Jensen's use of theta functions in P6lya [1927] to derive necessary and sufficient conditions for the Riemann hypothesis to hold. §1. 7 Some applications of the q-Saalschiitz formula are contained in Carlitz [1969b] and Wright [1968]. §1.9 Formulas (1.9.3) and (1.9.8) were rediscovered by Gustafson [1987a, Theorems 3.15 and 3.18] while working on multivariable orthogonal polynomials. §1.11 Also see Jackson [1917, 1951] and, for fractional q-integrals and q-derivatives, AI-Salam [1966] and Agarwal [1969b]. Toeplitz [1963, pp. 53-55] pointed out that around 1650 Fermat used a q-integral type Riemann sum to evaluate the integral of xk on the interval [0, b]. AI-Salam and Ismail [1994] evaluated a q-beta integral on the unit circle and found corresponding systems of biorthogonal rational functions. Ex.1.2
The q-binomial coefficient
[~L' which is also called the Gaus-
sian binomial coefficient, counts the number of k dimensional subspaces of an n dimensional vector space over a field GF(q), q a prime power (Goldman and Rota [1970]), and it is the generating function, in powers of q, for partitions into at most k parts not exceeding n - k (Sylvester [1882]). It arises in such diverse fields as analysis, computer programming, geometry, number theory, physics, and statistics. See, e.g., Aigner [1979], Andrews [1971a, 1976], M. Baker and Coon [1970], Baxter and Pearce [1983], Berman and Fryer [1972], Dowling [1973], Dunkl [1981]' Garvan and Stanton [1990], Handa and Mohanty [1980], Ihrig and Ismail [1981]' Jimbo [1985, 1986], van Kampen [1961]' Kendall and Stuart [1979, §31.25]' Knuth [1971, 1973], P6lya [1970], P6lya and Alexanderson [1970], Szego [1975, §2.7]' and Zaslavsky [1987]. Sylvester [1878] used the invariant theory that he and Cayley developed to prove that the coefficients of the Gaussian polynomial
[~L =
'2;ajqj
are unimodal. A constructive proof
was recently given by O'Hara [1990]. Also see Bressoud [1992] and Zeilberger [1989a,b, 1990b]. The unimodality of the sequence
([~] q : k = 0,1, ...
,n)
is
explicitly displayed in Aigner [1979, Proposition 3.13], and Macdonald [1995, Example 4 on p. 137]. Ex.1.3 Cigler [1979] derived an operator form ofthe q-binomial theorem. MacMahon [1916, Arts. 105-107] showed that if a multiset is permuted, then
36
Basic hypergeometric series
the generating function for inversions is the q-multinomial coefficient. Also see Carlitz [1963a], Kadell [1985a], and Knuth [1973, p. 33, Ex. 16]. Gasper derived the q-multinomial theorem in part (ii) several years ago by using the q-binomial theorem and mathematical induction. Andrews observed in a 1988 letter that it can also be derived by using the expansion formula for the q-Lauricella function
Notes
37
Ex.1.32 Masuda et al. [1991] showed that the matrix elements that arise in the representations of certain quantum groups are expressible in terms of little q-J acobi polynomials, and that this and a form of the Peter-Weyl theorem imply the orthogonality relation for these polynomials. Pade approximants for the moment generating function for the little q-Jacobi polynomials are employed in Andrews, Goulden and D.M. Jackson [1986] to explain and extend Shank's method for accelerating the convergence of sequences. Pade approximations for some q-hypergeometric functions are considered in Ismail, Perline and Wimp [1992].
2 SUMMATION, TRANSFORMATION, AND EXPANSION FORMULAS
2.1 Well-poised, nearly-poised, and very-well-poised hypergeometric and basic hypergeometric series
The hypergeometric series (2.1.1) is called well-poised if its parameters satisfy the relations 1 + al
= a2 + b1 = a3 + b2 = ... = ar+1 + bTl
(2.1.2)
and it is called nearly-poised if all but one of the above pairs of parameters (regarding 1 as the first denominator parameter) have the same sum. The series (2.1.1) is called a nearly-poised series of the first kind if 1 + al
i= a2 + b1 = a3 + b2 = ... = ar+l + bTl
(2.1.3)
and it is called a nearly-poised series of the second kind if 1 + al
= a2 + b1 = a3 + b2 = ... = ar + br- 1 i= ar+l + br·
(2.1.4)
The order of summation of a terminating nearly-poised series can be reversed so that the resulting series is either of the first kind or of the second kind. Kummer's summation formula (1.8.2) gives the sum of a well-poised 2Fl series with argument -1. Another example of a summable well-poised series is provided by Dixon's [1903] formula
3F2 [ a, b, c; 1 + a - b, 1 + a - c; 1] r(1 r(1 Re (1 c
+ ~a)r(1 + a - b)r(1 + a - c)r(1 + ~a - b - c) + a)r(1 + ~a - b)r(1 + ~a - c)r(1 + a - b - c)'
(2.1.5)
+ ~a - b - c) > 0, which reduces to Kummer's formula (1.8.2) by letting
---+ -00.
If the series (2.1.1) is well-poised and a2 = 1 + ~al' then it is called a very-well-poised series. Dougall's [1907] summation formulas F. [ 7 6
1
a, + ~a, b, c, d, e, -n . ~a, 1 + a - b, 1 + a - c, 1 + a - d, 1 + a - e, 1 + a + n '
(1 + a)n(1 + a - b - c)n(1 + a - b - d)n(1 + a - c - d)n (1 + a - b)n(1 + a - c)n(1 + a - d)n(1 + a - b - c - d)n 38
1] (2.1.6)
2.1 Well-, nearly-, and very-well-poised series
39
when the series is 2-balanced (i.e, 1 + 2a + n = b + c + d + e), and F [
s
4
1
a, + ~a, b, c, d . ~a, 1 + a - b, 1 + a - c, 1 + a - d'
r(1 r(1
+ a - b)r(1 + a + a)r(1 + a - b -
+a +a -
c)r(1 c)r(1
1]
d)r(1 + a - b - c - d) b - d)r(1 + a - c - d)'
(2.1. 7)
when Re (1 + a - b - c - d) > 0, illustrate the importance of very-well-poised hypergeometric series. Note that Dixon's formula (2.1.5) follows from (2.1.7) by setting d = ~a. Analogous to the hypergeometric case, we shall call the basic hypergeometric series '" [a l ,a2, ... ,ar+ l . ] r+ 1 'f'r b1, ... , b ' q, Z r
(2.1.8)
well-poised if the parameters satisfy the relations qal = a2 bl = a3 b2 = ... = ar+lb r ; 1
(2.1.9)
1
very-well-poised if, in addition, a2 = qai, a3 = -qai; a nearly-poised series of the first kind if qal -=I=- a2 bl = a3 b2 = ... = ar+lb n
and a nearly-poised series of the second kind if qal = a2 bl = a3 b2 = ... = arb r - l
-=I=-
ar+lbr .
(2.1.10)
In this chapter we shall be primarily concerned with the summation and transformation formulas for very-well-poised basic hypergeometric series. To help simplify some of the displays involving very-well-poised r+lcPr series which arise in the proofs in this and the subsequent chapters we shall frequently replace
by the more compact notation (2.1.11) In the displays of the main formulas, however, we shall continue to use the r+l cPr notation, since in most applications one needs to know the denominator parameters. In all of the very-well-poised 6cPS, 8cP7 and lOcP9 series in Appendix II and Appendix III the parameters aI, a4, ... , ar+l and the argument z satisfy the very-well-poised balancing condition (a4as···ar+l)z= (±(alq)~r-3
(2.1.12)
with either the plus or minus sign, where (alq)~ is the principal value of the square root of alq. Thus, we will call a r+l Wr series very-well-poised-balanced (or, for brevity, VWP-balanced) if (2.1.12) holds. A special case of this condition was stated at the end of §2.4 in the first edition of this book. It follows from (2.1.12) that a VWP-balanced series is balanced (I-balanced) when z = q,
Summation, transformation, and expansion formulas
40
a balanced r+l Wr series is VWP-balanced when r is even, but it is not necessarily VWP-balanced when r is odd since then we only know that the squares of both sides of (2.1.12) (with z = q) are equal (see the comments in the first paragraph of §2.6). When r + 1 = 2k is an even integer, which is usually the case, (2.1.12) simplifies to the condition (a4a5'" a2k)z = (alq)k-2.
(2.1.13)
Notice that if we set a2k = ±(alq)!, then the 2kW2k-l series reduces to a series that satisfies (2.1.12) with r = 2k - 2 and the corresponding plus or minus sign. Since any well-poised r+l¢r series that satisfies the relations in (2.1.9) can be written as a very-well-poised series in the form
2k-l W 2k-2
1
1
1
1
r+5 W r+4(al; al, -al, qal, -qal, a2, a3,"" ar+l; q, z),
(2.1.14)
if follows that the very-well-poised balancing condition for the series in (2.1.14) is consistent with the well-poised balancing condition (ala2'" ar+dz
= -(±(alq)!y+l
(2.1.15)
for a well-poised r+l¢r series to be well-poised-balanced (WP-balanced). In particular, the 2¢1 series in the Bailey-Daum summation formula (1.8.1) is WP-balanced. Clearly, every VWP-balanced basic hypergeometric series is WP-balanced and, by the above observations, every WP-balanced basic hypergeometric series can be rewritten to be a VWP-balanced series of the form in (2.1.14). 2.2 A general expansion formula
Let a, b, c be arbitrary parameters and k be a nonnegative integer. Then, by the q-Saalschiitz formula (1. 7.2) 3¢2
(q-k, aqk, aq/bc; aq/b,aq/c; q,q) (C,ql-k/b;q)k
(b,c;q)k
(aq/b, cq-k fa; q)k
(aq/b, aq/c; q)k
so that ~
(b, c, q-n; qh
~ (q, aq/b, aq/c; q)k
k=O =
A k
Ln Lk
(/b k Ak aq c, aq k ,q -k.: q ) j (q.-n.) ,q k qj (b) ~ k=Oj=O (q, aq/b, aq/c, q)j(q, q)k aq
_ ~~ (aq/bc, aqi+j,q-i- j ; q)j (q-n;q)i+j . (bc)i+j -~~ ~ ~H j=Oi=O (q,aq/b,aq/c;q)j(q;q)i+j aq ~ (aq/bc, aqj, q-n; q) .
= ~
j=O
J
(q, aq/b, aq/c; q)j
.
(-I)J q -
(j) 2
(2.2.1)
2.3 A summation formula for a terminating n-j (qj-n, aq2j ;q)i
X" ~
(q,aqj;q)i
q
(bC)i+j
-i. J
-
aq
41
41>3
(2.2.2)
A+·
• J'
where {Ak} is an arbitrary sequence. This is equivalent to Bailey's [1949] lemma. Choosing (2.2.3)
X
r+ 2 1>r+ 1
[
aq 2j ,a1 qj ,a2qj , ... , arq j ,qj-n . ..
.
.,
b1qJ, b2qJ, ... , brqJ, br+ 1qJ
bcz ] q, ------:::j:l . aqJ
(2.2.4)
This is a q-analogue of Bailey's formula [1935, 4.3(1)]. The most important property of (2.2.4) is that it enables one to reduce the r+41>r+3 series to a sum of r+21>r+1 series. Consequently, if the above r+21>r+1 series is summable for some values of the parameters then (2.2.4) gives a transformation formula for the corresponding r+41>r+3 series in terms of a single series. 2.3 A summation formula for a terminating very-well-poised 41>3 series Setting b = qa!, c = -qa! and ak obtain from (2.2.4) that 1
41>3
[
=
bk , k
=
=
aqn+l, we
1
] a, qa 2" , -qa 2" , q-n ! ! n+1 ;q,z a 2 , -a 2 , aq
~ (-q-1,q-n;q)j(a;qhj(
= ~
1
1
j=O (q, a2", -a2", aqn+1; q)j
x
1,2, ... , r, br+ 1
21>1
). _(j) qz Jq 2
(aq2j,qj-n;aqHn+\q,_zq1-j).
(2.3.1)
If we set z = qn so that the above 41>3 series is VWP-balanced, then the 21>1 series on the right of (2.3.1) can be summed by means of the Bailey-Daum summation formula (1.8.1), which gives A,
(
2'1'1 aq
2j
,q
j-n. j+n+1. _ 1+n- j ) _ , aq , q, q -
( _q. q) (aq2j+1 aq2n+2. q2) '00 , '00 ( aqn +J·+1 ,-q1+ n _.) . J; q 00
(2.3.2) Hence, using the identities (1.2.32), (1.2.39) and (1.2.40), and simplifying, we obtain the transformation formula 1 1 n] a, qa2", -qa2" , q-n [ 41>3 ! ! n+1; q, q a 2 ,-a 2 ,aq (aq, -q; q)n
(qa !2 , -qa !.) 2 ,q n
A,
2'1'1
(-n -1 -n ) q , -q ; -q ; q, q .
(2.3.3)
42
Summation, transformation, and expansion formulas
Clearly, both sides of (2.3.3) are equal to 1 when n = O. By (1.5.3) the 2¢1 series on the right of (2.3.3) has the sum(ql-n; q)n (_q-l j (_q-n; q)n when n = 0,1, .... Since (ql-n; q)n = 0 unless n = 0, it follows that
r
1
4¢3
[
1
a, qa"2 , -qa"2 , q-n n] 1 1 n+l; q, q a 2 , -a 2 , aq
= 6n ,o ,
(2.3.4)
where m=n, m
(2.3.5)
i- n,
is the Kronecker delta function. This summation formula will be used in the next section to obtain the sum of a 6¢5 series. 2.4 A summation formula for a terminating very-well-poised 1
6¢5
1
series
1
Let us now set al = qa"2, a2 = -qa"2, b1 = a"2, b2 ak = bk , for k = 3,4, ... , r. Then (2.2.4) gives 1
rf..
[
6'i"5
=
=
1
-a"2, br+1 = aqn
+1
and
1
a, qa"2, -qa"2, b, c, q-n ] 1 1 +1 ; q, z a"2, -a"2, aqjb, aqjc, aqn
~ (aqjbc,qa!,-qa!,q-n;q)j(a;qhj (_bacqz)j q_(~) ~ j=O ( q,a 12 ,-a 12 ,aq jb ,aq j c,aq n+1.) ,q j X4¢3
[
aq2j,qJ+la!,_qj+la!,qj-n. bCZ] ·1 ·1 . 1 ,q'------:::t=1. qJ a"2 , _qJ a"2 , aqJ+n+ aqJ
If z = aqn+1 jbc, then we can sum the above obtain the summation formula rf..
6'i"5
[
4¢3
(2.4.1)
series by means of (2.3.4) to
a, qa!, -qa!, b, c, q-n . a qn +1 ] 1 1 +1 , q, - a"2, -a"2, aqjb, aqjc, aqn bc
(aqjbc,qa!,-qd,q-n;q)n (a;qhn (_1)nqn(n+l)/2 (q, a!, -a!, aqjb, aqjc, aqn+l; q)n (aq, aqjbc; q)n (aqjb, aqj c; q)n .
(2.4.2)
Note that the above 6¢5 series is VWP-balanced and that this summation formula reduces to (2.3.4) when bc = aq. In the next two sections, like climbing the steps of a ladder, we will use (2.4.2) to extend it to a transformation formula and a summation formula for very-well-poised S¢7 series. 2.5 Watson's transformation formula for a terminating very-well-poised
S¢7
series
We shall now use (2.4.2) to prove Watson's [1929a] transformation formula for a terminating very-well-poised S¢7 series as a multiple of a terminating
43
2.6 Jackson's sum balanced
4¢3
series:
8
[
a 2q2 +n ]
a qa! -qa! b c d e q-n ~
1
,
,
,
,
,
;q, - - -
,
a"2, -a"2, aq/b, aq/c, aq/d, aq/e, aqn+1
bcde
] _ (aq, aq/ de; q)n ¢ [ q-n, d, e, aq/bc . - (aq/d, aq/e; q)n 4 3 aq/b, aq/c, deq-n /a' q, q .
(2.5.1)
It suffices to observe that from (2.2.4) we have A,
8'1'7
=
[
t
. a 2 q2+n] +1' q, - b d a"2, -a"2, aq/b, aq/c, aq/d, aq/e, aqn c e
j=O
.1 2 , b,c, d ,e,q -n a,qa .2 1 ,-qa
1
1
(~q/bc, ;a!, -qa!, d, e, q-n; q)j (a; qhj
(_ a qn +1 )j q-a) (q, a"2, -a"2, aq/b, aq/c, aq/d, aq/e, aqn+1; q)j de aq2j,qj+1a!,-qj+1a!,dqj,eqj,qj-n
x 6¢5 [ qJ.a "2,-qJ 1 . 1 ·+1 ·+1 .+ +1; q, a"2, aqJ /d,aqJ /e,aqJ n
aql+ n - j ] de '
(2.5.2)
which gives formula (2.5.1) by using (2.4.2) to sum the above 6¢5 series. Clearly, the 8¢7 series in (2.5.1) is VWP-balanced and the transformation formula (2.5.1) reduces to the summation formula (2.4.2) when bc = aq.
2.6 Jackson's sum of a terminating verywell-poised balanced
8¢7
series
The 8¢7 series in (2.5.1) is balanced when the six parameters a, b, c, d, e and n satisfy the condition (2.6.1) which is the very-well-poised balancing condition. For such a series Jackson [1921] showed that 1
S¢7
[
1
a, qa"2 , -qa"2 , b, c, d, e, q-n 1
1
a"2, -a"2, aq/b, aq/c, aq/d, aq/e, aqn+1 (aq,aq/bc, aq/bd, aq/cd;q)n (aq/b, aq/c, aq/d, aq/bcd; q)n '
] ;q,q
(2.6.2)
when n = 0,1,2, .... This formula follows directly from (2.5.1), since the 4¢3 series on the right of (2.5.1) becomes a balanced 3¢2 series when (2.6.1) holds, and therefore can be summed by the q-Saalschiitz formula. Notice that the very-well-poised series in (2.6.2) is also balanced if a 2qn+1 = -bcde i=- 0, but then it is not VWP-balanced and it is not summable as a quotient of products of q-shifted factorials. Hence, not every balanced terminating very-well-poised S¢7 series is summable. Formula (2.6.2) is a q-analogue of (2.1.6), as can be seen by replacing a, b, c, d, e by qa, qb, qC, qd, qe, respectively, and then letting q ----> 1. It should be observed that the series S¢7 in (2.6.2) is balanced, while the limiting series 7F6 in (2.1.6) is 2-balanced. The reason for this apparent discrepancy is that the appropriate q-analogue of the term (1 + ~ah/(~ah = (a + 2k)/a in the 7F6 series is not (qa!; qh/(a!; qh = (1 - a! qk)/(l - a!)
Summation, transformation, and expansion formulas
44
but (qa!, -qa!; qh/(a!, -a!; q)k, which introduces an additional q-factor in the ratio of the products of the numerator and denominator parameters. Analogous to (2.1.12), one could call the very-well-poised 7F6 series in (2.1.6) VWPbalanced if (b + e + d + e - n) + 1 = 2(a + 1), so that then every terminating very-well-poised 7F6(1) series is summable whenever it is VWP-balanced.
2.7 Some special and limiting cases of Jackson's and Watson's formulas: the Rogers-Ramanujan identities Many of the known summation formulas for basic hypergeometric series are special or limiting cases of Jackson's formula (2.6.2). For example, if we take d ---+ 00 in (2.6.2) we get (2.4.2). On the other hand, taking the limit a ---+ 0 after replacing d by aq/d gives the q-Saalschiitz formula (1.7.2). Let us now take the limit n ---+ 00 in (2.6.2) to obtain the following summation formula for a non-terminating VWP-balanced 6¢5 series 1
A,
[
6'/-'5
1
a, qa'2 , -qa'2 , b, e, d . aq ] 1 1 , q, -b d a'2, -a'2, aq/b, aq/e, aq/d c
(aq,aq/be,aq/bd,aq/ed;q)oo (aq/b,aq/e,aq/d,aq/bed; q)oo,
(2.7.1)
where, for convergence, it is required that laq/bedl < 1. This formula is dearly a q-analogue of Dougall's 5F4 summation formula (2.1.7), and it reduces to (2.3.4) when be = aq and d = q-n. If cd = aq, then (2.7.1) reduces to the summation formula 4 W3 (a; b; q, l/b) = 0, where Ibl > 1. Setting d = a! in (2.7.1), we get a q-analogue of Dixon's formula (2.1.5) 4¢3
a, -qa'2, 1 b, e [
1
qa'21
-a'2, aq/b, aq/e
;q,-
1
be
(aq, aq/be, qa! /b, qa! /e; q)oo (aq/b, aq/e, qa!, qa! /be; q)oo'
(2.7.2)
provided Iqa! /bel < 1. Watson [1929a] used his transformation formula (2.5.1) to give a simple proof of the famous Rogers-Ramanujan identities (Hardy [1937]):
f f
L
=
n=O (q; q)n
qn(n+l) = (q, q4, q5; q5)oo,
n=O (q; q)n
where Iql
< 1. First let
(q2,q3,q5;q5)oo, (q; q)oo
b, e, d, e
---+ 00
(q; q)oo
(2.7.3)
(2.7.4)
in (2.5.1) to obtain
~ (a; q)k (1 - aq2k) (q-n; qh q2k 2 (a 2 qn)k
6 =
(q;q)k (I-a) (aqn+l;q)k n (q-n. q ) k (aq;q)nL ' k (_aqn+l) l(k-l)!2. k=O (q; qh
(2.7.5)
2.8 Bailey's transformation formulas
45
Since the series on both sides are finite this limiting procedure is justified as long as the term-by-term limits are assumed to exist. However, our next step is to take the limit n -+ 00 on both sides of (2.7.5), which we can justify by applying the dominated convergence theorem. Thus we have 1+
f k=l
(aq; q)k-l(l - aq2k) (_1)k a 2k qk(5k-l)/2 (q;q)k
q -) . = (aq; q)oo "a -(-. 00
k k2
~
(2.7.6)
k=O q, q k
In view of Jacobi's triple product identity (1.6.1) the series on the left side of (2.7.6) can be summed in the cases a = 1 and a = q by observing that
00
(2.7.7) and 00
2:(1- q2k+l)(_1)k qk(5k+3)/2 k=O 00
(2.7.8) n=-oo
The identities (2.7.3) and (2.7.4) now follow immediately by using (2.7.6). For an early history of these identities see Hardy [1940, pp. 90-99]. 2.8 Bailey's transformation formulas for terminating 5¢4
and
7¢6
series
Using Jackson's formula (2.6.2), it can be easily shown that
(a, b, c; qh (q,aq/b, aq/c; q)k
(Abc/a; q)k (qa 2/Abc; qh x
t
j=O
(A;q)j(1- Aq2j)(Ab/a, Ac/a, aq/bc; q)j (q; q)j(l - A)(aq/b, aq/c, Abc/a; q)j (2.8.1)
where A is an arbitrary parameter. If {Ad is an arbitrary sequence, it follows that
f
(a,b,c;qh A k=O (q, aq/b, aq/c; qh k
Summation, transformation, and expansion formulas
46
=
t
f=
(Abc/a; qh Ak (A; q)j(l - Aq2j) (Ab/a, Ac/a, aq/bc; q)j k=O (qa 2/ Abc; qh j=O (q; q)j(l - A)(aq/b, aq/c, Abc/a; q)j
x (a;q)k+j(a/A;q)k-j
=
f=
(2::y
(Aq; q)k+j(q; q)k-j
A
(A; q)j(l - Aq2j) (Ab/a, Ac/a, aq/bc; q)j(a; qhj (~)j j=O (q;q)j(1- A)(aq/b,aq/c,qa2/Abc;q)j(Aq;qhj A 2·
00
.
,,(aqJ,a/A,AbcqJ/a;qh A x ~ (q,Aq2Hl,a2qH1/Abc;q)k Hk,
(
) 2.8.2
provided the change in order of summation is justified (e.g., if all of the series terminate or are absolutely convergent). It is clear that appropriate choices of A and Ak will lead to transformation formulas for basic hypergeometric series which have at least a partial wellpoised structure. First, let us take Ak = qk(d,q-n;qh/(aq/d,a 2q-n/A2;q)k and A = 2 qa /bcd, so that the inner series on the right of (2.8.2) becomes a balanced and terminating 3¢2. Summing this 3¢2 series and simplifying the coefficients, we obtain Bailey's [1947a,c] formula A.
5~4
[
a, b, c, d, q-n ] aq/b,aq/c,aq/d,a2q-n/A2;q,q
_ (Aq/a,A2q/a;q)n -
(,
'2 /
2
)
Aq, A q a ; q n
[A,qA!,-qA!,bA/a,CA/a,dA/a, 12 ¢11
1
1
A2, -A2, aq/b, aq/c, aq/d,
1
a!, -a!, (aq)!, -(aq)!, A2qn+1 la, q-n 1
1
1
1
Aq/a2, -Aq/a2, A(q/a)2, -A(q/a)2, aq-n / A, Aqn+1
;
q, q , (2.8.3)
where A = qa 2/bcd. Note that the 5¢4 series on the left is balanced and nearly-poised of the second kind, while the 12¢11 series on the right is balanced and VWP-balanced. Note also that a terminating nearly-poised series of the second kind can be expressed as a multiple of a nearly-poised series of the first kind by simply reversing the series. By proceeding as in the proof of (2.8.3), one can obtain the following variation of (2.8.3) 5¢4
[
q-n, b, c, d, e ] q1-n /b, q1-n / C, q1-n / d, eq-2n / A2 ; q, q
_ (A2qn+l,Aq/e;q)n ¢ [A,qA!,-qA!,Abqn,ACqn,Adqn, - (A2qn+1/e, Aq; q)n 12 11 A!, -A!, q1-n /b, q1-n /c, q1-n /d, q-n/2, _q-n/2, q(1-n)/2, _q(1-n)/2, e, A2qn+1 / e ] AqHn/2, _ Aq Hn/2, Aq(1+n)/2, _Aq(1+n)/2, Aq/e, eq-n / A; q, q , (2.8.4)
where A = ql-2n /bcd.
2.9 Bailey's transformation formula
47
a)(aq/d, Next, let us choose Ak = qk(1 - aq2k)(d, q-n; q)k/(1 a 2q2-n / )...2; q)k and A = qa 2/bcd in (2.8.2) so that the inner series on the right side takes the form
x
5¢>4
[
1
aq2j,qJ+1a!,_qJ+1a!,a/A,qj-n "1
"1
qJ a '5. , _qJ a '5. , Aq
2 "+1 J
2"
,a qJ -n
+2
2 ;
/A
q, q .
This 5¢>4 series is a special case of the 5¢>4 series on the left side of (2.8.3); in fact, the 12¢>11 series on the right side of (2.8.3) in this case reduces to a terminating 8¢>7 series, which we can sum by Jackson's formula (2.6.2). Carrying out the straightforward calculations, we get Bailey's [1947c] second transformation formula
x
12¢>11
[
A, qA! , -qA! ,b.A/ a, CA/ a, dA/ a, (aq)! , -( aq)! , 1
1
1
1
A'5. ,-A'5., aq/b, aq/c, aq/d, A(q/a) '5. ,-A(q/a)'5.,
1
qa!, -qa!, A2 qn-1 la, q-n
'/.1 '/.1 2-n/"A,Aqn+1; q, q A a ,-A a ,aq 2
2
(2.8.5)
,
where A = qa 2 /bcd. 2.9 Bailey's transformation formula for a terminating 1O¢>9 series One of the most important transformation formulas for basic hypergeometric series is Bailey's [1929] formula transforming a terminating 1O¢>9 series, which is both balanced and VWP-balanced, into a series of the same type: 1O¢>9
[
1
a, qa!, -qa!, b, c, d, e, j, Aaqn+1 /ej, q-n 1
1
a '5. , -a '5. ,aq/b, aq/c, aq/d, aq/e, aq/ j, ejq-n /A, aqn+
_ (aq, aq/ej, Aq/e, Aq/ j; q)n - (aq/ e, aq/ j, Aq/ ej, Aq; q)n
1O¢>9
;q,q
1
[A, qA!, -qA!, Ab/a, Ac/a, Ad/a, A! , -A! , aq/b, aq/ c, aq/ d,
e,j, Aaqn+1/e j,q-n, ] Aq/e, Aq/ j, ejq-n la, Aqn+1 ; q, q ,
(2.9.1)
where A = qa 2 /bcd. To derive this formula, first observe that by (2.6.2) 8¢>7
[
A, qA!, -qA!, Ab/a, Ac/a, Ad/a, aqm, q-m 1
1
A'5. ,-A'5., aq/b, aq/c, aq/d, Aq
1 -m
la, Aqm+
1 ;
1
q, q
Summation, transformation, and expansion formulas
48
(b, c, d, Aq; q)m (aq/b, aq/ c, aq/ d, a/ A; q)m '
(2.9.2)
and hence the left side of (2.9.1) equals ~ (a; q)m(l - aq2m)(e, I, Aaqn+l /el, q-n; q)m(a/ A; q)m m ~ m=O (q;q)m(l- a)(aq/e,aq/I,elq - n/A,aqn+l;q)m(Aq;q)m q
f
(A;q)j(l- Aq2j)(Ab/a,Ac/a,Ad/a,aqm,q-m;q)j j x j=O (q; q)j(l - A)(aq/b, aq/c, aq/d, Aql-m la, Aqm+l; q)j q
=
tf
(a; q)m+j(1- aq2m)(e, I, Aaqn+l /el, q-n; q)m m m=Oj=O (q;q)m-j(l- a)(aq/e,aq/I,elq-n/A,aqn+l;q)m q x (a/A; ~)m-j(A; :)j(l - Aq2j)(Ab/a, Ac/a, Ad(a; q)j (~)j (Aq, q)m+j(q, q)j(l - A)(aq/b, aq/c, aq/d, q)j A
=
t
~A; q)j(l - Aq2j)(Ab/a, Ac/a, Ad/a, e, I, Aaqn:~el, q~:l~)j . j=O (q,q)J(l- A)(aq/b,aq/c,aq/d,aq/e,aq/I,elq /A,aq ,q)J
aq)j (aq·qh· x (\ (A:) J /\ q, q 2J
8W7
( 2··· .1 . ) aq J;eqJ,lqJ,a/A,Aaqn+J+ /el,qJ-n;q,q , (2.9.3)
where the 8 W 7 series is defined as in §2.1. Summing the above 8 W 7 series by means of (2.6.2) and simplifying the coefficients, we obtain (2.9.1). It is sometimes helpful to rewrite (2.9.1) in a somewhat more symmetrical form: lOW9 (a; b, c, d, e, I, g, h; q, q)
(aq, aq/el,aq/eg, aq/eh, aq/Ig, aq/lh, aq/gh, aq/elgh; q)oo (aq/e,aq/I,aq/g,aq/h,aq/elg, aq/elh, aq/egh,aq/Igh; q)oo x lO W g (qa 2/bcd; aq/bc, aq/bd, aq/cd, e,J, g, h; q, q) , (2.9.4) where at least one of the parameters e, I, g, h is of form q-n, n and
=
0,1,2, ... , (2.9.5)
2.10 Limiting cases of Bailey's transformation formula
lO
A number of the known transformation formulas for basic hypergeometric series follow as limiting cases of the transformation formula (2.9.1). If we let b, c, or d ---; 00 in (2.9.1), we obtain Watson's formula (2.5.1). On the other hand, if we take the limit n ---; 00, we get the transformation formula for a nonterminating 8
8
1
2.10 Limiting cases of
XS¢7
[
1O¢9
transformation formula
A,qA~,-qA~,Ab/a,Ae/a,Ad/a,e,f 1
1
A2, -A2, aq/b, aq/e, aq/d, Aq/e, Aq/ f
49
aq ]
;q,- , ef
(2.10.1) where A = qa 2/bed and max (Iaq/efl, IAq/efl)
< 1.
(2.10.2)
The convergence of the two series in (2.10.1) is ensured by the inequalities (2.10.2) which, of course, are not required if both series terminate. For example, if f = q-n, n = 0,1,2, ... , then (2.10.1) becomes ¢
[
S 7
.1 a, qa .2 1 , -qa 2, b,e, d ,e, q -n . a 2 qn+2] a~, -a~, aq/b, aq/e, aq/d, aq/e, aqn+1 ,q, bcde
(aq, Aq/e; q)n (aq/e, Aq; q)n
x
S¢7
[
A, qA~, -qA~, Ab/a, Ae/a, Ad/a, e, q-n. 1
1
A2, -A2, aq/b, aq/e, aq/d, Aq/e, Aqn+
1'
a qn +1 ] q, - - . e (2.10.3)
This identity expresses one terminating VWP-balanced S¢7 series in terms of another. Using (2.5.1) we can now express (2.10.3) as a transformation formula between two terminating balanced 4¢3 series: rI.
4'1'3
[q-n,a,b,e ] d,e,f ;q,q
_ (e/a, f fa; q)n n (e, f; q)n a
rI.
4'1'3
[
q-n, a, d/b, d/e . ] d, aql-n Ie, aql-n / f' q, q , (2.10.4)
where abc = defqn-l. This is a very useful formula which was first derived by Sears [1951c], and hence is called Sears' 4¢3 transformation formula. It is a q-analogue of Whipple's [1926b] formula F 4
3
[-n, d,a,
b, e,
e.
f'
1]
_ (e - a)n (f - a )n F [ -
(e)n(f)n
4
3
-n, a, d - b, d - e . d,l+a-e-n,l+a-f-n'
1]
, (2.10.5)
where a + b + e + 1 = d + e + f + n. Use of (2.5.1) and (2.10.1) also enables us to express a terminating balanced 4¢3 series in terms of a nonterminating S¢7 series. For example, if b, c, or din (2.10.1) is of the form q-n,n = 0,1,2, ... , then the series on the left side of (2.10.1) terminates, but that on the right side does not. In particular,
Summation, transformation, and expansion formulas
50 setting d
=
q-n, and then replacing e and j by d and e, respectively, we obtain (a; b, e, d, e, q-n; q, a 2qn+2 /bede) (aq, aq/de, a 2qn+2/bed, a 2qn+2/bee; q)oo (aq/d,aq/e, a 2qn+2/be, a 2qn+2/bede; q)oo
8W 7
x 8 W 7 (a 2qn+1 /be; aqn+1 /b, aqn+1 /e, aq/be, d, e; q, aq/de) , (2.10.6) provided laq/del < 1 to ensure that the nonterminating series on the right side converges. Use of (2.5.1) then leads to the formula 4
¢ [q-n, a, b, e. ] _ (deqn / a, deqn /b, deqn / e, deqn / abc; q)oo 3 d, e, j , q, q - (deqn, deqn / ab, deqn / ae, deqn /be; q)oo
x
[deqn-l,
rI..
8'1-'7
q(deqn-1)!,
-q(deqn-1)!,
1
(deqn-1)2,
-(deqn-1)!,
a, b, e, dqn, eqn de] ;q,- , deqn / a, deqn /b, deqn / e, e, d abc
(2.10.7)
provided dej = q1-nabe and Ide/abel < l. As another limiting case of (2.9.1) Bailey [1935, 8.5(3)] found a nonterminating extension of (2.5.1) that expresses a VWP-balanced 8¢7 series in terms of two balanced 4¢3 series. First iterate (2.9.1) to get 10 W g
(a; b, e, d, e, j, a 3 qn+2 /bedej, q-n; q, q)
(aq,aq/de, aq/dj,aq/ej; q)n (aq/d,aq/e, aq/j,aq/dej; q)n
x 10W9 (dejq-n-1/ a ; aq/be, d, e, j, bdejq-n-1 /a 2 , edejq-n-1 /a 2 , q-n; q, q) . (2.10.8) Clearly, the 10 W g on the left side of (2.10.8) tends to the 8¢7 series on the left side of (2.10.1) as n ---+ 00. However, the terms near both ends of the series on the right side of (2.10.8) are large compared to those in the middle for large n, which prevents us from taking the term-by-term limit directly. To circumvent this difficulty, Bailey chose n to be an odd integer, say n = 2m + 1 (this is not necessary, but it makes the analysis simpler), and divided the series on the right into two halves, each containing m + 1 terms, and then reversed the order of the second series. The procedure is schematized as follows: 2m+1
L
k=O
Ak
=
m
2m+1
k=O
k=m+1 m
L Ak + L m
=L k=O
Ak
+L
A2m+1-k'
rI..
---+ 00
(and hence n
a, qa 12 , -qa 12 , b, e, d, e, j . a 2q2 ] 1 ,q,-a 2 , -a 2 , aq/b, aq/e, aq/d, aq/e, aq/ j bedej 1
(2.10.9)
k=O
where {Ad is an arbitrary sequence. Letting m follows from (2.10.8) that 8'1-'7 [
Ak
---+
(0), it
2.10 Limiting cases of
1O¢9
transformation formula
51
_ (aq, aq/de, aq/df, aq/ef; q)oo ¢ [ aq/bc, d, e, f . ] - (aq/d, aq/e, aq/ f, aq/def; q)oo 4 3 aq/b, aq/c, def /a' q, q (aq,aq/bc,d,e,f,a 2q2/bdef, a2q2/cdef;q)00 +~~--~~~--~--~~~~~~~7-~-(aq/b,aq/c,aq/d, aq/e,aq/f, a2q2/bcdef,def/aq; q)oo aq/de,aq/df,aq/ef,a2q2/bcdef ] (2.10.10) x 4¢3 [ a2 q2/ bdef, a2 q2/ cdef,aq2def / ; q, q , where la 2q2/bcdefl < 1, if the 8¢7 series does not terminate. Note that if either b or c is of the form q-n, n = 0,1,2, ... , then the 8¢7 series on the left side terminates but the series on the right side do not necessarily terminate. On the other hand if one of the numerator parameters (except a2q2/bcdef) in either of the two 4¢3 series in (2.10.10) is of the form q-n, then the coefficient of the other 4¢3 series vanishes and we get either (2.5.1) or (2.10.7). If aq/bc, aq/ de, aq/ df or aq/ ef equals 1, then (2.10.10) reduces to the 6¢5 summation formula (2.7.1). If, on the other hand, aq/cd = 1 then the 8¢7 series in (2.10.10) reduces to a 6¢5 which, via (2.7.1), leads to the summation formula
(aq,aq/be,aq/bf,aq/ef;q)oo (aq/b,aq/e,aq/f,aq/bef;q)oo _ (aq,c/e,c/f,aq/ef;q)oo ¢ [aq/bc,e,f. ] - (c, aq/e, aq/ f, clef; q)oo 3 2 aq/b, efq/c' q, q (aq,aq/ef,e, f,aq/bc,acq/bef; q)oo +~~-=~--~~--~=-~~~ (aq/e, aq/f,ef/c, c, aq/b, aq/bef;q)oo c/ e, c/ f, aq/bef ;q,q] . x 3¢2 [ cq/ef,acq/bef
(2.10.11)
Solving for the first 3¢2 series on the right and relabelling the parameters we get the following nonterminating extension of the q-Saalschiitz formula A,
3'1'2
[a,e,b,f c ; q, q]
(q/e, f la, f /b, f /c; q)oo (aq/e, bq/e, cq/e, f;q)oo (q/e, a, b, c, qf Ie; q)oo (e/q, aq/e,bq/e, cq/e, f; q)oo aq/e,bq/e,cq/e ] x 3¢2 [ q2/ e,q f/ e ; q, q ,
(2.10.12)
where ef = abcq. Sears [1951a, (5.2)] derived this formula by a different method. If a, b, or c is of the form q-n, n = 0,1,2, ... , then (2.10.12) reduces to (1.7.2). A special case of (2.10.12) which is worth mentioning is obtained by setting c = 0, f = 0, and then replacing e by c to get 2¢1
(
)
(q/ c, abq/ c; q)oo
a, b; c; q, q = (aq / c, bq/ c; q) 00
(q/c, a, b; q)oo (c / q, aq / c, bq/ c; q)00
2¢1
(/ / 2/ ) aq c, bq c; q c; q, q .
(2.10.13)
52
Summation, transformation, and expansion formulas
If a or b is of the form q-n, n = 0,1,2, ... , then (2.10.13) reduces to (1.5.3). In general, a 2¢1 (a, b; e; q, q) series does not have a sum which can be written as a ratio of infinite products. However, we can still express (2.10.13) as the summation formula for a single bilateral infinite series in the following way. First, use Heine's transformation formula (1.4.1) to transform both 2¢1 series in (2.10.13): 2¢1
(a, b; e; q, q) = (b, ( aq; q)oo ) e, q; q
00
= (b, aq; q)oo
2¢1
( /
)
e b, q; aq; q, b
f
(e/b; q)n bn, (e,q;q)oo n=O (aq;q)n
(2.10.14)
,,( / b / . 2/. ) _ (aq/e,bq2/e;q)00 ~ (q/a;q)n (aq)n 2'/'1 aq e, q e, q e, q, q (2/ .) ~ (b 2/ . ) q e, q, q 00 n=O q e, q n e
.
(2.10.15) Next, note that
f
f
(q~a; :)n (aq)n = _.:: 1 - bq/e (l/a;.q)n (aq)n. n=o(bq/e,q)n e q I-a n=l(bq/e,q)n e
(2.10.16)
Using (2.10.14) - (2.10.16) and the identity (1.2.28) in (2.10.13) we obtain
f n=-CX)
(e/b;q)nb n = (e, q/e, abq/e, q; q)oo , (aq; q)n (b, aq, aq/e, bq/e; q)oo
(2.10.17)
which is Ramanujan's sum (see Chapter 5 and Andrews and Askey [1978]). However, the conditions under which (2.10.17) is valid, namely, Iql < 1, Ibl < 1 and laq/el < 1, are more restrictive then those for (2.10.13). Note that (2.10.17) tends to Jacobi's triple product identity (1.6.1) when a = 0 and b ---+ O. We shall give an alternative derivation of this important sum in Chapter 5 where bilateral basic series are considered. As was pointed out by AI-Salam and Verma [1982a], both (2.10.10) and (2.10.12) can be conveniently expressed as q-integrals. Thus (2.10.12) is equivalent to
l
b
(qt/a, qt/b, et; q)oo dqt (dt, et, jt; q)oo = b(1 _ q) (q, bq/a, alb, e/d, e/e, e/ j; q)oo , (ad, ae, aj, bd, be, bj; q)oo a
where e
=
(2.10.18)
abdej, while (2.10.10) is equivalent to
l
a
=
b
(qt/a, qt/b, et, dt; q)oo dqt (et,Jt, gt, ht; q)oo b(1 _ q) (q, bq/ a, alb, ed/ eh, ed/ j h, ed/ gh, be, bd; q)oo (ae, aj, ag, be, bj, bg, bh, bed/h; q)oo x 8 W 7 (bed/hq; be, bj, bg, e/h, d/h; q, ah),
(2.10.19)
2.11 Bailey's three-term transformation formula for
8¢7
series
53
provided cd = abe j gh and Iah I < 1. By substituting c = abdej into (2.10.18), letting j ----+ 0 and then replacing a, d, e by a, cia, d/b, respectively, we obtain Andrews and Askey's [1981] formula
l
b (qt/a, qt/b; q)oo dqt a (ct/a,dt/b;q)oo b(l - q)(q, alb, bq/a, cd; q)oo (c, d, bc/a, ad/b; q)oo
(2.10.20)
In view of (1.3.18), (2.10.20) is a q-extension of the beta-type integral
j
( + d)a+b-l (1 + t/c)a-l(l - t/d)b-l dt = B(a b)":""c_.,..:..-:;---:;--c ' ca-1db- 1 ' d
(2.10.21)
which follows from (1.11.8) by a change of variable. The notational compactness of (2.10.18) and (2.10.19) is advantageous in many applications (see, e.g., the next section). In addition, the symmetry of the parameters in the q-integral on the left side of (2.10.19) implies the transformation formula (2.10.1).
2.11 Bailey's three-term transformation formula for VWP-balanced 8¢7 series The q-integral representation (2.10.19) of an 8¢7 series can be put to advantage in deriving Bailey's [1936] three-term transformation formula for VWPbalanced 8¢7 series: A,
a,qa!,-qa!,b,c,d,e,j . a 2q2 ] 1 ,q, -b-d-j a"2, -a"2, aq/b, aq/c, aq/d, aq/e, aq/ j c e (aq,aq/de,aq/dj,aq/ej,eq/c, jq/c, b/a, bej/a; q)oo (aq/d,aq/e,aq/j,aq/dej,q/c,ejq/c, be/a,bj/a; q)oo [
8'1-'7
1
X8¢7
[
ej /c, q(ej /c)!, -q(ej /c)!, aq/bc, aq/cd, ej la, e, j. bd] 1 1 ,q,(ej /C)"2, -( ej /c)"2, bej / a, dej la, aq/c, jq/ c, eq/c a
b (aq, bq/a, bq/c, bq/d, bq/e, bq/ j, d, e, j; q)oo a (aq/b,aq/c,aq/d,aq/e,aq/j,bd/a,be/a,bj/a,dej/a;q)oo (aq/bc, bdej/a 2, a2q/bdej;q)oo x --'------'-:------;-:--::--'------:-----::--,,----'--;-----;------'----(aq/dej, q/c, b2q/a; q)oo a2q2 ] b2/a, qba-!, -qba-!, b, bc/a, bd/a, be/a, bj /a. 1 1 ,q, b d j , x 8¢7 [ ba-"2, -ba-"2, bq/a, bq/c, bq/d, bq/e, bq/ j c e
+-~~~~~~~~~~~~~~~~~~-~
(2.11.1) where Ibd/al < 1 and la 2q2/bcdejl < 1. To prove this formula, first observe that by (2.10.19) 8W7
(a; b, c, d, e, j; q, a 2q2 /bcdej)
54
Summation, transformation, and expansion formulas
aq-dej (aq, d,e,j,aq/be, aq/de, aq/dj,aq/ej;q)oo adejq(l - q) (q, aq/b, aq/e, aq/d, aq/e, aq/ j, dej /aq, aq/dej; q)oo def (t/a, qt/dej, aqt/bdej, aqt/edej; q)oo d I x aq (t/de, t/dj, t/ej, aqt/bedej; q)oo qt. (2.11.2) Since
I
bdef fa aq
j(t) dqt +
I
def
bdef fa
j(t) dqt
=
Idef j(t) dqt, aq
(2.11.3)
where
j(t) = (t/a, qt/dej, aqt/bdej, aqt/edej; q)oo (t/de, t/dj, t/ej, aqt/bedej; q)oo '
(2.11.4)
and
bdeffa I aq j(t) dqt bdej(l - q)(q, bdej /a 2, a2q/bdej, bq/d, bq/e, bq/ j, bq/a, bq/e; q)oo a(aq/de, aq/dj,aq/ej,bd/a, be/a, bj/a, q/c, b2q/a; q)oo x 8 W 7 (b 2/a; b, be/a, bd/a, be/a, bj fa; q, a2q2/bedej) , (2.11.5)
I
def
bdeffa
j(t) dqt = dej(l - q)(q, aq/b, aq/e, eq/e, jq/e, bej la, dej fa; q)oo (d, e, j, be/a, bj la, aq/be, q/e, ejq/e; q)oo X 8 W 7 (ej /e; aq/be, aq/ed, ej la, e, j; q, bd/a) , (2.11.6)
we immediately get (2.11.1) by using (2.11.5) and (2.11.6) in (2.11.3). The advantage of our use of the q-integral notation can be seen by comparing the above proof with that given in Bailey [1936]. The special case qa 2 = bedej is particularly important since the series on the left side of (2.11.1) and the second series on the right become balanced, while the first series on the right becomes a 6¢5 series with sum
(aq/ee,aq/ej,ejq/e,q/e;q)oo (aq/e, eq/e, jq/e, aq/eej; q)oo' provided laq/eejl
< 1. This gives Bailey's summation formula: 1
A,
1
[a,qa"2,-qa"2,b,e,d,e,j ] 1 1 ;q,q a"2, -a"2 , aq/b, aq/ e, aq/ d, aq/ e, aq/ j (aq, e, d, e, j, bq/a, bq/e, bq/d, bq/e, bq/ j; q)oo b a (aq/b, aq/e, aq/d, aq/e, aq/ j, be/a, bd/a, be/a, bj / a, b2q/a; q)oo b2/a, qba-!, -qba-!, b, be/a, bd/a, be/a, bj /a X8¢7 [ 1 1 ;q,q ba-"2, -ba-"2, bq/a, bq/e, bq/d, bq/e, bq/ j (aq, b/a, aq/ed, aq/ee,aq/ej,aq/de, aq/dj,aq/ej; q)oo (2.11. 7) (aq/e,aq/d,aq/e,aq/j,be/a,bd/a,be/a,bj/a;q)oo '
8'1-'7
1
2.12 Bailey's four-term transformation formula for balanced
101>9
series
55
where qa 2 = bedej, which is a nonterminating extension of Jackson's formula (2.6.2). This can also be written in the following equivalent form
I
(qt/a, qt/b, t/a!, -t/a!, qt/e, qt/d, qt/e, qt/ j; q)= d 1 1 qt (t, bt/a, qt/a'l, -qt/a'l, et/a, dt/a, et/a, jt/a; q)= b(l - q)(q, alb, bq/a, aq/ed, aq/ee, aq/ ej, aq/de, aq/dj, aq/ej; q)= (b, e, d, e, j, be/a, bd/a, be/a, bj fa; q)=
b
a
(2.11.8) 2.12 Bailey's four-term transformation formula for balanced 10 1>9 series Let us start by replacing a, b, e, d, e, and j in (2.11.8) by A, bqn, Ae/a, Ad/ a, Ae/ a and a/bqn, respectively, to obtain
l
bqn
(qt/ A, tql-n /b, t/ A! , -t/ A!, aqt/eA, aqt/dA, aqt/eA, btqn+l / a; q)= d 1 1 qt A (t, btqn /A, qt/A'l, -qt/A'l, et/a, dt/a, et/a, atq-n /bA; q)= b(l - q)(q, A/b, bq/ A, bq/e, bq/d, bq/e, e, d, e; q)= (a/ A, b, Ae/ a, Ad/a, Ae/ a, alb, be/a, bd/ a, bel a; q)= (b,be/a,bd/a,be/a;q)n (Aq)n (2.12.1) x (bq/a,bq/e,bq/d,bq/e;q)n ~ , where n = 0,1,2,.... Let j, g, h be arbitrary complex numbers such that la 3q3/bedejghl < 1. Set p = a3q2/bedejgh, multiply both sides of (2.12.1) by (b 2/a; q)n(1- b2q2n /a)(bj la, bg/a, bh/a; q)n (ap)n (q;q)n(1- b2/a) (bq/j, bq/g, bq/h;q)n A and sum over n from 0 to
Then the right side of (2.12.1) leads to b(l - q)(q, A/b, bq/ A, bq/e, bq/d, bq/e, e, d, e; q)= 00.
(a/ A, b, Ae/ a, Ad/a, Ae/ a, alb, be/a, bd/ a, bel a; q)= x 1OW9 (b 2/a; b,be/a, bd/a, be/a, bf/a, bg/a, bh/a; q,pq) . (2.12.2) The left side leads to two double sums, one from each of the two limits of the q-integral. From the upper limit, bqn, we get
b(l - q) (q, bq/ A, abq/ eA, abq/ dA, abq/ eA, b2q/ a; q)= (b, a/ A, b2q/ A, be/ a, bd/ a, be/a; q)= x (b 2/a; q)n(1- b2q2n /a)(bj la, bg/a, bh/a; q)n (a:)n n=O (q; q)n (1 - b2/ a) (bq/ j, bq/ g, bq/ h; q)n /\
f
x
~
(b 2/ A; qhn+j (1- b2q2n+2j / A)(b, be/a, bd/a, be/a; q)n+j(a/ A; q)j j q j=O (q; q)j(l- b2/A)(bq/A, abq/c>.., abq/dA, abq/eA; q)n+j(b 2q/a; qhn+j
~
b(l - q)(q, bq/A, abq/c>.., abq/dA, abq/eA, b2q/a; q)= (b, a/ A, b2q/ A, be/a, bd/ a, bel a; q)= ~ (b 2/A;q)m(1- b2q2m/A)(b,be/a, bd/a, be/a, a/A; q)m m x ~ q m=O (q; q)m(1- b2/A)(bq/A, abq/c>.., abq/dA, abq/eA, b2q/a; q)m X SW7
(b 2/a; bj/a, bg/a, bh/a, b2qm/ A,q-m; q,pq) .
(2.12.3)
56
Summation, transformation, and expansion formulas
Let us now assume that p
=
1, that is, a 3 q2 = bedeigh.
(2.12.4)
Then, by Jackson's formula (2.6.2), the s(P7 series in (2.12.3) has the sum (b2q/a,aq/ig,aq/ih,aq/gh;q)~
(bq/ i, bq/ g, bq/ h, aq/ i gh; q)~
and the expression in (2.12.3) simplifies to b(l - q)(q, bq/ A, abq/eA, abq/dA, abq/eA, b2q/a; q)oo (b, a/ A, b2q/ A, be/ a, bd/ a, be/a; q)oo X lOW9 (b 2/A; b,be/a, bd/a, be/a, bi/A, bg/A, bh/A; q,q) ,
(2.12.5)
since, by (2.12.4), aq/ig = bh/A, aq/ih = bg/A, and aq/gh = bf/A. We now turn to the double sum that corresponds to the lower limit, A, in the q-integral (2.12.1). This leads to the series -A(l - q)(q, aq/e, aq/d, aq/e, Aq/b, bAq/a; q)oo (b, Aq, alb, eVa, dVa, eVa; q)oo ~ Wfa; q)n(1- b2q2n /a)(b, bi la, bg/a, bh/a, b/ A; q)n n X ~ q n=O (q; q)n(1- b2/a)(bq/a, bq/ i, bq/g, bq/h, bAq/a; q)n X SW7
(A; bqn, eVa, dVa, eA/a, aq-n /b; q, q)
-A(l - q)(q, aq/e, aq/d, aq/e, Aq/b, bAq/a; q)oo (b, Aq, a/b, eA/a, dA/a, eA/a; q)oo
x x
~ (A; q)~(1- Aq2~)(b, eVa, dVa, eVa, alb; q)~ ~
~=O (q; q)~(1- A)(Aq/b, aq/e, aq/d, aq/e, bAq/a; q)~ SW7
q
~
(b2/a;bq~,bq-~/A,bi/a,bg/a,bh/a;q,q).
(2.12.6)
The last s(P7 series in (2.12.6) is balanced and nonterminating, so we may use (2.11.7) to get
s W 7 (b 2fa; bq~, bq-~ /A, bi la, bg/a, bh/a; q, q) (b 2q/a, Aq~+l / i, Aq~+l /g, Ag~+l /h, aq/ ig, aq/ ih, aq/gh, aq~ /b; q)oo (bAq~+l la, bq/ i, bq/g, bq/h, a/A, iq~, gq~, hq~; q)oo 2 + aq~ (b q/a, bq-~ /A, bi la, bg/a, bh/a, aq~+l /b, aq~+l / i; q)oo bq/ i, bq/g, bq/h, a/ A, iq~, gq~; q)oo (aq~+l /g, aq~+l /h, Aq2~+1; q)oo (hq~, aq2~+1; q)oo b
X
(bql-~ la, bAq~+l la,
X SW7 (aq2~; bq~, iq~, gq~, hq~,
a/A; q, q).
Use of this breaks up the double series in (2.12.6) into two parts: -A(l - q)( q, b2q/ a, Aq/b, Aq/ i, Aq/ g, Aq/h, bi / A, bg / A; q)oo (b, i, g, h, bq/ i, bq/ g, bq/h, Aq; q)oo
(2.12.7)
2.12 Bailey's four-term transformation formula for balanced
101>9
series
57
(bh/A,aq/c,aq/d,aq/e;q)= (a/A,cA/a,dA/a, eA/a; q)= x lOW9 (A; b, cA/a, dA/a, eA/a, j,g, h; q, q) aA(l - q)(q, b/ A, Aq/b, b2q/a, aq/b, aq/c, aq/d, aq/e, aq/ j, aq/ g; q)=
x-';--c-:-----:---:-=-;'------'--:-:-=-';-----'-----:=-:-:---'---"':---'--'-
(b, a/b,aq, cA/a, dA/a, eA/a, bq/a, bq/j,bq/g,bq/h; q)= (aq/h, bj/a, bg/a, bh/a;q)= x ..:........::-'----':,......::.,:,----'::--=....:....,,-c---,:-'-.:.....::..:...(a/ A, j, g, h; q)=
x
~ (a;q)n(1- aq2n)(b,j,g,h,a/A;q)n n ~ q n=O (q;q)n(1- a)(aq/b,aq/j,aq/g, aq/h, Aq; q)n
x 8Wd A; cA/a, dA/a, eA/a, aqn, q-n; q, q).
(2.12.8)
Summing the last 81>7 series by (2.6.2) we find that the sum over n in (2.12.8) equals lOW9(a;b,c,d,e,j,g,h;q,q) which is, of course, balanced by virtue of (2.12.4). Equating the expression in (2.12.2) with the sum of those in (2.12.5) and (2.12.8), and simplifying the coefficients, we finally obtain Bailey's [1947b] four-term transformation formula 1 1 A-. [ a,qa'2,-qa'2,b,c,d,e,j,g,h ] 10'1'9 a'2,-a'2,aq 1 1 / / / / / / / ; q, q b,aq c,aq d,aq e,aq j,aq g,aq h (aq,b/a,c,d,e,j,g,h,bq/c,bq/d;q)= 2 + (b q/a,a/b,aq/c, aq/d,aq/e,aq/j,aq/g, aq/h, bc/a, bd/a; q)= (bq/ e, bq/ j, bq/ g, bq/ h; q)= x~-,;---,--:-=::..,::--:-,~,-,-:-:-:--:-.:....::..:,.(be/a, bj la, bg/a, bh/a; q)= b2la, qba-~, -qba-~, b, bc/a, bd/a, be/a, bj la, bg/a, bh/a x 101>9 [ 1 1 ;q,q ba- '2, -ba- '2, bq/ a, bq/ c, bq/ d, bq/ e, bq/ j, bq/ g, bq/ h
1
(aq, b/a, Aq/j,Aq/g, Aq/h, bj/A,bg/A,bh/A; q)= (Aq,b/A, aq/j,aq/g, aq/h,bj/a, bg/a, bh/a;q)=
x
101>9
[
A,qA~,-qA~,b,Ac/a,Ad/a,Ae/a,j,g,h 1
1
;
A'2, -A'2, Aq/b, aq/c, aq/d, aq/e, Aq/ j, Aq/g, Aq/h
1
q, q
(aq,b/a,j,g,h,bq/j,bq/g,bq/h,Ac/a,Ad/a;q)=
+ (b 2q/A,A/b,aq/c,aq/d,aq/e,aq/j,aq/g,aq/h,bc/a,bd/a;q)= (Ae/a,abq/Ac,abq/Ad,abq/Ae;q)= x --'------'-:-::----:---',--::-.,-------:-----'-;-------:-:,----;-'------:------'--(be/a, bj la, bg/a, bh/a; q)=
x
101>9
[
b2 /A, qbA-~, -qbA-~, b, bc/a, bd/a, be/a, bj /A, bg/A, bh/A 1
1
;
bA-'2, -bA-'2, bq/A, abq/cA, abq/dA, abq/eA, bq/ j, bq/g, bq/h
1
q, q .
(2.12.9)
I
In terms of the q-integrals this can be written in a more compact form:
a
b
(qt/a, qt/b, ta-~, -ta-~, qt/c, qt/d, qt/e, qt/ j, qt/g, qt/h; q)= d 1
1
(t, bt/a, qta-'2, -qta-'2, ct/a, dt/a, et/a, jt/a, gt/a, ht/a; q)= a (b/a,aq/b, Ac/a, Ad/a, Ae/a, bj/A, bg/A, bh/A;q)= A (b/A, Aq/b, c, d, e, bj la, bg/a, bh/a; q)=
qt
Summation, transformation, and expansion formulas
58
1 b
x
A
1
1
(qt/ A, qt/b, tA -"2, -tA -"2, aqt/d, aqt/dA, aqt/e>.., qt/ f, qt/ g, qt/h; q)oo d 1 1 qt, (t, btl A, qtA -"2, -qtA -"2, ct/ a, dt/ a, et/ a, ft/ A, gt/ A, ht/ A; q)oo (2.12.10)
where A = qa 2/ede and a3q2 = bedefgh. Exercises 2.1 Show that
3
q~!, -1a! ;q, t]
a 2 , -a 2
2.2 Show that, for max(ltl, laql)
= (1 - aqt 2)
(a~q\q)oo, It I < 1. t; q 00
< 1,
a,qa!,-qa!,b. ] _ (aq,bt;q)oo (-1.. ) 4
2
(a;q)k(l - aq2k)(b,e,a/be;q)k k q I: (q;q)k(l -a)(aq/b,aq/e,beq;q)k n
--;---'------,--'----,--'-_--:-:------'-:c::'-------;--'---::------'---;--
k=O
for n
(aq,bq, eq,aq/be; q)n -;-'---_--:-::-_-,------'-,,----------'-,_ (q,aq/b,aq/e,beq;q)n
0,1, ....
=
2.6 Show that
q-n b e _q1-n /be ] 4
0,
=
{
= 2m+ 1,
(q, b2, e2; q2)rn (be; qhrn 2 n= m, (b, e; q)2rn (b'2e'2; q'2)rn '
where n, m = 0,1,2, .... (Bailey [1941], Carlitz [1969a]) 2.7 Derive Jackson's terminating q-analogue of Dixon's sum:
2
where n [1969a])
b, q1-2n /b,
= 0, 1,2, ....
e . q2-n] _ (b,e;q)n(q,be;qhn q1-2n /e' q, be - (q, be; q)n(b, e; qhn'
(See Jackson [1921,1941]' Bailey [1941]' and Carlitz
Exercises
59
2.8 Show that [
4¢3
=
1 -b'i 1 ] q-n, a, b'i, I - n ! b; q, q q a2, -q 2 a2, I-n! 2
= 2m+ 1, } n = 2m ,
0, { (q, bqja; q2)m (bq, qja; q2)m'
n
where n, m = 0,1, .... (Andrews [1976a]) 2.9
Prove that
2.10 The q-Racah polynomials, which were introduced by Askey and Wilson [1979], are defined by Wn(x;a,b,c,N;q)
=
[
4¢3
q-n abqn+1 q-X cqx-N ] ' b' -N ;q,q, aq, cq, q
where n = 0,1,2, ... ,N. Show that (aqjc,bq;q)n n ( -1) Wn ( x; a, b, c, N; q) = ( b ) c Wn N - x; b, a, c ,N; q . aq, cq;qn 2.11 The Askey- Wilson polynomials are defined in Askey and Wilson [1985] by Pn(x;a,b,c,d I q)
= a-n(ab,ac,ad;q)n
4¢3
[
q-n abcdqn-1 ae iO ae- iO ] ' b 'd' ;q,q , a ,ac,a
where x = cos (). Show that (i) Pn(x; a, b, c, d I q) = Pn(x; b, a, c, d I q), (ii) Pn( -x; a, b, c, d I q) = (-l)n pn (x; -a, -b, -c, -d I q). 2.12 Show that 1
1
1
1
2
lOWg(a;b'i,-b'i,(bq)'i,-(bq)'i,ajb,a qn (aq,a 2qjb 2;q)n (aq jb ,a2 q jb',q) n ,n = 0,1,2, ....
+1
jb,q-n;q,q)
60
Summation, transformation, and expansion formulas
2.13 If A = qa 2/bcd and IqA/al
(i)
¢ [ 4 3
a,b,c,d . qA2] _ (Aq/a,A2q/a;q)oo aq/b,aq/c,aq/d,q, a2 - (Aq,A2q/a 2;q)oo
x lOW9 ( •• ) 11
A.. [ 4 '/"3
< 1, prove that
(A;a~,-a~,(aq)~,-(aq)~'Ab/a'Ac/a'Ad/a;q,qA/a),
a, b, c, d -Aq] aq/b, aq/ c, aq/ d; q, ------;;-
(aq, -q, Aqa-~, -Aqa-~; q)oo (Aq, -Aq/a, qa!, -qa!; q)oo X 8 W7
a~, -a~, Ab/a, Ac/a, Ad/a; q, -q)
(A;
.
2.14 (i) Show that
which is a q-analogue of Bailey [1935, 4.5(1.3)]. (ii) Using (i) in the formula (2.8.2) prove the following q-analogue of Bailey [1935,4.5(4)]: 1
A.. [ 6 '/"5
a, qa'2, b, c, d, q-n ] ;q,q a'2,aq/b,aq/c,aq/d,a 2ql-n/ A2 1
(A/a, A2 la, -Aqa-~; q)n (Aq, A2 /a 2, -Aa-~; q)n 1
x
12¢U
[
1
A, qA'2, -qA'2, Ab/a, Ac/a, Ad/a, 1
1
A'2, -A'2, aq/b, aq/c, aq/d,
where A = qa 2/bcd. This formula is equivalent to Jain's [1982, (4.6)] transformation formula. 2.15 By taking suitable q-integrals of the function f(t) _ (qt/b, qt/c, aqt/bc, t q2/bcdef; q)oo (at, qt/bcd, qt/bce, qt/bcf; q)oo '
Exercises
61
prove Bailey's [1936, (4.6)] identity
(aq/d,aq/e,aq/j,q/ad,q/ae,q/aj;q)= a (qa2, ab, ac, bfa, cia; q)= x 8W7 (a 2; ab, ac, ad, ae, aj; q, q2/abcdej) -1
+ b- 1 (bq/d, bq/e, bq/ j, q/bd, q/be, q/bj; q)= (qb 2,ba,bc,a/b,c/b;q)= x 8W7 (b 2;ba,bc,bd,be,bj;q,q2/abcdej) -1 (cq/d,cq/e,cq/j,q/cd,q/ce,q/cj;q)= +c (qc2, ca, cb, a/c, b/c; q)= 2 x 8W7 (c ;ca,cb,cd,ce,cj;q,q2/abcdej)
= 0, provided Iq2/abcdejl < 1. 2.16 (i) Let S(A, p" v, p) = (A, q/ A, p" q/ p" v, q/v, p, q/ p; q)=. Using Ex. 2.15, prove that
S(XA, x/ A, p,v, p,/v) - S(xv, x/v, Ap" p,/ A) p, = >..S(xp" x/ p" AV, A/v), where x, A, p" v are non-zero complex numbers. (Sears [1951c,d], Bailey [1936]) (ii) Deduce that
(q!, q!; q)= S(Aq!, Aq~, -p,q!, -p,q~)
= (q! Ap" q! / Ap" q! A/ p" q! p,/ A; q)= - q! (A + A-I - P, - p,-1 ) (qAp" q/ Ap" qA/ p" qp,/A; q)=. (Ismail and Rahman [2002b]) 2.17 Show that
A, qA!, -qA!, a, b, c, -c, Aq/C2 Aq] 1 2; q, --b A2, -A2, Aq/a, Aq/b, Aq/C, -Aq/C, c a (Aq, C2/A; q)=(aq, bq, c2q/a, c2q/b; q2)= (Aq/a, Aq/b; q)=(q, abq, c2q, c2q/ab; q2)='
8(P7 [
(i)
1
where A = -c(ab/q)! and IAq/abl (ii)
8¢7
< 1, and
[-c, q( -c)!, -q( -c)!, a, q/a, c, -d, -q/d ] 1 1 ;q,C (-C)2, -( -C)2, -cq/a, -ac, -q, cq/d, cd (-c, -cq; q)=(acd, acq/d, cdq/a, cq2/ad; q2)= (cd, cq/d, -ac, -cq/a; q)=
Icl < 1.
Verify that (i) is a q-analogue of Watson's summation formula (Bailey [1935,3.3(1)]) while (ii) is a q-analogue of Whipple's formula (Bailey [1935, 3.4(1)]). (See Jain and Verma [1985] and Gasper and Rahman [1986]).
62
Summation, transformation, and expansion formulas
2.18 In the 6(/15 summation formula (2.7.1) let b, c, d prove Euler's [1748] identity
----t
L (_l)n { qn(3n-1)/2 + qn(3n+1)/2}
Then set a
00.
=
1 to
00
1+
=
(q; q)oo.
n=1 2.19 Show that 10 W g (a; b, c, d, e, f, g, q-n; q, q)
=
(aq,aq/ce,aq/de,aq/ef,aq/eg,b;q)n n e (aq/c, aq/d,aq/e, aq/f,aq/g, b/e;q)n x lOW9 (eq-n /b; e, aq/bc, aq/bd, aq/bf, aq/bg, eq-n la, q-n; q, q) ,
= bcdefg
where a3qn+2
and n
= 0,1,2, ....
2.20 Prove that 10 W g
(a; b, c, d, e, f, g, q-n; q, a3qn+3/bcdefg)
= (aq,aq/fg;.q)n
t
(q-n,f,g,aq/de~~)j~j
(aq/f,aq/g,q)n j=O (q,aq/d,aq/e,fgq
x for n
4¢3
[
/a,q)j
q-j, d, e, aq/bc ] / 0/ ; q, q , aq b,aq c,deq-J a /
0,1,2, ....
=
2.21 Show that lOW9 (a; b, c, d, e, f, g, h; q, a 3q3/bcdefgh) =
f .
(A; q)n(l - Aq2n) (Ab/a, Ac/a, Ad/a, e,j, g, h; q)n\a q; qhn. n=O (q, q)n(1- A)(aq/b, aq/c, aq/d, aq/e, aq/ f, aq/g, aq/h, q)n(Aq, qhn
x (:;;:) n SW7 (aq2n; a/A,eqn, fqn,gqn, hqn; q,a3q3/bcdefgh) , where A = qa 2/bcd and la 3q3/bcdefghl
<
1.
2.22 Prove that
f .
(a; q)n(1- aq2n)(b, c, d, e; q)n . (_ a 2q2 ) n qG) n=O (q, q)n(1- a)(aq/b, aq/c, aq/d, aq/e, q)n bcde =
(aq,aq/de;q)oo d, aq / e; q) 00
( aq /
Deduce that
A,
3'1-'2
[aq/bc,d,e. aq ] I /d I 1 /b aq /c' q, de ,aq e < . aq,
f
~a; q)n(1- aq2n)(d, e; q?n (_ adqe)n qG) n=O (q, q)n(1- a) (aq/d, aq/e, q)n (aq, aq/de; q)oo (aq/d, aq/e; q)oo .
Exercises
63
2.23 Prove that
t
(ab, ae, ad; q)j qj j=O (abed,aqz, aq/z; q)j
=
(1 - z/a)(l - abez) (1 _ bz)(l _ ez) S W7 (abez; ab, ae, be, qz/d, q; q, dz) (ab, ae, ad; q)n+1 (1 - aqn+1 / z) (1 - abezqn+1) (abed, aqz, aq/ z; q)n+1 (1 - a/ z) (1 - abez) x SW7 (abezqn+1; abqn+1, aeqn+\ be, qz/d, q; q, dz) .
2.24 Show that ¢ [ 5 4 X
a, b, e, d, e . ] _ ()..q/a, )..q/e, q)..2/a, q/ j; q)oo aq/b, aq/e, aq/d, f' q, q - ()..q, aq/ j, eq/ j, aq/ )..j; q)oo 12Wll ()..;a!,-a!, (aq)!,-(aq)!,)"b/a,)..e/a,)"d/a,e,aq/j;q,q) (a,e,a/)..,q/j,)..q2/j;q)oo (j/q,)..q,aq/j,aq/)..j,eq/j;q)oo
x
f f (). ; j=Ok=O
~)j(l - )..q2j) ()"b/a, )..e/a, )"d~a; q)j (q,q)j(l- )..)(aq/b,aq/e, aq/d, q)j
x (f/q,aq/j;q)j (aqj+1/j, aq1- j /)..j,eq/j;q)k qHk, ()..q2 / j,)..j fa; q)j (q, q2-J / j, )..q2+J / j; q)k where ).. = qa 2/bed and j = ea 2/)..2. Note that this reduces to (2.8.3) when e = q-n, n = 0,1,2, .... 2.25 By interchanging the order of summation in the double sum in Ex. 2.24 and using Bailey's summation formula (2.11.7), prove Jain and Verma's [1982, (7.1)] transformation formula A-.
[
5'1'4
+
a, b, e, d, e ] aq/b, aq/e, aq/d, j; q, q
(a,b,e,d,e,q/j,aq2/bj,aq2/ej,aq2/dj;q)oo (aq/b,aq/e,aq/d,j/q,aq/j,bq/j,eq/j,dq/j,eq/j;q)oo
X 5¢4
eq/ j, aq/ j, bq/ j, eq/ j, dq/ j ] [ q2 / j, aq2/bj, aq2/ej, aq2/dj; q, q
()..q/a,)..q/e, q)..2/a, q/j;q)oo ()..q,aq/j,eq/j,aq/)..j;q)oo x 12Wll ()..;a!,-a!,(aq)!,-(aq)!,)"b/a,)..e/a,)"d/a,e,aq/j;q,q)
(a,e,)"b/a,)..e/a,)"d/a,q/j,a2q2/)"bj,a2q2/)..ej,a2q2/)"dj,aq3/j2;q)oo (aq/b,aq/e,aq/d,aq/j,bq/j,eq/j,dq/j,eq/j,)..j/aq,a 2q3/)..j2;q)oo 12 W ll (a 2q2 / )..j2; qa~ / )..j, -qa~ / )..j, (qa) ~ / )..j, -(qa) ~ / )..j,
+~~~~~~~~~~~~~~~~~~~~~~
X
)..q/a, aq/ j, bq/ j, eq/ j, dq/ j; q, q),
64
Summation, transformation, and expansion formulas where the parameters are related in the same way as in Ex. 2.24. Note that this is a nonterminating extension of (2.8.3) and that the first 5¢4 series on the left is a nearly-poised series of the second kind while the second 5¢4 series is a nearly-poised series of the first kind.
2.26 If a = q-n, n = 0,1,2, ... , prove that ¢ [ 3
2
x
a, b, c . aqx] _ (ax; q)(X) aq/b, aq/c' q, bc - (x; q)(X) 5¢4
[
a!, -a!, (aq)!, -(aq)!, aq/bc aq/b,aq/c,ax,q/x
] ;q,q .
(Sears [1951a, (4.1)], Carlitz [1969a, (2.4)]) 2.27 Show that ] ~ [q-n,c,ab/c,a 1, ... ,ar . r+3'1'r+2 a , b, b1,···, br ,q, z
(c,ab/c;q)n (a,b; ) q n
~ (q-n, cia, c/b; qh k ~ [qk-n, c, a1, ... , a r ] x k=O ~ ( -lb- 1 1-n ) q r+2'1'r+1 k b b ; q, z . q, C, ca q; q k cq , 1,···, r
2.28 Show that r+3¢r+2
[
] a,b1, ... ,br,c,q-n /b /b / n+1 ; q, qz aq 1, ... , aq To aq C, aq
(aq/cz,q1-n/ c;q)n ( aq/ c, q 1-n/ cz; q )n
n (q-n/CZ ,l/Z,q-n/a ,q-n;q)k (1- q2k-n/CZ ) (a qn +1 )k (q, aq/cz, q/CZ, q1-n /C; qh (1 - q-n /cz) c k=O a,b1, ... ,br,czq-k,qk-n ] [ X r+3¢r+2 n+1-k ; q, q . / / k+1/ cZ,aq aq b1, ... ,aq br,aq
X
L
2.29 Show that
2.30 Iterate (2.12.9) to prove Bailey's [1947b, (8.1)] transformation formula: loVVg(a;b,c,d,e,j,g,h;q,q)
+
(aq,c,d,e,j,g,h,b/a,bq/c,bq/d, bq/e; q)(X) (qb 2/a, bc/a, bd/a, be/a, bj/a, bg/a,bh/a, a/b, aq/c, aq/d,aq/ e;q)(X)
x ((bq/j,bq/g,bq/h;q)(X) /j / /h) aq ,aq g, ag ; q
(X)
10 VVg
(b 2 /
b b / bd/ b / bj/ b / bh/ ) a;, c a, a, e a, a, 9 a, a; q, q
Exercises
65
(aq,b/a,g, bq/g,aq/ch,aq/dh,aq/eh,aq/fh,bch/a, bdh/a;q)oo (bhq/g, bh/a,g/h, aq/h, aq/c, aq/d, aq/e,aq/f,bc/a, bd/a; q)oo
x
(beh/a, bfh/a; q)oo (be/a,bf/a;q)oo 10VVg(bh/g; b,aq/cg, aq/dg, aq/eg, aq/fg, bh/a, h; q,q) h,bq/h,aq/cg, aq/dg,aq/eg,aq/fg,bcg/a,bdg/a;q)oo (aq, b/a,__ (bgq/h,bg/a, h/g,aq/g, aq/c, aq/d,aq/e, aq/f,bc/a, bd/a; q)oo
+~~~
x
~~~~~~~~~~~~~~~~~~_
(beg/a, bfg/a; q)oo (be/a, bf fa; q)oo lOVVg(b9/h; b, aq/ch, aq/dh, aq/eh, aq/ fh, bg/a, g; q, q),
where a3q2 = bcdefgh. 2.31 By using the q-Dixon formula (2.7.2) prove that the constant term in the Laurent expansion of
(XdX2' Xd X3; q)al (X2/X3, qX2/X1; q)a2 (qx 3/X1, qX3/X2; q)a3 is where a1, a2 and a3 are nonnegative integers. (Andrews [1975a]) 2.32 Use (2.10.18), the q-binomial theorem, and the generating function in Ex. 1.29 to derive the formula 2iO 2iO C ( e.(31 ) = 2i sin e ((3, (3, (3e , (3e- ; q)oo n cos, q ( 2 2 '0 2 '0 1 - q q, (3 ,e ' ,e- , ; q) 00
((32;q)n q; q n
x ()
l
e - ie
eie
(qteiO,qte-iO;q)oo n _ '0 ) t dqt, (3te', (3te ' ; q 00
('0
0
< e < 1f.
(Rahman and Verma [1986a]) 2.33 (i) Prove that
6+2kVVS+2k(a; b, alb, d, e1, ... ,ek, aqnl +1 /e1, ... , aqnk +1 /ek; q, q1-N /d)
= (q, aq, aq/bd, bq/d; q)oo Ilk (aq/bej, bq/ej; q)nj , ( bq,aq / b,aq / d,q /d;q ) 00 j=l ( aq / ej,q / ) nj ej;q
k
= 1,2, ... ,
where n1, ... , nk are nonnegative integers, N = n1 +-. +nk, and q1-N /dl 1 when the series does not terminate. (Gasper [1997]) I
(ii) Deduce that
(be iO , be-iO;p)oo (aqe iO , aqe- iO ; q)oo 00
=
~
(k+1) , , 1 - a 2q2k (a 2q)k (_l)kq 2 (ae'O, ae-'o; q)k(abqk, bq-k /a;p)oo 1- a 2 (q;q)k (q,qa 2;q)00 (aqeiO,aqe-iO;q)k
when 0 < p < q, or p = q and (Ismail and Stanton [2003a]).
Ibl < lal.
<
66
Summation, transformation, and expansion formulas
2.34 Derive the formulas
where Re "( > Re A > 0 in (i) and (ii), and Re (A, v, "( + J-l- A - v) > 0 in (iii). These formulas are q-integral analogues of Erdelyi's [1939, equations (17), (11), and (20), respectively] fractional integral representations for 2Fl series. (Gasper [2000]) 2.35 Derive the following discrete extensions of the formulas in Ex. 2.34:
(Gasper [2000]) 2.36 Extend the formulas in Ex. 2.35 to:
(i) 4¢3(a, (3, vqn, q-n; ,,(, 8, p; q, q)
Notes =
67
b/)..,vqh;q)n)..n b,)..vqh;q)n x
~ ()..vh,)..,8/1,j0!/3,/1pI0!/3, vq n,q-n;qh()..vqh;qhk (8(J)k ~ (q, vqh, p, 8, )..q1-n h, )..vqn+1 h; q)k()..vh; qhk
k=O
/1V
x 4cP3(/110!, /11/3, )..vqk h, q-k;).., 8/110!/3, /1pIO!/3; q, q)
x 4cP3(0!/31/1, /11).., vqk+n, qk-n; "(I).., 8qk, pqk; q, q),
(ii) 4cP3(0!, /3, vqn, q-n; "(, 8, p; q, q) =
b/)..,vqh;q)n)..n b,)..vqh;q)n ~ X
k=O
()..vl,,(,)..,pI0!,81(J, vqn,q-n;q)k()..vql,,(;q)2k (q, vqh, 8, P(J IO!, )..q1-n h, v)..qn+1 h; q)k()..vh; qhk
x 4cP3(0!1(J, /3, )..vqk h, q-k;).., 81(J, p; q, q) X 4cP3((J, (31).., vqnH, qk-n, "(I).., 8qk, (Jpqk IO!; q, q), (iii) A,
r+2 'f's+l
[q-n,V,a1,a2, ... ,ar . ] ' q, z "(, b1, b2,···, bs
/1 v,"( I v,q k-n 1-k ] [ q-k ,/1,a1,···,ar ] X3cP2 [I "(/1I)..V,q1-nlv;q,q I).. r+2cPs+1 )..,b 1,b2, ... ,bs ;q,z , where O!/3vq
= 8"(p in (i) and (ii). (Gasper [2000]) Notes
§2.5 Some applications of Watson's transformation formula (2.5.1) to mock theta functions are presented in Watson [1936, 1937]. §2.7 For additional proofs of the Rogers-Ramanujan identities, identities of Rogers-Ramanujan type, applications to combinatorics, Lie algebras, statistical mechanics, etc., see Adiga et al. [1985], Alladi [1997], Alladi, Andrews and Berkovich [2003], Alladi and Berkovich [2002]' Andrews [1970b, 1974b,c, 1976, 1976b, 1979b, 1981a, 1984b,d, 1986, 1987a,b, 1997, 2001], Andrews, Askey, Berndt et al. [1988], Andrews and Baxter [1986, 1987], Andrews, Baxter, Bressoud et al. [1987], Andrews, Baxter, and Forrester [1984]' Andrews, Schilling and Warnaar [1999], Bailey [1947a, 1949, 1951], Baxter [1980-1988], Baxter and Andrews [1986], Baxter and Pearce [1983, 1984], Berkovich and McCoy [1998], Berkovich, McCoy and Schilling [1998], Berkovich and Paule [2001a,b], Berkovich and Warnaar [2003], Berndt [1985-2001]' Borwein and Borwein [1988], Bressoud [1980a, 1981a,b, 1983a], Bressoud, Ismail and Stanton [2000], Burge [1993], Dobbie [1962]' Dyson [1988], Fine [1988], Foda and Quano [1995], Garrett, Ismail and Stanton [1999], Garsia and Milne [1981]' Garvan [1988],
68
Summation, transformation, and expansion formulas
Lepowsky [1982]' Lepowsky and Wilson [1982]' McCoy [1999], Misra [1988], Paule [1985a], Ramanujan [1919]' Rogers and Ramanujan [1919]' Schilling and Warnaar [2000], Schur [1917]' Sills [2003a,b,c], Slater [1952a], Stanton [2001a,b], Warnaar [1999-2002a, 2003a-2003e], and Watson [1931]. §2.9 Agarwal [1953e] showed that Bailey's transformation (2.9.1) gives a transformation formula for truncated 8¢7 series, where the sum of the first N terms of an infinite series is called a truncated series. §2.10 Many additional transformation formulas for hypergeometric series are derived in Whipple [1926a,b]. §2.11 Additional transformation formulas for 8¢7 series are derived in Agarwal [1953c]. Ex.2.6 Also see the summation formulas for very-well-poised series in Joshi and Verma [1979]. Ex. 2.31 This exercise is the n = 3 case of the Zeilberger and Bressoud [1985] theorem that if Xl, ... , X n , q are commuting indeterminates and al, ... ,an are nonnegative integers, then the constant term in the Laurent expansion of
II
(xi/Xj; q)ai (qXj/Xi; q)aj
l~i<j~n
is equal to the q-multinomial coefficient
(q; q)al +·+a n (q; q)al ... (q; q)a n
•
This was called the Andrews' q-Dyson conjecture because Andrews [1975a] had conjectured it as a q-analogue of a previously proved conjecture of Dyson [1962] that the constant term in the Laurent expansion of
II l~i#-j~n
(1 _
Xi) ai Xj
is equal to the multinomial coefficient
(al+···+an)! al!'" an! The n = 4 case of the Andrews' q-Dyson conjecture was proved independently by Kadell [1985b]. Additional constant term results are derived in Baker and Forrester [1998, 1999], Bressoud [1989], Bressoud and Goulden [1985], Cherednik [1995], Cooper [1997a,b], Evans, Ismail and Stanton [1982]' Forrester [1990], Kadell [1994, 1997, 2000], Kaneko [1996-2001], Macdonald [1972-1998b], Morris [1982]' Opdam [1989], Stanton [1986b, 1989], Stembridge [1988], and Zeilberger [1987-1994].
3 ADDITIONAL SUMMATION, TRANSFORMATION, AND EXPANSION FORMULAS
3.1 Introduction In this chapter we shall use the summation and transformation formulas of Chapters 1 and 2 to deduce additional transformation formulas for basic hypergeometric series which are useful in many applications. In §3.2 and §3.3 we shall obtain q-analogues of some of Thomae's [1879] 3F2 transformation formulas, typical among which are p. [-n, a, b .1] 3
2
C,
d'
= (d - b)n p. [ (d)n
3
2
-n, c - a, b .1] c,1 + b - d - n' ,
(311) ..
n = 0,1,2, ... ,
a,b,c ] = f(d)f(e)f(s) D [d-a,e-a,s.l] (312) 3F2 [ d,e;1 f(a)f(s+b)f(s+c) 3r 2 s+b,s+c' , .. s
= d+e-
a - b - c, and
[a,d,b,ec.' 1]
p. 3
2
f(1 - a)r(d)r(e)f(c - b) f(d - b)f(e - b)f(1 + b - a)f(c) x
D
3r 2
[b,1b+- bd_+c,1,1b+- be_+a; 1 1]
(3 1 3) + 1·dem (b) ; c, ..
where the symbol "idem (b; c)" after an expression means that the preceding expression is repeated with band c interchanged. The main topic of this chapter, however, will be the q-analogues of a large class of transformations known as quadratic transformations. Two functions J(z) and g(w) are said to satisfy a quadratic transJormation if z and w identically satisfy a quadratic equation and J(z) = g(w). Among the important examples of quadratic transformation formulas are
(l+z)
a
2
Fl(a,b;l+a-b;-z)
=
2Fl
(a2'-2--b;l+a-b;(I+z)2 a+ 1 4z ) , (3.1.4)
a
(1 - z) 2Fl(a, b; 2b; 2z) (1 - z)a
2Fl (2a,
=
2Fl
a + b; 2a + 2b; z) =
(a a+ 1 b + -;1 ( Z2) )2' -, - - ;
2
2Fl
69
2
2
l-z
1 ( a, b; a + b + -2;
Z2) 4(z-l) ,
(3.1.5) (3.1.6)
70
Additional summation, transformation, and expansion formulas 1
1
2FI(2a, 2b; a + b + "2; z) = 2FI(a, b; a + b + "2; 4z(1 - z)),
(3.1. 7)
(1 - z)a 3F2 [1 + a _ab~i c+ a _ c; z] = 3
Po [~a,~(a+1),1+a-b-c._ 4z ] 2 1 + a _ b, 1 + a - c ' (1 - z)2 '
(l-z)a+1 1+z
-'--------'----- 4 F 3
_
[ ~ (a
[
(3.1.8)
a,l+~a,b,c ] ;z "2a, 1 + a - b, 1 + a - c I
+ 1), 1 + ~ a, 1 + a -
b - c. 4Z] (3.1.9) , - . l+a-b,l+a-c (l-z)2 The above definition of a quadratic transformation cannot be directly applied to basic hypergeometric series. For example, the a = q-n case of the identity in Ex. 2.26 is a q-analogue of the a = -n case of (3.1.8), but it does not fit into the above definition of a quadratic transformation. So we shall just say that a transformation between basic hypergeometric series is "quadratic" if it is a q-analogue of a quadratic transformation for hypergeometric series. It will be seen that one important feature of the quadratic transformations derived for basic hypergeometric series in the following sections is that the series obtained from an r
3F2
3.2 Two-term transformation formulas for 3
In general, a convergent 3
4
_ (e/a,de/bc;q)n
() 3.2.1
3.2 Two-term transformation formulas for
31>2
71
series
with n = 0,1,2, .... Keeping n fixed and choosing special or limiting values of one of the other parameters leads to transformation formulas for terminating 31>2 series. Let us consider this class of formulas first. Case (i)
Letting c ----t
°
in (3.2.1) we get
[q-n,a,b. ] _ (e/a;q)n n d , e ,q, q - (e,.q) n a
A,
3'1-'2
[q-n,a,d/b. bq ] d , aq 1-n/' e q, e .
A,
3'1-'2
(3.2.2)
Note that the series on the left is of type I and that on the right is of type II. Formula (3.2.2) is a q-analogue of (3.1.1). Case (ii) A,
3'1-'2
Letting a
----t
°
in (3.2.1) gives
[q-n,b,c. ] _ (de/bc;q)n (bC)n d ,e ' q, q (e,.q) n d
If we let c 21>1
A,
3'1-'2
°
[q-n,d/b,d/C. ] d , de /b c ' q, q.
(3.2.3)
in (3.2.3) we obtain
----t
(q-n, d/b; d; q, bq/e)
=
(-ltq-G)(e;q)n e-n
31>2
[q-;:~,O;q,q],
which may be written in the form (Ex. 1.15(i))
)_ (abz/c;q)oo ( .. 21>1 a, b, c, q, z - (bz/c; q)oo
31>2
[a,c/b,o. ] c, cq/bz' q, q ,
(3.2.4)
where a = q-n, n = 0, 1,2, .... Case (iii)
Let c
----t 00
in (3.2.1). This gives Sears' [1951c, (4.5)] formula
[q-n,a,b. de qn ] _ (e/a;q)n d ,q, b - ( . ) ,e a e, q n
A,
3'1-'2
A,
3'1-'2
[q-n,a,d/b. ] d 1-n/' q, q . , aq e
(3.2.5)
Note that there is no essential difference between (3.2.2) and (3.2.5) since one can be obtained from the other by a change of parameters. Case (iv)
Replacing a by aqn in (3.2.1) and simplifying, we get
q-n , aqn, b, c ] 41>3 [ d ,e, abcq /d e ;q,q
= (aq/e, de/bc; q)n (bC)n (e, abcq/de; q)n
d
1>
4 3
[q-n,aqn,d/b,d/C. ] d, de/bc, aq/e ,q, q .
Set d = 'xc and then let c ----t 00. In the resulting formula we replace ,x, e and abq/'xe by c, d and e, respectively, to get A,
3'1-'2
[
q-n , aqn, b . de] d ,e ,q'--b a
= (aq/d, aq/e; q)n (de)n (d, e; q)n
aq
31>2
[q-n,aqn,abq/de;q,fJ.] aq/d, aq/e b
(3.2.6)
72
Additional summation, transformation, and expansion formulas
which is a transformation formula between two terminating
series of type
3¢2
II. Let us now consider the class of transformation formulas that connect two nonterminating 3¢2 series. Case (v) In (3.2.1) let us take n ----; limiting process gives the formula
00.
A straightforward term-by-term
_ (e/a,de/be;q)oo [a,d/b,d/e.~] [ a,b,e.,q, ~] (3.2.7) b c- e(, de / a be; q) 00 3¢2 d, de / d, e a be ' q, a . Apart from the general requirement that no zero shall appear in the denominators of the two 3¢2 series, the parameters must be restricted by the convergence conditions: Ide/abel < 1 and le/al < 1. This formula is a q-analogue of the Kummer-Thomae-Whipple formula 3¢2
3
.1] , (3.2.8)
F [a,b,e. 1] = r(e)r(d+e-a-b-e) F [a,d-b,d-e 2 d,e' r(e-a)r(d+e-b-e) 3 2 d,d+e-b-e'
where Re (e - a) Case (vi)
> 0 and Re (d + e - a - b - c) > o.
Iterating (3.2.1) once gives 4¢3
[d
q-n, a, b, e ] b I-n/d ;q,q ,e, a eq e
_ (b, del ab, de/be; q)n (d, e, de/abc; q)n
4¢3
Let us assume that max (Ibl, Ide/abel) obtain Hall's [1936] formula A.
3~2
[
[q-n, d/b, e/b, del abc. ] de/ab, de/be, ql-n /b' q, q .
(3.2.9)
< 1. Then, taking the limit n ----;
00,
we
a, b, e . de ] d,e ,q, abc
= (b,de/ab,de/be;q)oo ¢ [d/b,e/b,de/abe. (d, e, de/abc; q)oo
3
2
de/ab, de/be ,q,
b] .
(3.2.10)
Note that this is a q-analogue of formula (3.1.2). Before leaving this section it is worth mentioning that by taking the limit n ----; 00 in Watson's formula (2.5.1), we get another transformation formula:
a q ] _ (aq/d, aq/e; q)oo aq/b, aq/e' q, de - (aq, aq/de; q)oo
¢ [aq/be, d, e. 3
2
x
~ ~
k=O
(a;q)k(l- aq2k)(b,e,d,e;qh (k) ( a 2q2 )k q 2 --(q;q)k(1 -a)(aq/b,aq/e,aq/d,aq/e;q)k bede' (3.2.11)
< 1. This is a q-analogue of the formula 3 F 2 [ 1 + a - b - e, d, e .1] = r(1 + a)r(1 + a - d - e) 1 + a - b, 1 + a - e' r(1 + a - d)r(1 + a - e)
provided laq/del
x
6F5
[
a,I+~a,b,e,d,e ] ; -1 , (3.2.12) "2a,1 + a - b, 1 + a - e, 1 + a - d, 1 + a - e 1
3.3 Three-term transformation formulas for
+a-
where Re (1
3¢2
series
73
d - e) > 0; see Bailey [1935, 4.4(2)].
3.3 Three-term transformation formulas for
3¢2
series
In (2.10.10) let us replace a,b,e,d,e,j by AqN,BqN,C,D,E,FqN, respectively, and then let N ----; 00. In the resulting formula replace C, D, E, Aq/ B and Aq/ F by a, b, e, d and e, respectively, to obtain A.
3'1'2
[a, b, e. ~] _ (e/b, e/e; q)oo d , e ,q, a be - (e, e /b e; q) 00 (d/ a, b, e, de/be; q)oo
A.
3'1'2
[d/a, b, e. ] d, beq / e ,q, q
[e/b, e/ e, de/abe
]
(3.3.1) + (d, e, be/e, de/abe; q)oo 3¢2 de/be, eq/be ; q, q , Ide/abel < 1, and be/e is not an integer power of q. This expresses a 3¢2
where series of type II in terms of a 3¢2 series of type 1. As a special case of (3.3.1), let a = q-n with n = 0,1,2, .... Then A.
3'1'2
[q-n, b, e . de qn ] d, e ' q, ----,;;;-
_ (e/b, e/e; q)oo
- (e, e /b e; q)
00
(b, e; q)oo
+ (e, be/e; q)oo
A.
3'1'2
[b, e, dqn . ] / ' q, q beq e, d
(de/be; q)n ¢ [e/b, e/ e, deqn /be. ] (d; q)n 3 2 eq/be, de/be ,q, q .
(3.3.2)
Setting n = 0 in (3.3.2) gives the summation formula (2.10.13). We shall now obtain a transformation formula involving three 3¢2 series of type II. We start by replacing a, b, e, d, e, j in (2.11.1) by AqN, BqN, C, D, E, FqN, respectively, and then taking the limit N ----; 00. In the resulting formula we replace C, D, E, Aq/ B, Aq/ F by a, b, e, d, e, respectively, and obtain 3¢2
b, e de] [ a,d,e ;q, abe
_ (e/b, e/ e, eq/ a, q/ d; q)oo ¢ [e, d/ a, eq/ e . bq ] - (e, eq/d, q/a, e/be; q)oo 3 2 eq/a, beq/e' q, d (q/ d, eq/ d, b, e, d/ a, de/beq, beq2 / de; q)oo ¢ [a q/ d, bq/ d, eq/ d . de ] - (d/q, e, bq/d, eq/d, q/a, e/be, beq/e; q)oo 3 2 q2/d, eq/d ,q, abe ' (3.3.3) provided Ibq/dl < 1, Ide/abel < 1 and none of the denominator parameters on either side produces a zero factor. If Iql < Ide/abel < 1, then A.
3'1'2
[a,d,eb, e .,q, abe de ]
_ (e/b, e/e, q/d, bq/a, eq/a, abeq/de; q)oo ¢ [q/a, d/a, e/a. abeq ] (e, e/be, q/a, bq/d, eq/d, beq/e; q)oo 3 2 bq/a, eq/a ,q, de
74
Additional summation, transformation, and expansion formulas
-
(b, c, ql d, dl a, eql d, delbcq, bcq2 I de; q)oo rP [aql d, bql d, cql d . de ] (e, elbc, qla, bqld, cqld, bcqle, dlq; q)oo 3 2 q2ld, eqld ,q, abc ' (3.3.4)
by observing that from (3.2.7)
C,dla,cqle. bq] 3rP2 [ cq I a, bcq I e ,q'-d _ (abcql de, bql a; q)oo rP [ql a, dl a, el a . abcq ] - (bcqle,bqld;q)oo 3 2 bqla,cqla ,q, de . If we set e = '\c in (3.3.3), let c ----+ 0 and then replace d and ,\ by c and abzlc, respectively, where Izl < 1, Ibqlcl < 1, then we obtain
2rPl (a, b; c; q, z )
=
(abzlc, qlc; q)oo ( I I ) 2rPl (CI a, cq I abz; cq I az; q, bqI c ) az c, q a; q 00
(b, qlc, cia, azlq, q2 laz; q)oo
(cI q, bqI c, qI a, az I c, cq I az; q)
,/, (aq I c, bqI c; q21 c; q, z. )
2'1'1
(3.3.5 )
00
Sears' [1951c, p. 173] four-term transformation formulas involving 3rP2 series of types I and II can also be derived by a combination of the formulas obtained in this and the previous section. Some of these transformation formulas also arise as special cases of the more general formulas that we shall obtain in the next chapter by using contour integrals.
3.4 Transformation formulas for well-poised 3rP2 and very-well-poised 5rP4 series with arbitrary arguments Gasper and Rahman [1986] found the following formula connecting a wellpoised 3rP2 series with two balanced 5rP4 series:
a, b, c aqx] 3rP2 [ aq I b, aq I c ;q, -bc _ (ax; q)oo ( ) x; q 00
-
,/, [a!, -a!, (aq)!, -(aq)!, aqlbc. ] ,q, q aqlb, aqlc, ax, qlx
5'1'4
(a,aqlbc,aqxlb,aqxlc;q)oo
+ (aqlb, aqlc, aqxlbc, X-I; q)oo x
,/, [xa!,-xa!,x(aq)!,-x(aq)!,aqxlbc ] ; q, q . aqxlb,aqxlc,xq,ax 2
5'1'4
(3.4.1)
Convergence of the 3rP2 series on the left requires that laqxlbcl < 1. It is also essential to assume that x does not equal q±j, j = 0,1,2, ... , because of the factors (x; q)oo and (x-\ q)oo appearing in the denominators on the right side of (3.4.1). Note that if either a or aqlbc is 1 or a negative integer power of q,
3.4 Transformation formulas of well-poised
3¢>2
and
5¢>4
series
75
then the coefficient ofthe second 5¢>4 series on the right vanishes, so that (3.4.1) reduces to the Sears-Carlitz formula (Ex. 2.26). An important application of (3.4.1) is given in §8.8. To prove (3.4.1) we replace d by dqn in (2.8.3) and then let n ----7 00. This gives ¢>
[
3 2
a, b, e . d] _ (bed/aq; q)oo aq/b, aq/e' q, ~ - (bed/qa 2; q)oo
x lim 12W11 (a 2q1 - n /bed; a!, -a!, (aq)!, -(aq)!,aql-n/be, n---+oo aql-n /bd, aql-n /ed, a 3q3-n /b 2e 2d 2 , q-n; q, q) . (3.4.2) To take the limit on the right side of (3.4.2) it suffices to proceed as in (2.10.9) to obtain
'" [d,-d,(aq)!,-(aq)!,aq/be ] ;q,q aq/b,aq/e,bed/aq,a 2q2/bed
=
5'1'4
bed (bed/ a 2 , bd/ a, ed/ a, aq/be, a; q)oo qa 2 (d/a,aq/b,aq/e,bed/a,a 2q2/bed;q)oo
x
5¢>4
[
d/a, bed/qa~, -bed/qa~ , bed/q! a~, -bed/q! a~ bd/a,ed/a,bed/a2,b2e2d2/qa2
]
;
q, q . (3.4.3)
Using this in (3.4.2) and replacing d by qxa 2/be, we get (3.4.1). If we now replace d by dqn in (2.8.5) and then let n ----7 00, we obtain the transformation formula '" [ a,qa!,-qa!,b,e. x(a q )!] 1 1 ,q, - b a"2, -a"2, aq/b, aq/e e
5'1'4
(1 - x 2 )(xq(aq)!; q)oo (x(aq)-!;q)oo
x
5¢>4
[
(aq)!, -(aq)!, qa!, -qd, aq/be 1
1
aq/b,aq/e,xq(aq)"2,q(aq)2/x
] ;q,q
(aq,aq/be,x(aq)!/b,x(aq)!/e;q)oo
+ (aq/b, aq/e, x(aq) "2 /be, (aq)2 Ix; q)oo 1
x
X,
5¢>4 [
1
-x, xq!, -xq! , x(aq)! /be 1
1
1;
x(aq)"2/b,x(aq)"2/e,x(q/a)"2,qx 2
] q, q .
In terms of q-integrals formulas (3.4.1) and (3.4.4) are equivalent to 3¢>2
[
a,b,e aqx] aq/b, aq/e; q, bc
(3.4.4)
76
Additional summation, transformation, and expansion formulas (a, aq/bc; q)oo s(l - q) (q, aq/b, aq/ c, q/ x, X; q)oo
x
1
(qu/xs, qui s, aqu/bs, aqu/cs, axu/ s; q)oo
8
8X
1
1
1
1
(ua"2 / s, -ua"2 / s, u(aq) "2 / s, -u(aq) "2 / s, aqu/bcs; q)oo
d qU,
(3.4.5) and 1 -qa"2, 1 b, c. a, qa"2,
[
5¢4
x(aq) "21
]
q, - b a"2, -a"2, aq/b, aq/c c 1
1
,
(1 - x 2) (aq, aq/bc; q)oo s(l - q)(q, aq/b, aq/c, x(aq)-~, q(aq)~ Ix; q)oo
x
1
(uq(aq) ~ / sx, qui s, aqu/bs, aqu/cs, uxq(aq) ~ / S; q)oo d
8
1
8x(aq)-~ (u(aq) "2 /
1
1
qU,
1
s, -u(aq) "2 / s, uqa"2 / s, -uqa"2 / s, aqu/bsc; q)oo
(3.4.6) respectively, where s -=I=- 0 is an arbitrary parameter. If we now set c = (aq)~ in (3.4.5), replace x by x/b(aq)~, and use (2.10.19), then we get 2
2¢1(a, b;aq/b; q,qx/b )
x
8¢7
[
(xq/b, aqx 2/b 2; q)oo
= (aqx /b ,qx2/F) ,q
00
ax/b, q(ax/b)~, -q(ax/b)~, x, a!, -a~, (aq)~, -(aq)~
qx] q, 2 (ax/b) "2 ,-(ax/b)"2, aq/b, xqa"2 /b, -xqa"2 /b, x(aq) "2 /b, -x(aq) "2 /b b 1
1
1
1
1
1;
(3.4.7) provided Iqx/b 2 < 1 when the two series do not terminate. Similarly, setting c = (aq)~ and replacing x by x/bq in (3.4.6) we obtain 1
¢ [a,qa~,-qa!,b . ....::...] _ (ax 2/b 2 ,x/qb;q)00 a~, -a~, aq/b ,q, qb 2 - (aqx/b, x 2/qb 2; q)oo
4 3
X 8¢7
[
ax/b, q(ax/b)~, -q(ax/b)~, (aq)~, -(aq)~, qa~, -qa~, x 1
1
1
1
1
1
(ax/b) "2 ,-(ax/b)"2, x(aq) "2 /b, -x(aq)"2 /b, xa"2 /b, -xa"2 /b, aq/b
x ] ;
q,
-2
qb
(3.4.8) provided Ix/qb 2
1
< 1 when the series do not terminate.
'
3.5 Quadratic transformations
77
3.5 Transformations of series with base q2 to series with base q If in Sears' summation formula (2.10.12) we set b = -c, e = -q, replace a by aqr, r = 0, 1, 2, ... , multiply both sides by (x 2 ,y2;q2)r b2r qr 2 2 2 (-q; q)r(x y b ; q2)r
and then sum over r from 0 to (-1,-q,ab,-ab,b 2 ;q)00 ( a, b, -a, - b, b, - b ; q) 00 _
-
we get
00,
A,.
a 2,x 2,y2 . 2 b2] 2b2 2 2b2 , q , q a, x y
[
3'1-'2
b -b· q) J" j (_q ' ab2., q) 00 00 (a " , ( b -b·) b2.) "q a" ,qooj=o q, q,a ,qJ
L (_
+ (-q,-ab2;q)00 ·) ( -a, - b, b, q
3¢2
~ (-a,-b,b;q)j
j x2 , y [ -q 2, 2 . 2 2] x 2y 2b2 q2-2 j jb 2 , q , q ' j
A,.
[
~ ( q, -q, -a b2., q ) J" q 3'1-'2 J=O
00"
2 2]
q-2 j ,x2,y2 . x 22b2 y , q2-2 j jb2 ,q,q
(3.5.1) assuming that Iqb 2 < 1 when the series on the left is nonterminating. Since the two 3¢2 series on the right side can be summed by the q-Saalschiitz formula (1.7.2) with the base q replaced by q2, it follows from (3.5.1) that 1
_ (-a, ab 2 ; q)oo (-1, b2 ; q)oo
-
+
A,.
5'1-'4
(a, -ab 2; q)oo (-1,b2;q)00
[a, bx, -bx, by, -by. ] -q, ab2 , bxy, -bxy' q, q
A,.
5'1-'4
[-a, -bx, bx, -by, by. ] -q,-ab2,-bxy,bxy,q,q .
(3.5.2)
Note that one of the terms on the right side of (3.5.2) drops out when a ±q-n, n = 0,1,2, .... Setting y = ab and using (2.10.10) gives 2¢1
( 2
2
2 2 4
2
2)
a , x ; a x b ; q , qb
X 8 W 7 (ab 2x 2 jq;
=
(b2,a2b2x2;q2)00(ab2,b2x2;q)00
= (a 2b 2, b2x,q 2. 2) (b 2 b2 2. ) 00 ,a x,q 00
a, x, -x, bx, -bx; q, ab 2),
(3.5.3)
where Iqb 2 < 1 and lab2 < 1 when the series do not terminate. Byapplying Heine's transformation formula (1.4.1) twice to the 2¢1 series above and replacing b by q! jb we find that 1
2¢1 (a 2 , b2 ; a 2q2 jb 2 ; q2,
1
x 2q2 jb4 )
(qa2x2jb2,q2a2x2jb4;q2)00(aqjb2,qx2jb2;q)00 (qx2jb2,q2x2jb4;q2)00(qa2jb2,aqx2jb2;q)00
78
Additional summation, transformation, and expansion formulas
(3.5.4) provided laq/b 2 < 1 and Ixq/b 2 < 1 when the series do not terminate. This formula was derived by Gasper and Rahman [1986], and a terminating version of it was given earlier by Verma [1980]. Application of the transformation formula (2.10.1) to the 8(P7 series on the right of (3.5.4) yields an equivalent formula 1
1
(qa 2x 2/b 2 , q2a 2x 2/b 4 ; q2)oo (-xq~ /b, -axq~ /b 3 ; q)oo (qx 2/b 2 , q 2x 2/b 4 ; q2)oo( -axq~ /b, -a2xq~ /b 3 ; q)oo x
8eP7 [
-xa2q~ /b 3 , q( -xa2q~ /b 3 ) ~ , 2
1
3
1
(-xa q2/b )2,
aq~ /b, -aqx/b 2 ,
1 -aq2/b, aqx/b 2 ,
2 -aq/b,
axq~/b,
-q( -xa2q~ /b 3 ) ~, a, 2
1
3
III
1
3
3
-(-xa q2/b )2, -axq2/b,
-xq2 /b . xq2 2 2 qa /b ,q'--b- ,
(3.5.5)
where Ixq/b 2 < 1 and Ixq~ /bl < 1 when the series do not terminate. It is clear that formula (3.5.5) is a q-analogue of the quadratic transformation formula 1
2Fl (a, b; 1 + a - b; x 2)
=
(1 - x) -2a 2Fl (a, a
1 2
+- -
b; 2a
+1-
2b; (
-4x I-x
)2).
(3.5.6) We shall now prove the following transformation formula due to Jain and Verma [1982]:
eP [ 10 9
a~,
a, q2a~, -q2a~, b, e, eq, d, dq, e, eq 2 a 3q3 ] -a~, aq2/b, aq2/e, aq/e, aq2/d, aq/d, aq2/e, aq/e; q , be2d 2e 2
= (aq/e, aq/d, aq/e, aq/ede; q)oo
5eP4
(
aq
)12, - (aq )12, aq /b ,ed e / a ; q, q
+
(aq2, a 3q3/e 2d 2e 2; q2)oo(e, d, e, a 2q2 /bede; q)oo (aq2/b,a3q3/be2d2e2;q2)oo(aq/e,aq/d,aq/e,ede/aq;q)oo
x
5eP4
[
1
[ (aq/b) ~, -(aq/b) ~,e, d, e
(aq, aq/ed, aq/ee, aq/de; q)oo
1
(a3q3/be2d2e2)~,-(a3q3/be2d2e2)~,aq/ed,aq/ee,aq/de ; q,
q
,
(a3q3/e2d2e2)~,-(a3q3/e2d2e2)~,a2q2/bede,aq2/ede
(3.5.7)
3.5 Quadratic transformations
79
with the usual understanding that if the 104>9 series on the left does not terminate then the convergence condition la3q3/bc2d2e21 < 1 must be assumed to hold. First we rewrite (2.10.12) in the form
(aq4n+l, aq/cd, aq/ce, aq/de; q)oo (cq2n, dq2n, eq2n, aql-2n / cde; q)oo
+ (a 2q2n+2/cde; q)oo (cdeq2n-l fa; q)oo
00
~
00
(cq2n, dq2n, eq2n; q)r
r
~ (q, aq4n+l, cdeq2n / a; q)r q
(aq/cd, aq/ce, aq/de; q)r qr 2 (q, a q2n+2 /cde, aq2-2n /cde; q)r
(c, d, e; qhn (aq/c, aq/d, aq/e; q)oo (c, d, e; q)oo (aq/c, aq/d, aq/e; qhn'
(3.5.8)
where n is a nonnegative integer. Using (1.2.39) and (1.2.40), multiplying both sides of (3.5.8) by
(a,b;q2)n(1-aq4n) ( a3q3 )n (q2, aq2/b;q2)n(1-a) bc2d 2e2 ' and summing over n from 0 to
00,
we get
loVVg(a;b,c,cq,d,dq,e,eq;q2,a3q3/bc2d2e2)
f
(aq, aq/cd, aq/ce, aq/de; q)oo (a, b; q2)n(1- aq4n) - (aq/c, aq/d, aq/e, aq/cde; q)oo n=O (q2, aq2/b; q2)n(1- a) (c, d ,e; q) 2n n(2n-l) ( aq 3 ) n x (cde/a; qhn(aq; q)4n q b
+
34>2
[ cq2n, dq2n, eq2n ] aq4n+1, cdeq2n /a; q, q
(c,d, e,a 2q2/cde; q)oo ~ (a,b;q2)n(1 _aq4n) (aq/ c, aq/ d, aq/ e, cde/ aq; q)oo (q2, aq2/b; q2)n (1- a)
t:o
(cde/aq; qhn ( a3q3 ) n x (a 2q2/cde;q)2n bc2d2e2
34>2
[aq/cd, aq/ce, aq/de ] a2q2n+2/cde,aq2-2n/cde;q,q . (3.5.9)
The first double series on the right side of (3.5.9) easily transforms to
f
(c,d,e;q)m qm 6 VV5 (a;b,ql-m,q-m;q2,aq2m+l/b) , m=O (q, aq, cde/a; q)m which, by (2.4.2), equals 54>4
[
c, d, e, (aq/b)!, -(aq/b)! 1
1
aq/b, cde/a, (aq)2, -(aq)2
;
1
q, q .
Similarly we can express the second double series on the right side of (3.5.9) as a single balanced 54>4 series. Combining the two we get (3.5.7). The special case of (3.5.7) that results from setting e = (aq)! is particularly interesting because both 54>4 series on the right side become balanced 44>3
80
Additional summation, transformation, and expansion formulas
series which, via (2.10.10), combine into a single we have the formula 8cP7
[
8cP7
series with base q. Thus
a, q2a~, -q2a~, b, e, eq, d, dq
2 a 2q2 ] q , b 2d2 a>, -a>, aq /b, aq /e, aq/e, aq /d, aq/d e 1
1
2
2
2
;
(aq, aq/be, aq/ed, -aq/ed, aq/db~, -aq/db~; q)oo (aq/b, aq/ e, aq/d, -aq/d, aq/edb~, -aq/edb~; q)oo
x
8cP7
[
-aid, q( -a/d)~, -q( -a/d)~, e, b~, -b~, (aq)~ /d, -(aq)~ /d
aq ] q, -b ' (-a/d) >, -( -a/d» , -aq/ed, -aq/db> , aq/db> , -(aq) > , (aq) > e 1
1
1
1
1
1
;
(3.5.10) where la 2q2/be2d 2
1
< 1 and
laq/bel
< 1 when the series do not terminate.
3.6 Bibasic summation formulas Our main objective in this section is to derive summation formulas containing two independent bases. Let us start by observing that when d = a/be Jackson's 8cP7 summation formula (2.6.2) reduces to the following sum of a truncated series
t
k=O
1 - aq2k (a, b, e, a/bc; q).k 1 - a (q, aq/b, aq/e, beq, qh
l
=
(aq, bq, eq, aq/be;?)n , (q, aq/b, aq/e, beq, q)n
where n = 0,1, .... Notice that this series telescopes, for if we set and (aq,bq,eq,aq/be;q)k O"k = (q,aq/b,aq/e,beq;q)k for k = 0,1, ... , and apply the difference operator Uk-l to O"k, then we get A
1...l.00k
=
~
defined by
(3.6.1) 0"-1
~Uk
(1 - aq2k)(a, b, c, a/bc; qh k q (1 - a)(q, aq/b, aq/e, beq; qh '
= 0
(3.6.2)
= Uk (3.6.3)
which gives (3.6.1), since n
L~Uk = Un - U-l
(3.6.4)
k=O
for any sequence {Uk}. These observations and the bibasic extension Tk
=
(ap,bp;p)k(eq,aq/be;q)k (q,aq/b; q)k(ap/c, bep;p)k
(3.6.5)
of O"k were used in Gasper [1989a] to show that A I...l.~=
(1 - apkqk)(1 - bpkq-k) (a, b;p)k(e, a/bc; q)k k q (1 - a)(1 - b) (q, aq/b; qh(ap/c, bep; ph '
(3.6.6)
81
3.6 Bibasic summation formulas which, by (3.6.4), gave the indefinite bibasic summation formula
t
k=O
(1 - apkqk)(l - bpkq-k) (a, b;ph(e, a/be; qh qk (1 - a)(l - b) (q, aq/b; q)k(ap/e, bep;p)k
(ap, bP;P)n(eq, aq/be; q)n (q,aq/b;q)n(ap/c,bep;P)n
(3.6.7)
for n = 0,1, .... Notice that the part of the series on the left side of (3.6.7) containing the q-shifted factorials is split-poised in the sense that aq = b(aq/b) and e(ap/e) = (a/be) (bep) = ap, while the expression on the right side is balanced and well-poised since
(ap) (bp) (eq) (aq/be)
q( aq/b) (ap/ e) (bep)
=
and The b ----;
(ap)q
°
(bp)(aq/b)
=
=
(eq)(ap/e)
=
(aq/be)(bep).
case of (3.6.7)
..[!--. 1 - apkqk (a;p)de; q)k -k - - --:----,-----:---,,-:-----,-----:-----,----:--e = ~
k=O
1- a
(q;q)k(ap/e;p)k
(ap;P)n(eq; q)n -n e (q;q)n(ap/e;P)n
(3.6.8)
is due to Gosper. To derive a useful extension of (3.6.7), Gasper and Rahman [1990] set
(ap,bp;p)k(eq, ad2q/be; q)k Sk = (dq, adq /b;) q k (adp / e, bep / d;p )k for k
= 0, ±1, ±2, ... , and /}.Sk
=
(3.6.9)
observed that
Sk - Sk-l (ap, bp;ph-l (eq, ad 2q/be; qh-l (dq,adq/b; q)k(adp/e, bep/d;p)k x {(I - apk)(l - bpk)(l - eqk)(l - ad 2qk /be)
-(1 - dqk)(l - adqk /b)(l - adpk /e)(l - bepk /d)} d(l - e/d)(l - ad/be)(l - adpkqk)(l - bpk /dqk) (1 - a)(l - b)(l - e)(l - ad 2/be) (a, b;p)k(e,ad2/be; q)kqk x (dq, adq /b;) q k (adp / e, bep /d;) p k
(3.6.10)
Since (3.6.4) extends to n
L
/}.Uk = Un - U-'m-l,
(3.6.11)
k=-'m where we employed the standard convention of defining
a'm n
L k='m
ak =
{
+ a'm+l + ... + an,
0,
-(an+l
+ an+2 + ... + a'm-d,
m:::::n, m
= n+ 1,
m
~
n+2, (3.6.12)
82
Additional summation, transformation, and expansion formulas
for n, m = 0, ±1, ±2, ... , it follows from (3.6.10) that (3.6.7) extends to the indefinite bibasic summation formula ~ (l-adpkqk)(l-bpk/dqk)
~
(1 - ad)(l - bid)
k=-rn
(a,b;ph(e,ad 2 /be;q)k k (dq, adq/b; q)dadp/e, bep/d;p)k q
(1 - a)(l - b)(l - e)(l - ad 2 /be) d(l- ad)(l - b/d)(l - e/d)(l - ad/be) 2 (e/ad, d/be;p)rn+l(l/d, b/ad; q)rn+l} { (ap, bp;p)n(eq, ad q/be; q)n x (dq, adq/b; q)n(adp/e, bep/d; P)n - (l/e, be/ad2 ; q)rn+l (l/a, l/b; P)rn+l (3.6.13) for n, m = 0, ±1, ±2, ... , by applying the identity (1.2.28). Observe that (3.6.7) is the case d = 1 of (3.6.13) and that the right side of (3.6.9) is balanced and well-poised since
(ap) (bp) (eq)( ad 2 q/be) = (dq) (adq/b) (adp/ c) (bep/ d) and
(ap)(dq)
=
(bp)(adq/b)
=
(eq)(adp/e)
=
(ad 2 q/be)(bep/d).
It is these observations and the factorization that occurred in (3.6.10) which motivated the choice of Sk in (3.6.9). If Ipl < 1 and Iql < 1, then by letting n or m tend to infinity in (3.6.13) we find that (3.6.13) also holds with n or m replaced by 00. In particular, this yields the following evaluation of a bilateral bibasic series ~
~ k=-oo
(1 - adpkqk) (1 - bpk / dqk) (a, b; ph (e, ad2 /be; q)k k (1 - ad)(l - bid) (dq, adq/b; q)k(adp/e, bep/d;ph q
(1 - a)(l - b)(l - e)(l - ad 2 /be) d(l - ad)(l - b/d)(l - e/d)(l - ad/be) 2 (e/ad, d/be;p)oo(1/d, b/ad; q)oo} { (ap, bp; p)oo (eq, ad q/be; q)oo x (dq, adq/b; q)oo(adp/e, bep/d; p)oo - (l/e, be/ad2 ; q)oo(1/a, l/b; p)oo ' (3.6.14) where Ipi < 1 and Iql < 1. In §3.8 we shall use the m =
°
case of (3.6.13) in the form
~ (1 - adpkqk)(l- bpk /dqk)
~
k=O
(1- ad)(l- bid)
(a, b;p)k(e, ad 2 /be; q)k k (dq, adq/b; qh(adp/e, bep/d;ph q
(1 - a)(l - b)(l - e)(l - ad2 /be) d(l - ad)(l - b/d)(l - e/d)(l - ad/be) (ap,bp;p)n(eq,ad 2 q/be;q)n X~'-----::---;-:,------'--,-----;-----::----,------'::-----;-:'------:-(dq, adq/b; q)n(adp/e, bep/d; P)n (1 - d)(l - ad/b)(l - ad/c) (1 - be/d) d(l - ad)(l - b/d)(l - e/d)(l - ad/be)·
(3.6.15)
There is no loss in generality since, by setting k = j - m in (3.6.13), it is seen that (3.6.13) is equivalent to (3.6.15) with n, a, b, e, d replaced by n +
3.6 Bibasic summation formulas
83
m, ap-m , bp-m, cq-m, dq-m, respectively. We shall also use the special case c = q-n of (3.6.15) in the form
t k=O
(1 - adpkqk)(l - bpk /dqk) (a, b;p)k(q-n, ad 2qn /b; qh k (1 - ad)(l - bid) (dq, adq/b; qh(adpqn, bp/dqn;ph q
(1 - d)(l - ad/b)(l - adqn)(l - dqn /b) (1 - ad)(l - d/b)(l - dqn)(l - adqn /b)' where n = 0,1, .... The d ----+ 1 limit case of (3.6.16) ~ (1 - apkqk)(l - bpkq-k)
~
(1- a)(l- b)
k=O
(3.6.16)
(a, b;ph(q-n, aqn /b; qh k _ 6 (3.6.17) (q, aq/b; q)dapqn, bpq-n;p)k q - n,O,
where 6n ,m is the Kronecker delta function and n = 0,1, ... , was derived independently by Bressoud [1988], Gasper [1989a], and Krattenthaler [1996]. If we replace n, a, band k in (3.6.17) by n - m, apmqm, bpmq-m and j - m, respectively, we obtain the orthogonality relation n
L
anjbjm
= 6n,m
(3.6.18)
j=m
with (-l)n+j(l- apJqj)(l- bpJq-j)(apqn,bpq-n;P)n_l anj = (q; q)n_j(apqn, bpq-n;p)j(bql-2n/a; q)n-j ,
(3.6.19) (3.6.20) This shows that the triangular matrix A = (anj) is inverse to the triangular matrix B = (b jm ). Since inverse matrices commute, by computing the jkth term of BA, we obtain the orthogonality relation
t
. k
(1 - apkqk)(l - bpkq-k) (apk+lqk+n, bpk+lq-k-n;p)j_k_l (q; q)n(q; q)j_k_n(aq2k+n /b; q)j-k-l
n=O
x
(1 - ~q2k+2n) (_ltqn(j-k-l)+e-~-n) = 6j,k,
(3.6.21)
which, by replacing j, n, a, b by n + k, k, ap-k-lq-k, bp-k-lqk, respectively, yields the bibasic summation formula
(1- ~) (1 - ~) t p
p
(aqk, bq-k;P)n_l(l - aq2k /b) k=O (q; q)k(q; q)n_k(aqk /b; q)n+l
(_l)kq(~)
=
6n 0 '
(3.6.22) for n = 0,1,... . The b ----+ limit case of (3.6.22) was derived in AI-Salam and Verma [1984] by using the fact that the nth q-difference of a polynomial in q of degree less than n is equal to zero. For applications to q-analogues of Lagrange inversion, see Gessel and Stanton [1983, 1986] and Gasper [1989a]. Formulas (3.6.17) and (3.6.22) will be used in §3.7 to derive some useful general expansion formulas.
°
84
Additional summation, transformation, and expansion formulas 3.7 Bibasic expansion formulas
One of the most important general expansion formulas for hypergeometric series is the Fields and Wimp [1961] expansion
(3.7.1) where we employed the contracted notation of representing al, ... , ar by aR , (al)n'" (ar)n by (aR)n, and n + al,"" n + ar by n + aR. In (3.7.1), as elsewhere, either the parameters and variables are assumed to be such that the (multiple) series converge absolutely or the series are considered to be formal power series in the variables x and w. Special cases of (3.7.1) were employed, e.g., in Gasper [1975a] to prove the nonnegativity of certain sums (kernels) of Jacobi polynomials and to give additional proofs of the Askey and Gasper [1976] inequalities that de Branges [1985] used at the last step in his proof of the Bieberbach conjecture. Verma [1972] showed that (3.7.1) is a special case of the expansion
oo oo (a)n+d,8)n+k B _ L (_x)n L L_ A n B n (xw)n n+k n! _ n!b+n)n k!b+2n+1h OO
n-O
n-O X
k
k=O
~ (-n)j(n+'Y)jA.w j ~ j=O
X
j!
() () a j j
,8
J
( ) 3.7.2
and derived the q-analogue
fAnBn (xw)n n=O (q; q)n
_Loo
-
L
(-x)n C2) oo (a,,8; q)n+k B k q n+k X n=O (q,'Yqn;q)n k=O (q,'Yq2n+l;qh X
t
(q-n,'Yqn;q)j Aj(wq)j. j=O (q,a,,8;q)j
(3.7.3)
To derive a bibasic extension of (3.7.3) we first observe that, by (3.6.17),
(3.7.4)
3.7 Bibasic expansion formulas
85
for m = 0,1, .... Hence, if Cr,m are complex numbers such that Cr,o = 1 for r = 0,1, ... , then B
r rX
00
=~ X
1_"W- 1q2r+2m ("(a- 1q2r;q)m("(pqr,apq-r;P)r 1 _ _W- 1q2r (q; q)m("(pqr+m, apq-r-m;P)r
q -mrBr+m Cr,mx r+ms:Um,O
00
00
= {;~
(1 _1'pnqn)(l _ apnq-n) (1 _1'a-1q2n+2k) (q; q)k(q; q)n("(pqn+k, apq-n-k;P)n
x ("(a-1qn+r+1; q)n+k-r-1 ("(pqr, apq-r;P)n_1 (q-n; q)r
x (_l)n B n+k C r,n+k-r xn+kqn(1+r-n-k l+G) (3.7.5) by setting j = n - rand m = n + k - r. Then by multiplying both sides of (3.7.5) by Arw r /(q; q)r and summing from r = to 00 we obtain Gasper's [1989a] bibasic expansion formula
°
f
n=O
AnBn (xw)n (q; q)n
=
f
n=O
(l-1'pnqn)(l - apnq-n) (-xtqn+G) (q; q)n
x'"' 00
1 - 1'a- 1q2n+2k
k
~ (q; q)k("(pqn+k, apq-n-k;P)n
X
Bn+kx
Ln
(-n q ; q) j (1'a -1 qn+j+1 ; q )n+k-j-1 j=O (q; q)j
x (1'pqj, apq-j; P)n-1AjCj,n+k-jw j qn(j-n-k),
(3.7.6)
where Cj,o = 1, for j = 0,1, .... Note that if p = q and Cj,m == 1, then (3.7.6) reduces to an expansion which is equivalent to
f
fAnBn (xw)n = (a, 1' qn+1/ a , a, (3; q)n n=O (q;q)n n=O (q,1' qn;q)n
(::.r a
("(q2n/a,qn+l"h/a,-qn+1yh/a,1/a,aqn,(3qn;q)kB k xL n+k X k=O (q, qn V1' /a, -q n V1' /a, 1'q2n+1; q)k 00
n
(-n n) q , 1'q ; q j A- (wq)j . ~ (q,1'qn+1/a ,q1-n/a ,a,(3;q)j J
X '"'
(3.7.7)
Verma's expansion (3.7.3) is the a ----7 00 limit case of (3.7.7). For basic hypergeometric series, (3.7.7) gives the following q-extension of (3.7.1)
86
Additional summation, transformation, and expansion formulas
rf,
x r+rn+2'f's+k+2
[q-j,,,(qj,aR,jM' ] "(qj+l jer, ql-j jer, bs ' eK ' q, wq ,
(3.7.8)
where we used a contracted notation analogous to that used in (3.7.1). Note that by letting er ----t 00 in (3.7.8) and setting m = 2, II = h = 0 we get the expansion rf, [aR,c r+t'f's+u b d T ; q, xw ]
s' u
X
r+2¢s+k [q-Jb',"(qj,aR;q,w q] ,
(3.7.9)
s,e K
which is equivalent to Verma's [1966] q-extension of the Fields and Wimp expansion (3.7.1). Other types of expansions are given in Fields and Ismail [1975]. AI-Salam and Verma [1984] used the b ----t 0 limit case of the summation formula (3.6.22) to show that Euler's transformation formula 00
Lanbnxn n=O
00
=
k
L(-l)k~! j(k) (x)b.kao,
(3.7.10)
k=O
where and
has the bibasic extension 00 00 k (1 n n) n A L AnBn(xw)n = L(apq\ph-l Xk L ( .) ap ~a )n n=O k=O n=O q, q k-n pq ,p n
v:..
87
3.7 Bibasic expansion formulas
m
oo(kk) . ap q ; p j B. (_)j () J+k x q . j=O q; q j
X "
~
(3.7.11)
The p = q case of (3.7.11) is due to Jackson [1910a]. In order to employ (3.6.22) to extend (3.7.10), replace n in (3.6.22) by j, multiply both sides by Bn+jx n +j (a/b)jqj2, sum from j = 0 to 00, change the order of summation and then replace k by k - nand j by j + k - n to obtain
(3.7.12) Next we replace a by apn+lqn, b by bpn+lq-n, multiply both sides by Anwn and then sum from n = 0 to 00 to get 00
"A B ( )n ~ n n XW n=O
=" 00
k
(
b
-k
)
apq ,pq ;p k-l k (a k/b') X k=O q, q k ~
k (1 - apnqn) (1 - bpnq-n)(aqk /b; q)n
X X
L n=O
() q; q k-n (apq k ,bpq - k;p) n
n An w
(apkqk,bpkq-k;p)j (_'!.-q2n)j+k-n b
~ (q; q)j(aq2k+l /b; q)j 00
x Bj+k( _x)jq(k-n)(j+k-n-l)+(~)+e+k2n+l). (3.7.13) This formula tends directly to (3.7.11) as b ---+ O. By replacing An, B n , x, w by suitable multiples, we may change (3.7.13) to an equivalent form which tends to (3.7.11) as b ---+
00.
In addition, by replacing An, B n , x, w by Anq2(;) ,
B n q-2(;) , bx/a, aw/b, respectively, we can write (3.7.13) in the simpler looking equivalent form 00 00 ( k b -k ) k+l " A B n n _ " apq ,pq ;p k-l (_ )k ( 2 ) ~ n nX W - ~ ( k/b) x q n=O k=O q, aq ;q k
X
k (1 - apnqn) (1 - bpnq-n)(q-k, aqk /b; q)n An wn (apqk, bpq-k;P)n
~
x" 00
~ j=O
'
(apkqk bpkq-k. p) , j B· j j (q, aq 2k+l/b') J+k X q . ,q J.
(3.7.14)
As in the derivation of (3.7.6), one may extend (3.7.14) by replacing Bj+k by Bj+kCn,j+k-n with Cn,o = 1 for n = 0,1, .... Multivariable expansions, which are really special cases of (3.7.6) and (3.7.14), may be obtained by
88
Additional summation, transformation, and expansion formulas
replacing An and Bn in (3.7.6) and (3.7.14) by multiple power series, see, e.g. Gasper [1989a], Ex. 3.22 and, in the hypergeometric limit case, Luke [1969]. For a multivariable special case ofthe AI-Salam and Verma expansion (3.7.11), see Srivastava [1984].
3.8 Quadratic, cubic, and quartic summation and transformation formulas By setting P = qj or q = p1, j = 2,3, ... , in the bibasic summation formulas of §3.7 and using summation and transformation formulas for basic hypergeometric series, one can derive families of quadratic, cubic, etc. summation, transformation and expansion formulas. To illustrate this we shall derive a quadratic transformation formula containing five arbitrary parameters by starting with the q = p2 case of (3.6.16)
n (1 - adp3k) (1 - b/dpk) (a, b;ph(p-2n, ad2p2n /b; p 2h 2k ~ (1 - ad)(l - bid) (dp2, adp2 /b;p2)dadp2n+l, bpl-2n /d;p)k p
(1 - d)(l - ad/b)(l - adp2n)(1 - dp2n /b) (1 - ad)(l - d/b)(l - dp2n)(1 - adp2n /b)'
(3.8.1)
where n = 0,1, .... Change p to q and d to c in (3.8.1), multiply both sides by
(ac 2/b; q2)n(c/b; q)zn C (q2; q2)n(acq; q)zn n and sum over n to get
(3.8.2) Setting
c _
(1 - ac2q4n /b)(d, e, j; q2)n(a 2q3/deJ)n n - (1 - ac2/b)(ac 2q2/bd, ac2q2 /be, ac2q2 /bj; q2)n'
it follows from (3.8.2) that
lOVVg(ac2/b;ac/b,c,cq/b,cq2/b,d,e,j;q2,a2c2q3/dej)
89
3.8 Quadratic, cubic, and quartic formulas
x
(d,e,J;q2)k(a 2c3q4jbde!)k q(~) (ac 2q2jbd,ac2q2jbe, ac 2q2jbJ; q2)k
x 8 W 7(ac 2q4k jb; cl jb, cqk+l jb, dq2k, eq2k, J q2\ q2, a 2c2q3 jde!). (3.8.3)
If we now assume that
a 2c2q = deJ,
(3.8.4)
then we can apply (2.11.7) to get 8W 7(ac 2q4kjb; cqkjb, cqk+ljb, dq2k, eq2k, J q2k; q2,q2)
(ac 2q4k+2jb, bJjac2q2k,abq2k+l, acqk+2jd;q2)oo (acq3k+2, acq3k+l, ac 2q2k+2 jbd, ac2q2k+2 jbe; q2)oo X (acqk+2je,acqk+ljd,acqk+lje,ac 2q2jbde;q2)oo
(beJ j ac2, bdJ j ac 2, J j acqk, J j acqk-l; q2)oo
+ bJq-2k(ac2q4k+2 jb, cqk jb, cqk+l jb, dq2k, eq2k; q2)oo ac2(ac2q2k+2jbJ,acq3k+2,acq3k+l,ac2q2k+2jbd;q2)oo X
(Jq2 je, J q2j d, bJq2-2k jac 2, bJqk+l jc, bJqk+2 jc; q2)oo (ac 2q2k+2jbe,beJjac2, bdJjac2, bJ2q2jac 2, Jjacqk, Jjacqk-l ;q2)oo
X 8 W 7(bJ 2 jac2; J q2k,
beJ jac 2, bdJ jac 2, J jacl, J jacl-\ q2, q2), (3.8.5)
which, combined with (3.8.3), gives
lOW9 (ac 2jb;c,d,e,J,acjb,cqjb,cq2jb;q2,q2) (ac 2q2jb,ac2q2jbde,abq,bJjac2;q2)oo(acqjd,acqje;q)oo (ac 2q2jbd, ac2q2 jbe, bdJ jac2, beJ jac2; q2)oo(acq, J jac; q)oo ~ x
(1 - acq3k)(1 - bjcqk)(a, b, cjb; q)k(d, e, J; q2h
2k
~ (1 - ac)(l - bjc) (cq2, acq2 jb, abq; q2)k(acqjd, acqje, acqj J; q)k q
+
bJ(ac2q2jb, d, e, J q2jd, J q2je, bJq2jac2; q2)oo ac2(ac 2q2jbJ,ac2q2jbe,ac2q2jbd, bdJjac2, beJjac2,bJ2q2jac2;q2)oo (bJqjc, cjb; q)oo x --';--------'------::-;-'---c-(acq, J j ac; q)oo ~ (1 - acq3k)(1 - bjcqk) X
~
(1 - ac)(l - bjc)
(a, b; qh(ac 2jbJ, J; q2)k 2k (cq2, acq2 jb; q2h(bJqjc, acqj J; q)k q
x 8 W 7(bJ 2jac2; J q2k, bdJ jac2, beflac2, J jacqk, J jacqk-\ q2, q2).
(3.8.6)
90
Additional summation, transformation, and expansion formulas
The last sum over k in (3.8.6) is ~ (bP lac2 ,j, bdf lac 2, bef lac2 , f lac, fqlac; q2)j(1 - bp q4 j lac2 ) q2j (q2, bfq2 lac2 ,jq2 Id, fq2 Ie, bfq 2 /c, bfqlc; q2)j(1 - bP lac 2 )
f;:o
~
(1 - acq3k)(1 - blcqk) (a, b; q)dac2 Ibfq2j ,jq2j; q2)k 2k ~ (1 - ac)(l - blc) (cq2, acq2/b; q2)k(bfq2J+1 Ic, acql-2j If; q)k q
X
=
~ (bP lac2 , f, bd!lac2 , bef lac2 , !lac, fqlac; q2)j(1 - bp q4 j lac 2 ) q2j
f;:o
(q2, bfq2 lac2 , fq2ld, fq2 Ie, bfq2/c, bfqlc; q2)j(1 - bP lac2 )
(1 - c)(l - aclb)(l - acl fq2j)(1 - bfq2j Ic) . . c(l - ac)(l- blc)(l - fq2) Ic)(l - aclbfq2J)
x
{
x
(a, b; q)00(Jq2j , ac2 Ibfq2j; q2)00 (c,aclb;q2)00(aclfq2j,bfq2jlc;q)00-
I}
(1 - c)(l - aclb)(l - f lac)(l - bf Ic) (1 - clb)(l - ac)(l - f Ic)(l - bf lac) x 10 W g(bf2 lac2 ; f, bdf lac2 , bef lac 2, bf lac, f Ic, fql ac, fq2 lac; q2, q2)
f
+
ac(l- ac)(l - clb)(l - f Ic)(l- bf lac) (a, b; q)oo(J, ac2/bf; q2)00 x-;----c;o---'---'--;;-'-:-:'-'-~_'__;_____'___c~'-:'--;---;- (cq2, acq 2 /b; q2)00(acql f, bfqlc; q)oo 00 (bP lac 2 , bdf lac 2 , bef lac2 , f Ic, bf lac; q2)j (1 - bp q j lac 2 ) ( 2: (q2, fq2 Id,jq2 Ie, bfq2 lac, fq 2/c; q2)j (1 - bP lac2 ) 4
X
fq2)j j2 q ab
--
j=O
(3.8.7) by the n lO W
00
case of (3.6.15). Thus,
2 2 2 2 bf(l - c)(l - aclb) g(ac Ib;c,d,e,f,aclb,cqlb,cq Ib;q ,q ) - ac2(1- !l c )(l- bflac)
x
X
=
(ac 2q2/b, d, e, fq2ld, fq2 Ie, bfq2 lac 2 ; q2)00 2 (bf2q2/ac , ac2q2/be,ac2q2/bd,bdflac2, beflac2 ,ac2q2 /bf; q2)00 (bf Ic, cqlb; q)oo (ac,fqlac;q)oo
W. (bP. f bdf bef f 10
9
2 2)
bf fq fq2. ac2' 'ac2'ac2'~'ac'ac'ac2,q ,q
3.8 Quadratic, cubic, and quartic formulas
x
91
(1, ac2/bf, ac 2q2 /b, d, e, f q2/d, f q2/e, bfq2 /ac 2; q2)00 (cq2, acq2/b, ac 2q2/bf,ac 2q2/be, ac 2q2/bd, bdf/ac2, bef/ac2,bf2q2/ac2;q2)00
x ~ (bP /ac 2, bdflac 2, bef /ac 2 , flc, bflac; q2)j
~
x
(q2, f q2/d, f q2/e, bfq2/ac, f q2/c; q2)j
(1- bpq4j /ac 2) ( -fq2)j j2 - q (1 - bj2 /ac 2) ab
(3.8.8)
when (3.8.4) holds. Now observe that since
f
(a, qva, -qva, c, d, e, f; q)j (_ a 2q2 )j q(~) j=O (q, va, -va, aq/c, aq/d, aq/e, aq/ f; q)j cdef _ (aq,aq/ef;q)oo - (aq/e,aq/f;q)oo
by the n
~ 00
3
¢ [aq/cd,e,f. aq ] 2 aq/c,aq/d,q, ef
(3.8.9)
limit case of (2.5.1), the sum over j in (3.8.8) equals
2 2]
(q2,bPq2/ac 2;q2)00 ¢ [f/c,bf/ac,ac2q2/bde. (bf q2/ ac , f q2/ C; q2)00 3 2 f q2/d, f q2/e ,q ,q
.
(3.8.10)
Hence, by setting e = a 2c2q/df in (3.8.8) we obtain the Gasper and Rahman [1990] quadratic transformation formula
lOVVg(ac2/b;f,ac/b,c,cq/b,cq2/b,d,a2c2q/df;q2,q2) (ac2q2/b,bf/ac2,ac/b,c,cq/b,cq2/b,bfq2/ac;q2)00 (bf2 q2/ac2, ac2/bf,ac2q2/bd,dfq/ab,bdf/ac2,abq/d,cq2; q2 )00
+~~~~~~--~~--~~~~~~~~~~
x (1q2/c, bf /c, bfq/c, fq2 / d, dPq/a 2c2, d, a 2c2q/df; q2)00 (acq2/b,f/c, bf/ac, aC,acq, fq/ac, f q2/ac;q2)00
(a, b, cq/b; q)oo (ac, ac/ f, fq/ac; q)oo
x
(f,d,a2c2q/df,bf/ac2,ac2q2/b,fq2/d,df2q/a2c2,q2;q2)00 (bf/ac, f/c, cq2, acq2/b, ac2q2/bd, dfq/ab, bdf/ac2, abq/d;q2 )00
x
3¢2
[
22]
flC,bf/ac,fq/ab f q2/d, dpq/a2c2 ; q ,q
(acq/d, df lac; q)00(ac2q2 /b, abq, bf /ac 2, fq/ab; q2)00 (acq, f lac; q)00(ac 2q2 /bd, dfq/ab, bdf /ac 2, abq/d; q2)00
x
f
k=O
1 - acq3k (a,b,cq/b;q)k(d,f,a 2c2q/df;q2) kqk 1 _ ac (cq2, acq2/b, abq; q2h(acq/d, acq/ f, df lac; q)k· (3.8.11)
92
Additional summation, transformation, and expansion formulas
Note that the first two terms on the left side of (3.8.11) containing the series can be transformed to another pair of 10 W g series by applying the four-term transformation formula (2.12.9). Since the 31>2 series in (3.8.11) is balanced it can be summed by (1.7.2) whenever it terminates. When c = 1 formula (3.8.11) reduces to the quadratic summation formula 10 W g
~ 1 - aq3k (a, b, q/b; q)k(d, j, a 2q/dj; q2)k k ~ 1 - a (q2, aq2/b, abq; q2h(aq/d, aq/ j, dj fa; q)k q
k=O
(aq, j/a, b, q/b; q)=(d,a 2q/dj,jq2/d, dj2q/a 2;q2)= (a/j,jq/a, aq/d,dj/a; q)=(aq2/b,abq, jq/ab,bj/a; q2)=
+~~~~--=-~~~~~~~~~~~~-=--
X 31>2
2 2]
j, bj la, jq/ab [ jq2/d,dj2q/a 2 ;q ,q
(aq, j/a; q)=(aq2/bd, abq/d, bdj/a, djq/ab; q2)= (aq/d, dj fa; q)=(aq2/b, abq, bj la, jq/ab; q2)= .
(3.8.12)
By multiplying both sides of (3.8.11) by (J lac; q)= and then setting j we obtain Rahman's [1993] quadratic transformation formula ~
~
k=O
=
ac
(a; q 2h(1 - aq3k)(d, aq/d; q2)k(b, c, aq/bc; q)k k (q;q)k(1 -a)(aq/d,d;q)k(aq2/b,aq2/c,bcq;q2)k q
_ ( aq 2,bq,cq, aq 2/bc;q2)= - (q, aq2/b, aq2/c, bcq; q2)=
1> 3
2
[b,c,aq/bc. 2 2] dq, aq2/d ,q ,q ,
(3.8.13)
provided d or aq/d is not of the form q-2n, n a nonnegative integer. Also, the case d = q-2n of (3.8.11) gives n 1 - acq3k (a, b, cq/b; q)k(J, a 2c2q2n+1 / j, q-2n; q2)k k 1 - ac (cq2, acq2/b, abq; q2h(acq/ j, Jlacq2n, acq2n+1; q)k q
~
(acq;q)2n(ac 2q2/bj,abq/j;q2)n (acq/j;q)2n(abq,ac 2q2/b;q2)n x 10 W g (ac 2/b; j, ac/b, c, cq/b, cq2/b, a 2c 2q2 n+l / j, q-2n; q2, q2) (3.8.14)
and the case b = cq2n+1 gives 2n 1 - acq3k (d, j, a 2c 2q/dj; q 2h(a, cq2n+l, q-2n; qh k 1 - ac (acq/d, acq/ j, dJlac; q)k(cq2, aql-2n, acq2n+2; q 2h q
~
(acq2,dq/ac, jq/ac, acq2/dj; q2)n (q/ac,acq2/d,acq2/j,djq/ac;q2)n x 10 W g (acq-2n-\ c, d, j, a 2c2q/dj, aq-2n-\ ql-2n, q-2n; q2, q2)
(3.8.15) for n = 0, 1, .... Similarly, the special case n (l-acq4k)(I-b/cq2k) (a,b;qh(q-3n,ac 2q3n/b;q3h 3k (1 - ac)(1 - b/c) (cq3, acq3/b; q3h(acq3n+1, b/cq3n-1; qh q
~
3.8 Quadratic, cubic, and quartic formulas (1 - e)(1 - ae/b)(1 - aeq3n) (1 - eq3n /b) (1 - ae)(1 - e/b)(1 - eq3n)(1 - aeq3n /b)'
=0 12 n
" , ... ,
93 (3.8.16)
of (3.6.16) is used in Gasper and Rahman [1990] to show that Gosper's sum (see Gessel and Stanton [1982]) F. [ 7 6
a, a + 1/2, b, 1 - b, e, (2a + 1)/3 - e, a/2 + 1 .1] 1/2, (2a - b + 3)/3, (2a + b + 2)/3, 3e, 2a + 1 - 3e, a/2'
2 r(e+!)r(e+~)r(~)r(~)
V3 r (2at2) r (2at 3 ) r (3c+;+l) r (3c+i- b) x r (2+2~-3C) r (3+2~-3C) sin ~(b + 1) r (2+2atb-3c) r (3+2a 3b-3c)
(3.8.17)
has a q-analogue of the form ~ 1 _ aeq4k (a,q/a;q)k(ae;q)2k(d,aeq/d;q3)k qk ~ 1 -ae (eq3,a2eq2;q3)k(q;q)2k(aeq/d,d;q)k
k=O (aeq2,aeq3,d/ae,dq/ae,adq,aq,q2/a,dq2/a;q3)00 (q,q2,dq,dq2,a2eq2,eq3,dq/a2e,d/e;q3)00 d(a,q/a,aeq;q)00(q3,d,aeq/d,d 2q2/ae;q3)00 ae(q,d,aeq/d;q)00(eq3,a2eq2,d/e,dq/a2e;q3)00
+~~~--~7-~~~~~--~~~-
x
die, dq/a2e 3 3] 2
(3.8.18)
and to derive the extension
(a,b,eq/b;q)00(q3,d,ae2q3/b,a2be/d,d2q3/a2be,bd/ae2;q3) 00 (ae,dq/ae,ae/d;q)00(eq3,aeq3/b,d/e,bd/ae,edq3/ab2,ab2/e;q3)00
(ab, dq/ ab; q)oo (bd/ ae 2, ae2q3/b; q3)00 (aeq, d/ ae; q)oo (ab 2/ e, edq3/ ab 2;q3)00 oo 1 - aeq4k (a, b; q)k(eq/b; qhk(d, a2be/d; q3)k k X q k=O 1 -ae (eq3,aeq3/b;q3)k(ab;q)2k(aeq/d,dq/ab;q)k
L
and some other cubic transformation formulas.
(3.8.19)
94
Additional summation, transformation, and expansion formulas The special case
n (1-acq5k)(1-bjcq3k) (a,b;qh(q-4n,ac 2q4njb;q4)k 4k {; (1 - ac) (1 - bj c) (eq4, aeq4 jb; q4 h (aeq4n+1, bj eq4n-1; q)k q
(1 - e)(l - aejb)(l - aeq4n) (1 - eq4n jb) (1 _ ae)(l _ ejb)(l _ eq4n)(1 _ aeq4n jb)'
n = 0,1,2, ... ,
(3.8.20)
of (3.6.16) was used in Gasper and Rahman [1990] to derive the quartic transformation formula
+
(1 - e)(l - acjb) (a 2b3 jcq2, eqjb; q)00(ae 2q4 jb, ab 3jc2q2; q4)00 (1 -a 2b2jeq2)(1 -ab3jeq2)(ab2jeq,ae;q)00(a3b5je2,e2q2jab3;q4)00
+
ab 2(a, b, eqjb; q)00(ae 2q4 jb, ab3 je2q2, a 2b2jq2; q4)00 eq2(1 -a2b2jeq2)(ae,ab2jeq2,eq3jab2;q)00(ab3jeq2,cq4,aeq4jb;q4)00
x
11>1
a2b2jeq2. 4 ab 3q2 ] [ a2b2q2 je' q , -e-
(ab; q)oo (abj q; q2)00 (ae 2q4 jb, ab3 j C2q2; q4)00 (aeq, ab 2jeq2; q)oo X
1 - aeq5k (a, b; q)k(eqjb, eq2 jb, eq3 jb; q3h(a 2b2jq2; q4)k k {; 1 - ac (eq4, aeq4 jb; q4)k (abq, ab, abj q; q2h (eq3j ab 2; qh q . 00
(3.8.21 ) When b = q2 j a the sum of the two 10 W g series in (3.8.21) reduces to a sum of two 8W7 series, which can be summed by (2.11.7) to obtain the quartic summation formula ~ 1 - aeq5k (a, q2 ja; q)k(aejq, ae, aeq; q3h(q2; q4h
~
1 - ae
l
(eq4,a2eq2;q4)k(q3,q2,q;q2)k(aejq;q)k q2 (a, q2 j a, acq; q)oo (q2; q4)00
(q,q3;q2)00(eq4,q2je,a2eq2,q4ja2e;q4)00 .
(3.8.22)
Additional quadratic, cubic and quartic summations and transformation formulas are given in the exercises.
3.9 Multibasic hypergeometric series
95
3.9 Multibasic hypergeometric series In view of the observation in §1.2 that a series 2::::"=0 Vn is a basic hypergeometric series in base q if Vo = 1 and Vn+IiVn is a rational function of qn, a series 2::::"=0 Vn will be called a bibasic hypergeometric series in bases p and q if vn+I / Vn is a rational function of pn and qn, and p and q are independent. More generally, we shall call a series 2::::"=0 Vn a multibasic (or m-basic) hypergeometric series in bases ql, ... , qm if v n+IiVn is a rational function of q'l, ... , q;;:', and ql,"" qm are independent. Similarly a bilateral series 2::::"=-00 Vn will be called a bilateral multibasic (or m-basic) hypergeometric series in bases ql, ... , qm if v n+I / Vn is a rational function of q'l, ... , q;;:', and ql,· .. , qm are independent. Multibasic series are sometimes called polybasic series. Since a multibasic series in bases ql,"" qm may contain products and quotients of q-shifted factorials (a; q)n with q replaced by q~l ... q~rn where k l , ... , k m are arbitrary integers, the form of such a series could be quite complicated. Therefore, in working with multibasic series either the series are displayed explicitly or notations are employed which apply only to the series under consideration. For example, to shorten the displays of many of the formulas derived in §3.8 we employ the notation if. 'l'
[al,"" ar : CI,I,···, cI,rl : ... : Cm,l,"" cm,rrn . . ] b1,0", bs: d 1,1,···,1,81:"': d dm,l,···, dm,srn ,q,ql,···,qm,z
to represent the (m
L00
(3.9.1)
+ I)-basic hypergeometric series ( al,···,ar;q ) n zn [ (_I)n G) ] l+s-r q
b n=O (q, l
x
IT
j=l
, ... ,
bs ; q)n
(Cj,I, ... ,Cj,rj;.qj)n [(_I)n q G)]sj-r (dj,l, ... ,dj,sj,qj)n J
j (3.9.2)
The notation in (3.9.1) may be abbreviated by using the vector notations: a = (al, ... ,ar ), b = (bl, ... ,bs), Cj = (Cj,I, ... ,Cj,rj)' d j (dj,l, ... ,dj,sj) and letting ] if. [ a : CI : ... : C m (3.9.3) 'l' b' d ' . d ; q, ql, ... , qm; Z
. I·····
m
denote the series in (3.9.2). If in (2.2.2) we set
Ak =
IT
(a; q)k zk (Cj,l, ... , Cj,rj; qj)k [( _1)kq}~)]Sj-rj , (q-n;q)k j=l (dj,l, ... ,dj,sj;qj)k
(3.9.4)
then we obtain the expansion
[
=
a, b, C : CI,I,···, cI,rl : ... : Cm,l,···, cm,rrn ; q, ql, ... , qm; z] aq/b, aq/ C : dl,l, ... , dl,Sl : ... : dm,l, ... , dm,srn
f=
(aq/bc;q)n(a;qhn (_ bCZ)n q-G) n=O (q,aq/b,aq/c;q)n aq
96
Additional summation, transformation, and expansion formulas
: C1,lqr,··· ,c1,rlqr: ... : : d1,lqr, .. ·, d1,Sl qr : ... : n , ... , Cm r qm n Cm 1qm ' -bcz q -n-1 p n ] d , n d , '" n"'q q 1, ..q ·, m, m,lqm"'"
m,s",qm
a
(3.9.5) where
m
=
Pn
II q;(Sj-r j ),
(3.9.6)
j=l
provided at least one Cj,rj is a negative integer power of qj so that the series terminate. Note that this formula is valid even if a = q-k, k = 0,1, ... , in which case the upper limit of the sum on the right side can be replaced by [k/2]' where [k/2] denotes the greatest integer less than or equal to k/2. Similarly, use of the expansion formula (2.8.2) leads to the formula
x
a q2n ,
X: c1,lqr,···,
'q2n+ I /\
.
.
cI,rl qr
: ... : cm,lq;::', ... , cm,r",q;::'
d1,1 qn1,···, d 1,81 qn. 1 .
.
d qn . ... dm,l qn m " " , m,srn m
] ;q,ql, ... ,qm;zPn ,
(3.9.7) where A. = qa 2 /bcd. In (3.9.7) the series on both sides need not terminate as long as they converge absolutely. Formulas (3.9.5) and (3.9.7) are multibasic extensions of formulas 4.3(1) and 4.3(6), respectively, in Bailey [1935].
3.10 Transformations of series with base q to series with base q2 Following Nassrallah and Rahman [1981]' we shall derive some quadratic transformation formulas for basic hypergeometric series by using the following special cases of (3.9.5) and (3.9.7):
[
2 b2 , C2.. al,"" an q -n a 2 , -aq, 2 2
2
2 2
2.
-a,aq/b,aq/c .b 1, ... ,br ,br+1
2 ; q , q;
]
z
97
3.10 Transformations of series n' = '"
~ . J=O
(a 2 q2/b2 c2 ,-aq2.,q2) j ( a 2.,q2) 2j ( al,···, aT) q -no,q) j -2(~). ( ( q2 ,-a, a 2 q2/b2 ,a2 q2/ c, 2. q2) J·(b 1,···, bT) br+l,.). qJ q
b2 2 _ ~ a 2 q2
)j
a 2q4j , _ aq 2j +2 : al qj, ... , arqj, qj-n
X<J? [ -aq2j : b1qj, ... , brqj, br +1qj
(3.10.1) and
a 2q4j , _ aq2j+2, a 2/ A: alqj, ... , arqj, qj-n
x<J? [ _ aq 2j , Aq4j+2 : b1 qj , ... , brqj, br +1 qj
;q',q;z],
(3.10.2)
respectively, where A = (qa 2 /bcd) 2. Let us first consider the r = 1 case of (3.10.1). If we set al = -aq/w, b1 = w, b2 = _ aqn+l, z = awqn+2/b2c2 and apply (1.2.40), the series on the right side of (3.10.1) reduces to a VWP-balanced 6¢5 series in base q, and hence can be summed by (2.4.2). This gives the transformation formula
_ (w/a,-aq;q)n (w,-q;q)n
¢ [ 5 4
2 2]
a,aq,a2q2/w2,a2q2/b2c2,q-2n . a2q2/b2,a2q2/c2,aql-n/w,aq2-n/w,q ,q
.
(3.10.3) Note that the above 5¢4 series is balanced and that the <J? series on the left side of (3.10.3) can be written as lO W 9( -a; a, b, -b, c, -c, -aq/w, q-n; q, awqn+2/b2c2).
Formula (3.10.3) is a q-analogue of Bailey [1935, 4.5(1)]. By reversing the series on both sides of (3.10.3) and relabelling the parameters, this formula can be written, as in Jain and Verma [1980], in the form
1> [ 10
9
a qa~ -qa~ b x -x y -y _q-n q-n
a3q2n+3] a~, -a~, aq/b, aq/x, -aq/x, aq/y, -aq/y, _ aq n+l, aqn+l ,q, - bx 2y2
'"
"
""
.
2 2]
_ (a 2q2, a 2q2 /X 2y2; q2)n ¢ [ q-2n, x 2, y2, -aq/b, - aq2/b . - (a 2q2/x 2, a 2q2 /y2; q2)n 5 4 x2y2q-2n /a2, a 2q2/b 2, -aq, -aq2' q ,q
.
(3.10.4)
98
Additional summation, transformation, and expansion formulas
For a nonterminating extension of (3.10.4) see Jain and Verma [1982]. Since the 5¢>4 series on the right side of (3.10.3) is balanced, it can be summed by (1.7.2) whenever it reduces to a 3¢>2 series. Thus, we obtain the summation formulas: a2,aq2,_aq2: -aq/w,q-n w qn - 1 ]
(3.10.5)
a2, _ aq2, b2 : _aqn /b 2 , q-n ] ;q2,q;q2 2 2 2 -a, a q2/b : b q l-n, _aqn+l
(3.10.6)
(3.10.7) and
(3.10.8) These are q-analogues of formulas 4.5(1.1) - 4.5(1.4) in Bailey [1935]. Since the series on the left sides of (3.10.5) and (3.10.6) can also be written as VWPbalanced 8¢>7 series in base q, which are transformable to balanced 4¢>3 series by Watson's formula (2.5.1), formulas (3.10.5) and (3.10.6) are equivalent to the summation formulas 4¢>3
[
a, qa!, -w/qa!, q-n .1
.1
a 2 ,w,-a 2 q
I-n
;
1
q, q
(w/aq, -a!, aq2 /w; q)n
(3.10.9)
(w, -a-!, aq/w; q)n
and
2
(1 + l/b)(l + a"2 qn /b)(a/b , qa"2 /b, l/b; q)n (1 + qn /b)(l + a! /b)(aq/b, a! /b, 1/b2; q)n ' 1
1
(3.10.10)
respectively, which are closer to what one would expect q-analogues of formulas 4.5(1.1) and 4.5(1.2) in Bailey [1935] to look like.
3.10 Transformations of series
99
It is also of interest to note that if we set c2 = aq in (3.10.3), rewrite the il> series on the left side as an 8¢7 series in base q and transform it to a balanced 4¢3 series, we obtain 4¢3
[
a, b, -w /b, q-n ] /b -b -n ; q, q aq , q ,w
_ (w/a,-aq/b;q)n (w,-q/b;q)n
¢ [ 4
3
2 2] (
a,a 2q2/ w 2,aq/b 2,q-2n . a2q2/b2,aql-n/w,aq2-n/w,q,q
,
)
3.10.11
which is a q-analogue of the c = (1 + a)/2 case of Bailey [1935, 4.5(1)]. Using (2.10.4), the left side of (3.10.11) can be transformed to give q-n, a, aq/b 2, -aq/w ] [ 4¢3 aq /b , -aq /b ,aq I-n/ w ; q, q
(3.10.12) This formula was first proved by Singh [1959] and more recently by Askey and Wilson [1985]. The latter authors also wrote it in the form 4¢3
=
a2,b2,c,d ] [ b! a q2, -ab! q2,-cd; q, q 4¢3
[
2b22d2 a, ,c , 2 2] 2b2 d d; q , q , qa , -c , -qc
(3.10.13)
provided that both series terminate. Now that we have the summation formulas (3.10.5)-(3.10.8), we can use them to produce additional transformation formulas. Set r = 3 and al = -a2 = qa!, a3 = -aq/w, bl = -b 2 = a!, b4 = _aqn+1 in (3.10.1). The il> series on the right side can now be summed by (3.10.5) and this leads to the following q-analogue of Bailey [1935, 4.5(2)]
(-aq, aq2 /w, w/aq; q)n (-q, aq/w, w; q)n X5¢4
[
2 2]
aq,aq2,a2q2/b2c2,a2q2/w2,q-2n a2q2/b2,a2q2/c2,aq2-n/w,aq3-n/w;q,q
.
(3.10.14)
Let us now turn to applications of (3.10.2). If we set r = 1, al = _Aqn+l/a, bl = a 2q-n /A, b2 = _aqn+l, z = q in (3.10.2), where A = (qa 2/bcd)2, then the il> series on the right side reduces to a balanced very-well-poised 8¢7 series in base q which can be summed by Jackson's formula (2.6.2). Thus, we obtain the transformation formula a2,_aq2,b2,c2,d2: _Aqn+l/a,q-n il> [ ;q2,q;q -a, a 2q2 /b 2, a 2q2 / c2, a 2q2 / d 2 : a 2q-n / A, _ aqn+1
1
100
Additional summation, transformation, and expansion formulas
(-aq, >..qj a; q)n(>..q2 j a 2; q2)n (-q, >..qja 2; q)n(>..q2; q2)n
This is a q-analogue of Bailey [1935, 4.5(3)]. Similarly, setting r = 3 and choosing the parameters so that the
series on the right side of (3.10.2) can be summed by (3.10.7), we get the following q-analogue of Bailey [1935,4.5(4)] [
2 2 d 2 . ->..qnja q-n a2 'aq2 ] " _aq2 b , c , ., 2 a, -a, a 2q2 jb2, a 2q2 j c2, a 2q2 j d 2 : a 2ql-n j >.., _ aq n+l ; q ,q; q
Exercises 3.1 Deduce from (3.10.13) that
provided that both series terminate. Show that this formula is a q-analogue of Gauss' quadratic transformation formula (3.1.7) when the series terminate. 3.2 Using the sum -+. 2 '/'1
(q-n ql-n. qb2. q2 q2) ,
,
"
2 = (b(b 2'..q2)) n q- (n) 2 , ,q
n
prove that
(i)
q)oo = ((Z; .) az, q
00
-+.
2'/'1
(
a, aq; qb2 ; q2,Z 2) , IZI < 1.
101
Exercises
2
a
2 2 2 ]11 22;q,azq,z<1. b Z
Show that (iii)
Formulas (i) and (ii), due to Jain [1981]' are q-analogues of (3.1.5) and (3.1.6), respectively. T. Koornwinder in a letter (1990) suggested part (iii) . 3.3 Show that 3rP2
[
] a, q/a, Z ;q,q c,-q
_ (-1, -qz/c; q)oo -
(/
.) -qc,-z,qoo
A,.
3'1'2
2 2] ,
[cia, ac/q, Z2. 2 ,q ,q c,O
when the series terminate. This is a q-analogue of 2Fl (a, 1 - a; c; z)
= (1 - zt- 1 2Fl ((c - a)/2, (c + a - 1)/2; c; 4z(1 - z)) ,
when the series terminate. 3.4 Show that
~ (q-n, b, -b; qh nk-(~) 2 ~ (b q q,; q) k
k=O
vanishes if n is an odd integer. Evaluate the sum when n is even. Hence, or otherwise, show that
Deduce that 4rP3
[q
-n ,q I-n ,a,aq 2 2 ] b2 d d ; q ,q q , , q
_ (d/a;q)n n - (d.) ,q n a
(Jain [1981])
A,.
4'1'2
[q-n,a,b,-b. _Cfc] b2 ,aq I-n/d ' q, d .
102
Additional summation, transformation, and expansion formulas
3.5 By using Sears' summation formula (2.10.12) show that
f
(a, aq/e; q)r Ar r=O (q, abq/e, acq/e; q)r (aq/e,bq/e,cq/e,abcq/e;q)oo (q/e,abq/e,acq/e,bcq/e;q)oo ~
t='o
x
+
(a, b, c; qhqk ~ (aqk; q)r A (q, e, abcq/e; q)k ~ (q, abcqk+l Ie; q)r r
(a, b, c, abcq2 /e 2; q)oo (e/q,abq/e, acq/e, bcq/e; q)oo
~ (aq/e, bq/e, cq/e; q)k k ~ (aq k+ 1 Ie; q)r A x (q, q2/e, abcq2 /e 2; q)k q ~ (q, abcqk+2 /e 2; q)r r,
t='o
where a, b, c, e are arbitrary parameters such that e -I=- q, and {Ar} is an arbitrary sequence such that the infinite series on both sides converge absolutely. 3.6 Prove that 3(P2
x
[a,
C]
b, d,
3¢2
[
e; q, q
aq/e,
(q/e,a,b,c,dq/e;q)oo
+ (e/q, aq/e, bq/e, cq/e, d; q)oo
bq/e, q
2/
e,
cq/e ] d q / e ; q, q
_ (q/e, abq/e, acq/e, d/a; q)oo ¢ [a, aq/e, abcq/de. ~] (d, aq/e, bq/e, cq/e; q)oo 3 2 abq/e, acq/e ,q, a '
<
where e -I=- q and Id/al
l.
3.7 Prove that
[
3¢2
a,aq/e,e/bc bcq] ;q,-abq/e,acq/e e
(aq, aq/e, bq/e, cq/e, -q; q)oo (q/e, abq/e, acq/e, bcq/e, -bcq/e; q)oo 00
x provided Ibcq/el 3.8 Assuming that
<
~
(a,b,c;q)k (ab2c2qk+2/e2;q2)00 k (q, aq, e; q)k(aqk+2; q2)00 q,
l.
Ixl < 1 and alb -I=- qj, 2¢1 (a,
j = 0, ±1, ±2, ... , prove that
b; c; q, x)
_ (b, cia, ax; q)oo (b/ a, c, x; q) 00
-
(a,c/b,bx;q)oo
+ (a / b, c, x; q)
'" [a, c/b, 0 . ] aq / b, ax ' q, q
3'1-'2
00
'" [b,c/a,o ] ; q, q . bq/a, bx
3'1-'2
Exercises
103
Show that this is a q-analogue of the formula
2F1(a, b; C; x)
r(c)r(b-a)
=r(b)r(c_a)(l-x)
-a
2F 1(a,c-b;a-b+1;(1-x)
-1
)
r(c)r(a - b) -b -1 +r(a)r(c-b)(l-x) 2F 1(b,c-a;b-a+1;(1-x) ). 3.9 Show that
[
3¢2
a,>..q,b >..2] >..,q>..2/b;q, ab2
1 - >.. + >../b(l - >../a) (>..2 /b 2, q>..2/ab; q)oo -'-------'c--c--'---=--,"--:----,,----''''-----I>.. 2/ ab 2 < l. (1 - >")(1 + >../b) (q>..2/b, >..2 /ab 2; q)oo' 1
3.10 Show that 8W7 (
->..; q>,,~, -q>,,~, a, b, -b; q, >..jab2)
(>..ja, >"q, ->..q, >../b2, >..q/ab, ->..q/ab;.q)oo , l>..jab2 < l. (>.., >"q/a, ->..q/a, >"q/b, ->..q/b, >../ab ,q)oo Show that this is a q-analogue of the formula 1
F 3
2
1+>..j2,>"/2,
[a,
b
.
1 + >.. - b'
r(>../2)r (1 + A;a) r(l r(l + >..j2)r (A;a) r(l
1]
+ >.. - b)r(>.. - a + >.. - a - b)r(>.. -
2b) Re(>..-a-2b) >0. 2b)'
3.11 Derive Jackson's [1941] product formula 2(Pl (a 2, b2; qa 2b2; q2,
=
4¢3
z)
2¢1 (a 2, b2; qa 2b2; q2,
] a2, b2, ab, -ab [ 2b2 b! b! ; q, z , a ,a q2, -a q2
qz)
Izl < 1, Iql < 1,
and show that it has Clausen's [1828] formula
[2a, 2b, a + b 1 ; Z ] , Izl < 1, 2a + 2b, a + b + "2 as a limit case. Additional q-analogues of Clausen's formula are given in §8.8. 1 )] 2 [2F1 ( a, b; a + b + -; z = 2
3 F2
3.12 Prove that IJ? [
wq2-n] b2 c2 . d _q1-n/ w q-2n ' _q2-n " ., . 2 • _q-n,q2-2n/b 2,q2-2n/ c2: _q1-n/d,w,q ,q, b2c2d
= (-l,w/d;q)nd n (-d,w; q)n d2,q2-2n/b2c2,q2-2n/w2,q-n,q1-n [ X 5¢4 q2-2n/b2,q 2-2n/ c, 2 dq1-n/ w, dq2-n/ w ; q ,q
2 2] .
104
Additional summation, transformation, and expansion formulas
3.13 If A = a 4q2 /b 2c2d 2 , show that [
a2 ' _q 2a" d2 . . -Aqn/a ] b2 c2, , q-n 2 2 _a,a2q2/b2,a2q2/c2,a2q2/d2: a2ql-n/A,_aqn+l;q ,q;q
(-aq, Aja; q)n(Aja 2 , Aq2/a; q2)n (-q, A/a 2; q)n(Aq2, A/a; q2)n X lOVVg(A; a,aq, Ab2/a 2 , Ac2/a 2 , Ad2/a2,A2q2n/a2,q-2n;q2,a2q3 /A)
and [
=
2a b2 c2 d 2 . _Aqn-l/a q-n a2 " q2a -q ] " ,. , 2 2 a, -a, a 2q2 /b 2, a 2q2 / c2 , a 2q2 / d 2 : a 2q2-n / A, _aqn+l ; q , q; q (-aq,Ajaq;q)n(A/a 2q2;q2)n(1- Aq2n-l/a) n q
(-q, A/qa 2; q)n(Aq2; q2)n(1- Aq-l/a) x lQVVg (A; aq, aq2, Ab2/a 2, AC2/a 2, Ad2/a 2, A2q2n-2 /a 2, q-2n; q2, a 2l/A).
(Nassrallah and Rahman [1981]) 3.14 Using (3.4.7) show that the q-Bessel function defined in Ex. 1.24 can be expressed as
for Ixl < 2. (Rahman [1987]) 3.16 Show that 1
4¢3
[
axq ] a, -qa"2, b, c 1 ; q, -b-a"2, aq/b, aq/c c
_ (l-xa~)(axq;q)oo () x; q 00
-
J.
5'1-'4
[a~,-qa~,(aq)~,-(aq)~,aq/bc. /
/
/
aq b, aq c, q x, axq
(1 - a~)(aq, aq/bc, axq/b, axq/c; q)oo
+~~~~~~--~~~~~-
(aq/b,aq/c,axq/bc,l/x;q)oo
x
J.
5'1-'4
[xa~,-xqd,x(aq)~,-x(aq)~,aXq/bc axq/b,axq/c,qx,aqx 2
] ;q,q .
'
] q, q
Exercises
105
(cz/ab; q2)00 ( /) ~ n (. 2) 2¢1 a,b;c;q,cz abq = ~anZ , n=O z, q 00 show that
3.17 If
(Singh [1959], Nassrallah [1982]) 3.18 If
(cqz/ab; q2)00 ( / ) ~ n 2¢1 a, b; c; q, cz ab = ~ anz ,show that ( . 2) z,q 00 n=O 2¢1(cq/a, cq/b; cq2; q2, z) 2¢1(a, b; c; q2, cz/ab)
(Singh [1959], Nassrallah [1982]) 3.19 If
(cqz/ab; q2)00 2¢1 (/ ) = ~ n a q, b; c/ q; q, cz /ab ~ anz ,show that ( . 2) z,q 00 n=O 2¢1(cq/a, c/bq; c; q2, z) 2¢1(a, b; c; q2, cz/ab)
(Singh [1959], Nassrallah [1982]) 3.20 Prove that
when max(lpl, Iql, Ibl) formulas
< 1, and extend this to the bibasic transformation
~ 1 - apkqk (a;ph(c/b; qh bk 1- a (q; qh(abp;ph
6 =
=
1 - c ~ (ap;ph(c/b; q)k (bq)k 1- b (q; q)k(abp;p)k
6
1 - c ~ (ap;p)k(cq/b; qh bk 1- abp (q;q)k(abp2;ph
6
= (1 - c)(ap;p)oo
f
(b;ph(cqpk; q)oo (ap)k (1 - b) (abp; p)oo k=O (p; p)k (bqpk; q)oo
when max(lpl, Iql, lapl, Ibl) (Gasper [1989a])
<
l.
106
Additional summation, transformation, and expansion formulas
3.21 Derive the quadbasic transformation formula ~ (1 - apkqk)(1 - bpkq-k)
t:a
(1 - a)(I- b)
(a, b;p)de, a/be; q)k (q, aq/b; q)k(ap/e, bep;p)k
x (CP- n /A, p- n / BC; Ph(Q-n, BQ-n /A; Qh k (Q-n /C, BCQ-n /A; Qh(p-n /A, p-n / B; P)k q (ap, bP;P)n(eq, aq/be; q)n (Q, AQ/ B; Q)n(AP/C, BCP; P)n (q, aq/b; q)n(ap/e, bep;P)n (AP, BP; P)n(CQ, AQ/BC; Q)n
x ~ (1 - APkQk)(1 - BpkQ-k)
t:a
(I-A)(I-B)
(A, B; P)k(C, A/ BC; Q)k (Q,AQ/B;Q)k(AP/C,BCP;Ph
x (ep-n /a,p-n /be;p)k(q-n, bq-n fa; qh Qk (q-n /e, beq-n fa; q)k(p-n /a,p-n /b;ph
for n = 0,1, .... Use it to derive the mixed bibasic and hypergeometric transformation formula
(C - A - n) k ( - B - C - n) k (/lB - jJ,A - n) k ( -n) k X (_ jJ,C
- n) k (jJ,B
+ jJ,C -
k
jJ,A - n) k ( - A - n) k ( - B - n) k q
(ap, bP;P)n(eq, aq/be; q)n n!(jJ,A + 1 - jJ,B)n(A + 1 - C)n (q, aq/b; q)n(ap/e, bep;P)n (A + l)n(B + l)n(jJ,C + l)n (B+C+l)n x -,----..,..-'----'--.".,,--(jJ,A + 1 - jJ,B - jJ,C)n
x ~ (A
t:a
+ k + k/jJ,)(B + k AB
kjJ,)
(A)k(B)k (jJ,Ch (jJ,A - jJ,B - jJ,C)k k!(jJ,A + 1 - jJ,B)k(A + 1 - Ch(B + C + l)k
(e/apn,l/bepn;p)k(q-n,b/aqn;q)k x (l/eqn,be/aqn;q)k(l/apn,l/bpn;p)k'
and the following transformation formula for a "split-poised" [ 104>9
104>9
series
a, qVa, -qVa, b, e, a/be, C/Aqn, 1/ Bcqn, B/Aqn, q-n ] Va, -Va, aq/b, aq/e, beq, l/cqn, BC/Aqn, 1/ Bqn, I/Aqn; q, q
(aq, bq, eq, aq/be, Aq/ B, Aq/C, BCq; q)n (Aq, Bq, Cq, Aq/ BC, aq/b, aq/e, beq; q)n [ X 104>9
A, qVA, -qVA, B, c, A/ BC, e/aqn, l/beqn, b/aqn, q-n ] VA,-VA,Aq/B,Aq/C,BCq,l/eqn,be/aqn,l/bqn,l/aqn;q,q .
(Gasper [1989a])
Exercises
107
3.22 Using the observation that, for arbitrary (fixed) positive integers ml, ... , mr, 00
L A(k l kl, ... ,kr=O
, ... ,
kr)Z~l ... Z~r
00
=L M
=0 kl ml +···+krmr=M
kl, ... ,kr?O
show that (3.7.14) implies the multivariable bibasic expansion formula 00
00
X
L j=O
(
apn qn , bp n q -no,p ) j n j j ~~j+nX q (q, aq2n+1 jb; q)j
(1 - apM qM) (1 - bpM q-M) (q-n, aqn jb; q) M
x k1ml +···+krmr=M:S:n
kl, ... ,kr?O
X
A(k l , ... , kr)(w m1 Zd k1 ... (w mr zr)kr,
which is equivalent to (3.7.14) and extends Srivastava [1984, (10)]. (Gasper [1989a]) 3.23 Prove the following q-Lagrange inversion theorem: If 00
Gn(x) = Lbjn xj , j=n
where bjn is as defined in (3.6.20), and if 00
f(x)
=
00
L /jx j j=O
=
L cnGn(x), n=O
then j
/j = Lbjncn n=O
and, vice versa,
n Cn
=
where anj is as defined in (3.6.19). (Gasper [1989a]) 3.24 Derive (3.8.19)-(3.8.22).
Lanj/j, j=O
108
Additional summation, transformation, and expansion formulas
3.25 Prove Gosper's formula 00
~
(a2b2c2q-\ q2)n(1- a 2b2c2q3n-l ) (abcd, abcd-\ q2)n(a 2 , b2 , c 2 ; q)n n (q; q)n(1- a 2b2c 2q-l ) (abcd-l, abed; q)n(b 2c 2q, e 2a 2q, a 2b2q; q2)n q
(a2b2c2q,a2q,b2q,c2q, dq/abe, bcdq/a, cdaq/b, dabq/c; q2)00 (q, b2c 2q, c2a 2q, a 2b2q, abcdq, adq/bc, bdq/ae, cdq/ab; q2)00 (a 2 , b2 , c 2 ; q)oo (a 2b2c2q; q2)00 (q; q)00(b 2e2q, c2a 2q, a 2b2q, abedq, abed-1q; q2)00
x
I:(1-
n=O
d2q4n+2) (bcdq/a,cdaq/b,dabq/c;q2)n (adq/bc, dbq/ac, cdq/ab; q2)n+l
(_~)n q(n+l)2. abc
(Rahman [1993]) 3.26 Show that ~ (abcq;q)k(l -abcq3k+l)(d,q/d;q)k(abq,bcq,caq;q2)k k ~ (q2;q2)k(1 -abcq)(abcq3/d,abcdq2;q2)k(cq,aq,bq;q)k q k=O abeq, q2Jabcq, -q2.jabcq, d, q/d, abq, bcq, eaq = 8 ¢7 [ yabcq,-yabcq,abcq r:::c:::. r:::c:::. 3 2 2 2 2 ; q ,q /d,abcdq ,cq ,aq ,bq
2 2]
+
(abcq3, abq, bcq, caq, d, q/ d; q2)00 q (q2, aq2, bq2, cq2, abcq3/ d, abcdq2; q2)00 (1 - aq) (1 - bq) (1 - cq)
X 4¢3
3.27 Prove
[
q2,abCq2 ,dq ,q2/d 3
b
3
aq , q ,cq
I:
;q
3
2,q2] .
(bedq-2; q3)n(1- bcdq4n-2)(b, c, d; q)n n2
n=O (q; q)n(1- bcdq-2) (cdq, bdq, bcq; q3)n q
X 4¢3
[
3 3]
q-n, ql-n, q2-n, bcdq3n
b
2
2
d
q ,eq , q
2
;
q ,q
(bcdq, bq, cq,dq;q3)00 (q,cdq, bdq, bcq; q3)00 .
(Rahman [1993]) 3.28 Show that ~ (bcdq-\ q3)n(1- bcdq4n-l )(b, c, d; q)n n2+n ~ (q; q)n(l - bedq-l) (edq2, bdq2, beq2; q3)n q X 4¢3
[
(bcdq2,bq2, cq2, dq2; q3)00 (q2,cdq2,bdq2,bcq2;q3)00 .
(Rahman [1993])
3 3]
q-n-l, q-n, ql-n, bcdq3n b d ; q ,q q,cq, q
Exercises
109
3.29 Derive the summation formulas:
(i)
~ (1 - aq5k)(a, b; q2)k(ab 2j q3; q3)k(q2 jb; q)daq3jb; q6)k (_ q2) k q(k!l)
6
(1 - a)(q3, aq3 jb; q3h(q5 jb 2; q2)k(ab, abq2; q4)k
b
(aq2,qa 3jb;q2)00(ab 2,q9jb3;q6)00 (ab, q5 jb 2; q2)00(q3, aq6 jb; q6)00 ' (ii) ~
6
(1 - acq5k)(a,q4ja;q2)k(q5jac;q3)k(acjq2;q)k (1 - ac)(cq3, a2cjq; q3h(a2c2j q3; q2)k
(a 2c2jq;q6)k ( -ac)k q (k+l) 2 (q4,q6;q4)k q2 (acq2,acjq;q2)00(q6,a3c2jq3,ac2q,a2c2jq,aq4,q8ja;q6)00 (q4,a2c2jq3;q2)00(cq3,cq6,a2cq2,a2cjq,acq,acq4;q6)00 ' (iii) ~ (1 - acq5k)(a, q2 ja; q2h(qjac; q3h(ac; q)k(a 2c2q; q6)k (_ac)kq(k!l) (1 - ac)(cq3,a2cq;q3)k(a2c2q;q2)k(q2,q4;q4)k x
6
(acq2,acq;q2)00(q6,a3c2q3,ac2q5,a2c2q,aq2,q4ja;q6)00 (q2,a2c2q;q2)00(cq3,cq6,a2cq4,a2cq,acq2,acq5;q6)00 (Rahman [1989b]) 3.30 Derive the quartic summation formula 00
~
(a, b; q)k(qjb, q2 jb, q3 jb; q3)k(a 2b2jq2; q4)k k 1- a (q4, aq4jb;q 4h(abq,ab,abjq;q2)k(q3jab 2;q)k q
1 _ aq5k
(aq, bq, 1jb; q)00(a 2b2q2; q4)00 2 q2 (ab, q3 j ab ; q)oo (abj q; q2)00 (q4, ab 3 j q2, aq4 jb; q4)00 a2b2jq2 ] x l¢l [ a 2b2q2 ;q4,ab3q2
+ ab3
(aq, ab 2j q2; q)oo (ab; q)oo (abj q; q2)00 (aq4 jb, ab3 j q2; q4)00 . (Gasper [1989a]) 3.31 Derive the cubic transformation formulas
(i) ~ 1- acq4k
6
(a, b; qh(cqjb; qhk(a 2bcq3n, q-3n; q3h k 3 1- ac (cq3, acq3 jb; q h(ab; qhk(ql-3n jab, acq3n+l; q)k q
(1 -acq2)(1 -abjq2)(1 -abq3n)(1 _ acq3n) (1 - acq3n+2) (1 - abq3n-2)(1 - ab)(l - ac)
110
Additional summation, transformation, and expansion formulas X n 1 _ aeq6k+2
t;
1- aeq2
(aq2,bq2,eq2/b,eq3/b,a2beq3n,q-3n;q3)k 3k (eq3,aeq3/b,abq3,abq2,q5-3n/ab,aeq3n+5;q3)k q ,
(ii) ~ (1 - aq4k)(1 - bq-2k)
~
(1 -a)(1 -b)
(a, 1/ab; q)k(abq; qhk(e, a/be; q3)k q3k (q3,a 2bq3;q3)k(q/b;q)2k(aq/e,beq;q)k
(aq,bq;q)n( eq3, aq3/be;q3)n (aq/e,beq;q)n(q3, aq3/b;q3)n x lOWg(a/b; q/ab 2, e, a/be, aqn+l, aqn+2, aqn+3, q-3n; q3, q3), where n = 0,1, .... (Gasper and Rahman [1990]) 3.32 Derive the cubic summation formula 00 1_a2q4k (b,q2/b;qh(a2/q;q)2k(e3,a2q2/e3;q3h k 1 - a2 (a 2q3/b, a 2bq; q3h(q2; qhk(a 2q/e3, e3/q; q)k q
t;
(bq2,q4/b,be3/q,e3q/b,e3/a2,e3q2/a2,a2q,a2q3;q3)00 (q2,q4,e3q,be3/a2,a2q3/b,a2bq,e3q2/a2b;q3)00 (b,bq,bq2,q2/b,q3/b,q4/b,a2/q,a2q,a2q3,e3/a2;q3)00 (q2,q4,e3/q,c3q,a2/e3,a2q/e3,e3q3/a2,c3q3/a2b,a3q3/b;q3)00 (e 6q/a 2; q3)00 [be 3/a 2, c3q2/a 2b X (a 2bq,be3/a 2;q3)00 2¢1 e6q/a 2 ;q ,q
3 3]
and show that it has the q ----; 1- limit case
a - 1/2, a, b, 2 - b, c, (2a + 2 - 3c)/3, a/2 + 1 .1] 3/2, (2a - b + 3)/3, (2a + b + 1)/3, 3c - 1, 2a + 1 - 3e, a/2'
Po [ 7
6
r (~) r
(~)
r (e -
~)
r (c + ~) r (2a-3b+3)
r(bt2)r(43b)r(b+~-1)r(3c-:+l)r(2atl)
x
r (~) r (2a-~c+3) r (2a-~c+l) r (2at 3 ) r (2a-3~-b+3) r (2a-3~+b+l)·
(Gasper and Rahman [1990]) 3.33 Derive the quartic transformation formula
t; 00
1 - a 2b2q5k-2 (a, b; q)k(ab/q, ab, abq; q3)k(a 2b2/q2; q4h k 1 - a 2b2/ q2 (ab 2q2, a2bq2; q4 h (abq, ab, ab/ q; q2)k (q; qh q
4 4]
¢ [ a . b _ (aq, b; q)00(a 2b2q2; q4)00 - (q; q)oo (abq; q2)00 (b, ab 2q2, a2bq2; q4)00 1 1 aq4' q , q
(Gasper and Rahman [1990]) 3.34 Show that
2 2] _- (-q,qa/e;q)n ( /).
.t, [ q-2n,e 2,a,qa . 4'f'3 q2 a 2 , eq -n , eq I-n , q , q
-aq, q c; q n
.
111
Notes 3.35 For k
=
f
1,2, ... , show that
f
(a; qk)n(b; qhn tn _ (b; q)oo(at; qk)oo (c/b; q)n(t; qk)n bn n=O (qk; qk)n(c; qhn - (c; q)oo(t; qk)oo n=O (q; q)n(at; qk)n . (Andrews [1966c]) 3.36 For the q-exponential function defined in (1.3.33) prove Suslov's addition formula where
£ (x,y;a)
=
q
with x = cos
(a 2. q2) 00 '
/4 L --(ae-''I't q "'" ( _q-2 I-n e'"0+"'" I-n"","o) ''I',_q-2 e''I'-';q 00
n2
(qa 2; q2)00 n=O (q; q)n
e, y = cos cP, 0 ::::; e : : ;
7f
and 0 ::::; cP ::::;
7f.
(Suslov [1997, 2003]) 3.37 Derive the quadratic transformation formula
(-a; q1/2)00 £(x;a)= (qa 2.2) ,q 00
A-.
2'/-'1
[q1/4 eiO, q1/4 e -iO. 1/2 ] 1/2 ,q ,-a. -q
(Ismail and Stanton [2002])
Notes §3.4 Bressoud [1987] contains some transformation formulas for terminating r+ 1 CPr (a1' a2, ... , ar+1; b1, ... , br ; q; z) series that are almost poised in the sense that bkak+1 = a1qbk with 15 k = 0,1 or 2 for 1 ::::; k ::::; r. 1fansformations for level basic series, that is r+1CPr series in which a1bk = qak+1 for 1 ::::; k ::::; r, are considered in Gasper [1985]. §3.5 For a comprehensive list of q-analogues of the quadratic transformation formulas in §2.11 of Erdelyi [1953], see Rahman and Verma [1993]. §3.6 Agarwal and Verma [1967a,b] derived transformation formulas for certain sums of bibasic series by applying the theorem of residues to contour integrals of the form (4.9.2) considered in Chapter 4. Inversion formulas are also considered in Carlitz [1973] and W. Chu [1994b, 1995] and, in connection with the Bailey lattice, in A.K. Agarwal, Andrews and Bressoud [1987]. §3.7 Jackson [1928] applied his q-analogue of the Euler's transformation formula (the p = q case of (3.7.11)) to the derivation of transformation formulas and theta function series. Jackson [1942, 1944] and Jain [1980a] also derived q-analogues of some of the double hypergeometric function expansions in Burchnall and Chaundy [1940, 1941]. §3.8 Gosper [1988a] stated a strange q-series transformation formula containing bases q2, q3, and q6. Krattenthaler [1989b] independently derived the terminating case of (3.8.18) and terminating special cases of some of the other summation formulas in this section. For further results on cubic and quintic summation and transformation formulas, see Rahman [1989d, 1992b, 1997].
n
112
Additional summation, transformation, and expansion formulas §3.9
For multibasic series containing bibasic shifted factorials of the form
(a; p, q)r,s = TI;:~ TI~:~(1- api qk) and connections with Schur functions and permutation statistics, see Desarmenien and Foata [1985-1988]. §3.10 Jain and Verma [1986] contains nonterminating versions of some of Nassrallah and Rahman's [1981] transformation formulas. Ex.3.11 q-Differential equations for certain products of basic hypergeometric series are considered in Jackson [1911]. Exercises 3.17-3.19 These exercises are q-analogues of the Cayley [1858] and Orr [1899] theorems (also see Bailey [1935, Chapter X], Burchnall and Chaundy [1949]' Edwards [1923]' Watson [1924]' and Whipple [1927, 1929]). Other q-analogues are given in N. Agarwal [1959], and Jain and Verma [1987]. Ex. 3.20 The formula obtained by writing the last series as a multiple of the series with argument bp is equivalent to the bibasic identity (21.9) in Fine [1988], and it is a special case of the Fundamental Lemma in Andrews [1966a, p. 65]. Applications of the Fundamental Lemma to mock theta functions and partitions are contained in Andrews [1966a,b]. Agarwal [1969a] extended Andrews' Fundamental Lemma and pointed out some expansion formulas that follow from his extension. Ex. 3.23 For additional material on Lagrange inversion, see Andrews [1975b, 1979a], Bressoud [1983b], Cigler [1980], Fiirlinger and Hofbauer [1985], Garsia [1981]' Garsia and Remmel [1986], Gessel [1980], Gessel and Stanton [1983, 1986], Hofbauer [1982, 1984]' Krattenthaler [1984, 1988, 1989a], Paule [1985b], and Stanton [1988].
4 BASIC CONTOUR INTEGRALS
4.1 Introduction Our first objective in this chapter is to give q-analogues of Barnes' [1908] contour integral representation for the hypergeometric function F ( 21
b. . ) a, ,c,z
=
where larg( -z)1
f(c)
_1 jiOO f(a+s)f(b+s)f(-s)(_ )8 d -ioo r(c+s) Z s, (4.1.1)
r(a)r(b)2ni
< n, Barnes' [1908] first lemma
~ jiOO r(a + s)r(b + s)r(c 2nz -ioo
s)f(d - s) ds
r(a + c)f(a + d)r(b + c)f(b + d) f(a + b + c + d)
(4.1.2)
and Barnes' [1910] second lemma
_1_jiOO f(a + s)r(b + s)f(c + s)r(1 - d - s)f( -s) ds 2ni -ioo f( e + s) f(a)f(b)f(c)f(1 - d + a)f(1 - d + b)f(1 - d + c) r(e - a)r(e - b)f(e - c) where e
= a + b+ c -
(4.1.3)
d + 1.
In (4.1.1) the contour of integration is the imaginary axis directed upward with indentations, if necessary, to ensure that the poles of f( -s), i.e. s = 0,1,2, ... , lie to the right of the contour and the poles of f( a + s )f(b + s), i.e. s = -a - n, -b - n, with n = 0,1,2, ... , lie to the left of the contour (as shown in Fig. 4.1 at the end of this section). The assumption that there exists such a contour excludes the possibility that a or b is zero or a negative integer. Similarly, in (4.1.2), (4.1.3) and the other contour integrals in this book it is assumed that the parameters are such that the contour of integration can be drawn separating the increasing and decreasing sequences of poles. Barnes' first and second lemmas are integral analogues of Gauss' 2Fl summation formula (1.2.11) and Saalschiitz's formula (1.7.1), respectively. In Askey and Roy [1986] it was pointed out that Barnes' first lemma is an extension of the beta integral (1.11.8). To see this, replace b by b - iw, d by d + iw and then set s = wx in (4.1.2). Then let w ----; CXJ and use Stirling's formula to obtain the beta integral in the form
I:
x++ c - 1 (1 -
x)~+d-ldx = 113
B(a
+ c, b + d),
(4.1.4)
114
Basic contour integrals
where Re(a + c) > 0, Re(b + d) > 0 and
_ {X
x+ -
o
if x 2: 0, if x < O.
(4.1.5)
It is for this reason that Askey and Roy call (4.1.2) Barnes' beta integral. Following Watson [1910]' we shall give a q-analogue of (4.1.1) in §4.2, that is, a Barnes-type integral representation for 2 (/JI (a, b; c; q, z). It will be used in §4.3 to derive an analytic continuation formula for the 2¢1 series. We shall give q-analogues of (4.1.2) and (4.1.3) in §4.4. The rest of the chapter will be devoted to generalizations of these integral representations, other types of basic contour integrals, and to the use of these integrals to derive general transformation formulas for basic hypergeometric series.
y
c .)
•
•
•
•
-a
____________________~~~------~~------.--------+-x 2
" 0
•
•
•
•
.J -b
Fig. 4.1
3
115
4.2 Watson's contour integral 4.2 Watson's contour integral representation for 2¢>1(a,b;c;q,z) series
For the sake of simplicity we shall assume in this and the following five sections that 0 < q < 1 and write q = e- w ,
w
> o.
(4.2.1)
This is not a severe restriction for most applications because the results derived for 0 < q < 1 can usually be extended to complex q in the unit disc by using analytic continuation. The restriction 0 < q < 1 has the advantage of simplifying the proofs by enabling one to use contours parallel to the imaginary axis. Extensions to complex q in the unit disc will be considered in §4.8. For 0 < q < 1 Watson [1910] showed that Barnes' contour integral in (4.1.1) has a q-analogue of the form
(4.2.2) where Iz I < 1, Iarg( - z) I < 1r. The contour of integration (denote it by C) runs from -ioo to ioo (in Watson's paper the contour is taken in the opposite direction) so that the poles of (ql +s; q) 00 / sin 1r s lie to the right of the contour and the poles of 1/( aqS, bqs; q)oo, i.e. s = w-1loga - n + 21rimw-1, s = w-1logb - n + 21rimw- 1 with n = 0, 1,2, ... , and m = 0, ±1, ±2, ... , when a and b are not zero, lie to the left of the contour and are at least some E > 0 distance away from the contour. To prove (4.2.2) first observe that by the triangle inequality,
and so I
(ql+s,cqs;q)oo I (aqS, bqS; q)oo 00 (1 + e-(n+H
:s;
11
(1-l a le-(n+
+ Icle-(n+ Re(s))w)
Re(s))w)
(1
Re(s))w)
(1 -Ible-(n+
Re(s))w) '
(4.2.3)
Hence the integral in (4.2.2) converges if which is bounded on C. Re[slog(-z) -log(sin1rs)] < 0 on C for large lsi, i.e. if larg(-z)1 < 1r. Now consider the integral in (4.2.2) with C replaced by a contour C R consisting of a large clockwise-oriented semicircle of radius R with center at the origin that lies to the right of C, is terminated by C and is bounded away
Basic contour integrals
116
from the poles (as shown in Fig. 4.2).
y
r--~
c •
• •
•
•
----------------~~--------.---------.---_r--~-x
o
2
R
R [cos e log Izl - sine arg( -z) - 7r1 sinel] -R [sine arg( -z) + 7r1 sinel] + 0(1).
+ 0(1)
•
3
•
•
•
1
•
Fig. 4.2 Setting
S
= Re iIJ , we Re [log =
~
Hence, when Izl
have for Izl
< 1 that
(~z)S]
SIn 7rS
< 1 and larg( -z)1 < 7r - 0, 0 < 0 < 7r, we have
(~z)S = 0 [exp( -oRI sinel)] (4.2.4) SIn 7rS on C R as R ---+ 00, and it follows that the integral in (4.2.2) with C replaced by CR tends to zero as R ---+ 00, under the above restrictions. Therefore, by applying Cauchy's theorem to the closed contour consisting of C R and that part of C terminated above and below by CR and letting R ---+ 00, we 1 . Ji~ obtain that - --2 ds equals the sum of the residues of the integrand at 'ITt -'/;00 n = 0,1,2, .... Since
...
. (q1+s, cqS;q) 7r(-z)S hm (s - n) 00 ---'-----'s-+n (aqs,bqS;q)oo sin7rs
4.3 Analytic continuation of 21>1 (a, b; C; q, z)
117
this completes the proof of Watson's formula (4.2.2). It should be noted that the contour of integration in (4.2.2) can be replaced by other suitably indented contours parallel to the imaginary axis. To see that (4.2.2) is a q-analogue of (4.1.1) it suffices to use (1.10.1) and the reflection formula 7r r(x)r(1 - x) = - . (4.2.5) SIn 7rX
to rewrite (4.2.2) in the form
21>1 (qa,qb;qc;q,z) =
fiOO
rq(c) 1 rq(a)rq(b)27ri
-ioo
r q(a+s)rq(b+s)r(-s)r(1+s)(_z)Sds. rq(c+s)rq(1+s) (4.2.6)
4.3 Analytic continuation of 21>I(a, b; C; q, z) Since the integral in (4.2.2) defines an analytic function of z which is singlevalued when larg( -z)1 < 7r, the right side of (4.2.2) gives the analytic continuation ofthe function represented by the series 21>I(a, b; c; q, z) when Izl < 1. As in the ordinary hypergeometric case, we shall denote this analytic continuation of 21>I(a,b;c;q,z) to the domain larg(-z)1 < 7r again by 21>I(a,b;c;q,z). Barnes [1908] used (4.1.1) to show that if larg( -z)1 < 7r and a, b, c, a - b are not integers, then the analytic continuation for Izl > 1 of the series which defines 2Fl(a, b; C; z) for Izl < 1 is given by the equation
r(c)r(b-a) -a -1 2Fl(a,b;c;z) = r(b)r(c-a)(-z) 2Fl(a,1+a-c; 1+a-b;z ) r(c)r(a - b) -b + r(a)r(c-b)(-z) 2Fl(b,1+b-c; 1+b-a;z-1), (4.3.1) where, as elsewhere in this section, the symbol "=" is used in the sense "is the analytic continuation of". To illustrate the extension of Barnes' method to 21>1 series we shall now give Watson's [1910] derivation of the following q-analogue of (4.3.1):
21>1 (a, b; C; q, z) (b, cl a; q)oo (az, ql az; q)oo ( I I I) = (bl ) ( I ) 21>1 a, aq C; aq b; q, cq abz c, a; q 00 z, q Z; q 00 (a, clb; q)oo (bz, qlbz; q)oo ( I I I) + (c, aIb) ; q 00 (z, qI Z; q)00 21>1 b, bq C; bq a; q, cq abz , provided that larg( -z)1
o.
(4.3.2)
< 7r, c and alb are not integer powers of q, and a, b, z =I=-
First consider the integral
h =
_1-/ 27ri
(ql+s, cq S;q)oo7r(_z)S ds (aqs, bqs; q)oo sin 7rS
(4.3.3)
118
Basic contour integrals
along three line-segments AI, A 2 , B, whose equations are: Al : Im(s) = ml,
A2
:
Im(s) = -m2,
B: Re(s) = -M,
(4.3.4)
where ml,m2,M are large positive constants chosen so that A l ,A2 ,B are at least a distance E > 0 away from each pole and zero of
(ql+s, cqs; q)oo
g(s)
= ( ) aqS, bqS; q
(4.3.5) 00
and it is assumed that AI, A 2 , B are terminated by each other and by the contour of the integral in (4.2.2), i.e. Re(s) = 0 with suitable indentations. From an asymptotic formula for (a; q)oo with q = e- w , W > 0 , due to Littlewood [1907, §12]' it follows that if Re(s) ---. -00 with Is - sol> E for some fixed E > 0 and any zero So of (qS; q)oo, then
Re[log(qS; q)ool
=
W
"2(Re(s))
2
W
+"2 Re(s) + 0(1).
(4.3.6)
This implies that
g(s)
=
0
(I :~ IRe(S)) ,
(4.3.7)
when Re(s) ---. -00 with s bounded away from the zeros and poles of g(s). By using this asymptotic expansion and the method of §4.2 it can be shown that if Izl > Icqjabl, then the value of the integral II in (4.3.3) taken along the contours AI, A 2, B tends to zero as ml, m2, M ---. 00. Hence, by Cauchy's theorem, the value of h, taken along the contour C of §4.2, equals the sum of the residues of the integrand at its poles to the left of C when Izl > Icqjabl. Set 0: = -w-lloga, f3 = -w-llogb so that a = q''',b = qf3. Since the residue of the integrand at -0: - n + 27rimw- l is (
-1 l-n
a
-1 -n
n+l
)
q ,ca q ,q ;q -1 -n (q, q, ba q ; q)oo
00
7rW
-l(
-z
)-a-n n(n+l)/2 q
x exp {2m7riw- l log( -z)} csc (2m7r 2iw- l - 0:7r) , we have 00
m=-oo
x
7rw- l (cja, qja; q)oo ( )-a (bj .) -z a, q, q
2¢1
( j j j) a, aq c; aq b; q, cq abz
00
+ idem (a; b).
(4.3.8)
Thus it remains to evaluate the above sums over m when larg(-z)1 < 7r. Letting c = bin (4.3.8) and using (4.2.2), we find that the analytic continuation of 2¢1(a,b;b;q,z) is 00
m=-oo
l qj x 7rw-( (a, )a;q)oo( -z )-a q,q;q 00
2¢1
(a, aq j b; aq j b; q, q jaz) .
4.4 q-analogues of Barnes' first and second lemmas
119
Since, by the q-binomial theorem, 2rP1
(a, b; b; q, z )
(az; q)oo
= (z;q )
00
and the products converge for all values of z, it follows that 00
m=-oo
(4.3.9)
w(q,q,az, q/az; q)oo 7r(a,q/a,z,q/z;q)oo . Using (4.3.9) in (4.3.8) we finally obtain (4.3.2). 4.4 q-analogues of Barnes' first and second lemmas Assume, as before, that 0< q < 1, and consider the integral (q1-c+s " q1-d+s. q) I _ _1 00 7rq S dS 2 - 27ri -ioo (qa+s,qb+s;q)oo sin7r(c-s)sin7r(d-s)'
fiOO
(4.4.1)
where, as usual, the contour of integration runs from -ioo to ioo so that the increasing sequences of poles of the integrand (i.e. c + n, d + n with n = 0,1,2, ... ) lie to the right and the decreasing sequences of poles (i.e. the zeros of (qa+s, qb+ s ; q)oo) lie to the left of the contour. By using Cauchy's theorem as in §4.2 to evaluate this integral as the sum of the residues at the poles c + n, d + n with n = 0,1,2, ... , we find that c (q q1+c-d. q) I 7rq , '00 A.. (a+c b+c. l+c-d. ) 2 -. ( d) (a+c ,q ,q ,q, q sm 7r c q , qb+c.) ,q 00 2'/-'1 q
+ idem (c; d).
(4.4.2)
Applying the formula (2.10.13) to (4.4.2), we get 1
fiOO
(q1-c+s, q1-d+s; q)oo
7rqs ds
(qa+s,qb+s;q)oo sin7r(c-s)sin7r(d-s) qC (q, q1+c-d, qd-c, qa+b+c+d; q) 00 sin 7r(c - d) (qa+c, qa+d, qb+ c, qb+d; q)oo
27ri
-ioo
(4.4.3)
which is Watson's [1910] q-analogue of Barnes' first lemma (4.1.2), as can be seen by rewriting it in terms of q-gamma functions. A q-analogue of Barnes' second lemma (4.1.3) can be derived by proceeding as in Agarwal [1953b]. Set c = nand d = c - a - bin (4.4.3) to obtain 1
27ri
fiOO
(q1-n+s,q1-c+a+b+s;q)00
-ioo
= csc 7r( C X
7rqs ds
(qa+s,qb+s;q)oo sin7rssin7r(c-a-b-s) (q1+a+b-c, qc-a-b, qC, q; q)oo a-
b) -'---,----,----------;----:---'-''''(qa, qb, qc-a, qC-b; q)oo
(_l)n (q(a,qb;)q)nqn(C-a-bl-G) qC;q n
(4.4.4)
120
Basic contour integrals
for n = 0,1,2, .... Next, replace c by d in (4.4.4), multiply both sides by
(_I)n qn (e-c)+G) (qC; q)n / (q, qe; q)n' sum over n and change the order of integration and summation (which is easily justified if Iqe-c+sl < 1) to obtain (ql+a+b-d, qd-a-b, qd, q; q) 00 csc 7f( d - a - b) --'--------;----;----;-----;--;--;---'---''-''(qa,qb,qd-a,qd-b;q)oo
x
(qa, l, qC; qd, qe; q, qd+e-a-b-c)
3¢2
jiOO
1
(ql+s,ql-d+a+b+s;q)oo
=
27fi -ioo (qa+s,qb+s;q)oo x 2¢1 (q-S, qC; qe; q, qe-c+s) ds
=
(qe-c; q)oo ~
jiOO
27f~
-ioo
(qe; q)oo
7fqs sin7fssin7f(d-a-b-s)
(ql+s, qe+s, ql-d+a+b+s; q) 00 (qa+s, qb+ s , qe-c+s; q)oo
7fqs ds x sin 7f s sin 7f (d - a - b + s )'
(4.4.5)
by the q-Gauss formula (1.5.1). Now take c = d. Then the series on the left of (4.4.5) can be summed by the q-Gauss formula to give, after an obvious change in parameters,
jiOO
(ql+S, qd+s, qe+s; q) 00 7fqs ds 27fi -ioo (qa+s, qb+ s , qc+s; q)oo sin 7fS sin 7f( d + s) (q, qd, ql-d, qe-a, qe-b, qe-c; q) 00 d =~7f , (qa, qb, qC, ql+a-d, ql+b-d, ql+c-d; q)oo 1
(4.4.6)
where d + e = 1 + a + b + c, which is Agarwal's q-analogue of Barnes' second lemma. This integral converges if q is so small that Re [s log q - log(sin 7fS sin 7f(d + s ))]
<0
(4.4.7)
on the contour for large lsi-
4.5 Analytic continuation of r+l¢r series By employing Cauchy's theorem as in §4.2, we find that if Izl larg(-z)1 < 7f, then .t..
r+l 'f'r
< 1 and
] [a l ,a2, ... ,ar+l . b1, ... , br ' q, Z
(aI, a2,···, ar+l; q)oo (q, bl , ... , br ; q)oo X
(2)jiOO 27fi
-ioo
(ql+S,b1qS, ... ,brqS;q)00 7f(-z)Sds (alqS, a2qs, ... , ar+lqS; q)oo sm 7fS
(4.5.1)
where, as before, only the poles of the integrand at 0,1,2, ... , lie to the right of the contour. As in the r = 1 case, the right side of (4.5.1) gives the analytic continuation of the r+l¢r series on the left side to the domain larg( -z)1 < 7f.
4.6 Contour integrals representing well-poised series
121
Also, as in §4.3, it can be shown that if Izl > Ib1 ... br q/a1 ... a r +11, then the integral I3 = 2;'i t,:,oo ... ds is equal to the sum of the residues of the integrand at those poles of the integrand which lie on the left of the contour. Set a1 = -w- 1 10g a1. Since the residue ofthe integrand at -a1 -n+27rimw- 1 IS
b a -1 q -n , ... , br a 1-1 q -n i q ) ( qn+1 ,a 1-1 q 1-n ,1 -1 _ _ (+1)/2 1 --'-----;------,-------------,-------,----------'---"oo=7rW (-z) <>1 nqn n 1 1 (q, q, a2 a l q-n, ... , a r+1 a l q-n i q)oo x csc(2m7r 2iw- 1 - al7r) exp {2m7riw- 1 10g( -z)},
by proceeding as in the proof of (4.3.2) and using (4.3.9) and (4.5.1) we obtain the expansion
(a2, ... , ar+1, bl/a1, ... , br/a1, a1 Z, q/a1 z i q)oo (b1, ... ,br,a2/a1, ... ,ar+1/al,z,q/Ziq)00 A,
X r+1'i/r
[al,a1q/bl, ... ,alq/br. qb1 ... br ] / / ' q, alq a2,···,alq ar+l za1···ar+1
(4.5.2)
+ idem (ali a2,···, ar+d, where the equality holds in the "is the analytic continuation of" sense. The symbol "idem (ali a2, ... , ar+d" after an expression stands for the sum of the r expressions obtained from the preceding expression by interchanging a1 with each ak, k = 2,3, ... , r + 1.
4.6 Contour integrals representing well-poised series Let us replace a, b, c, d and e in (4.4.6) by a + n, b + n, c + n, d + nand e + 2n, respectively, where (4.6.1) e = 1 + a + b + c - d, and transform the integration variable s to s - n, where n is a non-negative integer. Then we get n+ioo (q1+s-n,qd+s,qe+s+n iq )00 1 7rqs-n ds 27ri n-ioo (qa+s , qb+ s , qc+s i q) 00 sin 7r s sin 7r (d + s)
I
= CSC7r
d
(q,qd+n,q1-d-n,qe-a+n,qe-b+n,qe-c+niq)00 ( b +n,qc+n,qe-a- b,qe-a-c,qe- b ) 00 . qa+n,q -c iq
(4.6.2)
The limits of integration n ± ioo can be replaced by ±ioo because we always indent the contour of integration to separate the increasing and decreasing sequences of poles. Thus, it follows from (4.6.2) that (a b c ) ( q,q d ,qI-d ,e e-a ,qe-b ,qe-c iq ) 00 csc(7rd) q ,q ,q iq n (l-d)n (qa, qb, qC, qe-b-c, qe-c-a, qe-a-b i q)oo (qe-a, qe-b, qe-c i q)n q =
1 fioo (q1+S, qd+s, qe+s i q)oo 7rq(n+l)s(q- Si q)n ds 27ri -ioo (qa+s, qb+ s , qc+s i q)oo sin 7rssin 7r(d + s)(qe+s i q)n'
(4.6.3)
122
Basic contour integrals
(4.6.4) where e = 1 + a + b + c - d, Izl < Iqdl and it is assumed that (4.4.7) holds. Therefore, if we can sum the series on the right of (4.6.4), then we can find a simpler contour integral representing the series on the left. In particular, if we let r = 4,e = 1 +A,al = ql+A/2 = -a2,a3 = qd,a4 = qe, _ q2-d-e+A , b1 -- qA/2 -- -b 2, b3 -- ql-d+A , b4 -- ql-e+A ,Z -
then we get a VWP-balanced 6¢5 series which can be summed by (2.7.1). This yields Agarwal's [1953b] contour integral representation for a VWP-balanced 8¢7 series: 8¢7
=
[
qA,ql+A/2,_ql+A/2,qa,qb,qc,qd,qe qA/2, _qA/2, ql+A-a, ql+A-b, ql+A-c, ql+A-d, ql+A-e ; q, q
B]
sin 7r(a + b + c - A)
x
7rqS ds
(4.6.5)
--------~~----------~
sin 7rS sin 7r( a + b + c - A
where B
= 2 + 2A -
+ s) ,
a - b - c - d - e, provided Re(B)
Re [s log q -log(sin7rssin7r(a + b+ c - A
> 0 and
+ s))] < o.
If we evaluate the integral in (4.6.5) by considering the residues at the poles of 1/ [sin 7r s sin 7r( a + b + c - A + s ) ] lying to the right of the contour, then we obtain the transformation (2.10.10) of a VWP-balanced 8¢7 series in terms of the sum of two balanced 4¢3 series. In addition, if we replace A, d, e and a by A, A+d-A, A+e-A and A+a-A, respectively, and take A+a+d+e = 1+2A in (4.6.5), then the integral in (4.6.5) remains unchanged. This gives Bailey's transformation formula (2.10.1) between two VWP-balanced 8¢7 series.
4.7 Contour integral analogue of Bailey's formula
123
4.7 A contour integral analogue of Bailey's summation formula By replacing A, a, b, c, d, e in (4.6.5) by a, d, e, f, b, c, respectively, we obtain the formula
s W7 (qa; qb, qC, qd, qe, qf; q, q)
=
sin n (d + e +
f - a)
(ql+a,qd,qe, qf,ql+a-d-e, ql+a-d-f, ql+a-e- f ; q)oo X
~~~~--~----~--~~------~--~~--~--~~--
(q, ql+a-b, ql+a-c, ql+a-d, ql+a-e, ql+a- f ,ql+a-d-e- f ; q)oo
1
x-
jiOO
2ni
(ql+S, ql+a-b+s, ql+a-c+s; q) 00 (qd+s, qe+s, qf+ s ; q)oo
-ioo
nqs ds
sin ns sin n(d + e +
f -
a + s)'
(4.7.1) provided the series is VWP-balanced, i.e.,
1 + 2a = b + c + d + e +
f.
(4.7.2)
Since 1 + 2(2b - a) = b + (b + c - a) + (b + d - a) + (b + e - a) + (b + f - a) by (4.7.2), it follows that (4.7.2) remains unchanged if we replace a, c, d, e, f by 2b - a, b + c - a, b + d - a, b + e - a, b + f - a, respectively, and keep b unaltered. Then (4.7.1) gives S W7
X
(q2b-a; qb, qb+c-a, qb+d-a, qb+e-a, qb+ f-a; q, q)
) ( q 1+2b-a ,qb+d-a ,qb+e-a ,qb+f-a ,q l+a-d-e ,q l+a-d-f ,q l+a-e-f.,qoo ~------~~~~--~~--~~~~~--~~~~--~------~=
(q,ql+b-a,ql+b-c, ql+b-d,ql+b-e,ql+b-f, qC; q)oo
1
x2ni
.
= SIn nc X
= sin nc
jiOO -ioo
(ql+b-a+s, ql+b-c+s, ql+s; q) 00
nqs ds
(qb+ d- a+s , qb+ e- a+s , qb+ f -a+s; q) 00 sin n s sin n( c - s)
(ql+2b-a, qb+d-a, qb+e-a, qb+f- a ; q)oo """"'--------,----:;-:-;-----:;-.,-;-------,-------'-'''''' (q, q1+ b- a , q1+ b- c , qC; q)oo
(ql+a-d-e,ql+a-d- f ,ql+a-e- f ;q)oo 1 (q1+b-d, q1+b-e , q1+b- f ; q) 00
2ni
jiOO
(q1+s;q)oo
-ioo
(qd+s; q) 00
(ql+a-c+s, ql+a-b+s; q) 00
nqs ds
(qe+s, qf+ s ; q)oo
sinn(a-b+s)sinn(c+b-a-s)'
x
(4.7.3)
where the second integral in (4.7.3) follows from the first by a change of the integration variable s ----; a-b+s. Combining (4.7.1) and (4.7.3) and simplifying, we obtain (ql+a-b, ql+a-c, ql+a-d, ql+a-e, ql+a- f ; q)oo (q1+a, qC, qd, qe, qf ; q) 00 X SW7
b-a
- q
(qa;qb,qc,qd,qe,qf;q,q) ) (q l+b-a ' ql+b-c , ql+b-d , ql+b-e , ql+b- f '. (qX l (ql+2b-a, qb+c-a, qb+d-a, qb+e-a, qb+f-a; q)oo
124
Basic contour integrals X 8W7
(q2b-a; qb, qb+c-a, qb+d-a, qb+e-a, l+f- a ; q, q)
) ( q I+a-d-e ,q I+a-e-f ,q I+a-d-f.,q(X) (q, qC, qb+c-a; q)(X) q) 1 ji(x) (q l+s , ql+a-b+s , ql+a-c+s.'(x)
x2ni
(qd+s, qe+s, qf+ s ; q)(X)
-i(X)
· nq sSIn n (a_ b) d s sin ns sin n(a - b + s) ,
(4.7.4) when (4.7.2) holds. Evaluating the above integral via (4.4.6), we obtain Bailey's summation formula (2.11.7). Agarwal's [1953b] formula 1 2ITi
i(X) (ql+S, q~a+s, _q~a+s, ql+a-b+s, ql+a-c+s; q) (X)
l,~
(
q"+", q1+j "+', _q1+ j "+', qb+., qo+,; q) oc
(ql+a-d+s,ql+a-e+s,ql+a-f+s;q)(X)
x
nqs ds
------~~--~--~--~----~--------~--~--7
(qd+ s , qe+s, qf+ s ; q)(X)
sin ns sin n(a - b + s)
b (q, ql+a-b, qb-a, ql+a-d-e, ql+a-e- f; q) (X) =cscn(a-)
(qb,qc,qd,qe,qf;q)(X)
( q I+a-d- f ,q I+a-c-d ,q I+a-c-e ,q I+a-c- f.,q ) (X)
x
(qb+c-a, qb+d-a, qb+e-a, qb+ f -a; q) (X)
,
(4.7.5)
where 1 + 2a = b + c + d + e + j, follows directly from (2.11.7) by considering the residues of the integrand of the above integral at the poles to the right of the contour, i.e. at s = n, b - a + n with n = 0,1,2, .... Thus (4.7.5) gives an integral analogue of Bailey's summation formula (2.11.7). The integral in (4.7.5) converges if q is so small that Re [s log q - log (sin n s sin n (a - b + s) )1 < 0 on the contour for large
(4.7.6)
lsi-
4.8 Extensions to complex q inside the unit disc The previous basic contour integrals can be extended to complex q inside the unit disc by using suitable contours. For 0 < Iql < 1, let (4.8.1) where WI = -log Iql > 0 and W2 = -Arg q. Thus q = e-(Wl +iW 2). Then a modification of the proof in §4.2 (see Watson [1910]) shows that if 0 < Iql < 1 and Izl < 1, then formula (4.2.2) extends to 2¢1
(a, b; c; q, z) (a, b; q)(X) (q,c;q)(X)
(4.8.2)
4.9 Other types of basic contour integral formulas
125
where C is an upward directed contour parallel to the line Re( s( WI + iW2)) = 0 with indentations, to ensure that the increasing sequence of poles 0, 1,2, ... , of the integrand lie to the right, and the decreasing sequences of poles lie to the left of C. Since the above integral converges if Re[s log( -z) - log (sin 7fs)] < 0 on C for large lsi, i.e., if (4.8.3) it is required that z satisfies (4.8.3) in order for (4.8.2) to hold. This restriction means that the z-plane has a cut in the form of the spiral whose equation in polar coordinates is r = eW1 (J/W2. Analogously, when 0 < Iql < 1, the contours in the q-analogues of Barnes' first and second lemmas given in §4.4 and the contours in the other integrals in §§4.4-4.7 must be replaced by upward directed contours parallel to the line Re(s(wl + iW2)) = 0 with indentations to separate the increasing and decreasing sequences of poles. 4.9 Other types of basic contour integrals
Let q = e- w with
W
> 0 and suppose that
(aIz, ... ,aAz, bl/z, ... ,bB/z;q)oo P () z = ~----------~~----~~~~ (CIZ, ... ,ccz , dl/z, ... ,dD/z;q)oo
(4.9.1)
has only simple poles. During the 1950's Slater [1952c,d, 1955] considered contour integrals of the form 1m
W == 1m(A,B;C,D) = -.
27f~
J-i7r i7r
/
W
P(qS)qmsds
(4.9.2)
/w
with m = 0 or 1. However, here we shall let m be an arbitrary integer. It is assumed that none of the poles of P(qS) lie on the lines Ims = ±7f/w and that the contour of integration runs from -i7f / W to i7f / wand separates the increasing sequences of poles in 11m sl < 7f/w from those that are decreasing. By setting if) = -sw the integral 1m can also be written in the "exponential" form 1m
= ~ J7r 27f
-7r
P (ei(J) eim(Jdf)
(4.9.3)
with suitable indentations, if necessary, in the contour of integration. Similarly, setting z = qS we obtain that (4.9.4) where the contour K is a deformation of the (positively oriented) unit circle so that the poles of 1/ (CIZ, ... , ccz; q)oo lie outside the contour and the origin and poles of 1/(dl/z, ... ,dD/z;q)oo lie inside the contour. Special cases of (4.9.3) and (4.9.4) have been considered by Askey and Roy [1986]. Although each of the above three types of integrals can be used to derive transformation formulas for basic hypergeometric series, we shall prefer to
126
Basic contour integrals
use mainly the contour integrals of the type in (4.9.4) since they are easier to work with, especially when the assumption in Slater [1952c,d, 1966] that 0< q < 1 is replaced by only assuming that Iql < 1, which is the case we wish to consider in the remainder of this chapter. 4.10 General basic contour integral formulas
Our main objective in this section is to see what formulas can be derived by applying Cauchy's theorem to the integrals Im in (4.9.4). Let Iql < 1 and let 15 be a positive number such that 15 -=1= Idjqnl for j = 1,2, ... , D, and 15 -=1= Icjlq-nl for j = 1,2, ... ,0 when n = 0,1,2, .... Also let ON be the circle Izl = blqlN, where N is a positive integer. Then ON does not pass through any of the poles of P(z) and we have that
IF (bqN)
(bqN) m-l I = I (alb, ... ,aAb, bd 15, ... ,bB/b; q)oo I (clb, ... ,ceb, dd b, ... , dD/b; q)oo
x
I (Clb, ... ,ceb,qb/bl, ... ,qb/bB;.q)N
X
bm-llbNq(Ntl)
(bl ... bBqm-l)NI (alb, ... , aAb, qb/d l , ... , qb/d D, q)N d l ··· d D D I
-
B
=
0
(I bl
~~ ~~~:-l IN IbN q(Ntl) ..
D I
-
B
)
.
(4.10.1) Since ON is of length O(lqIN) it follows from (4.10.1) that if D and
> B or if D = B (4.10.2)
then
lim
N--+oo
1
eN
p(z)zm-ldz
=
O.
(4.10.3)
Hence, by applying Cauchy's residue theorem to the region between K and ON for sufficiently large N and letting N ----+ 00, we find that if D > B or if D = B and (4.10.2) holds, then Im equals the sum of the residues of p(z)zm-l at the poles of 1/ (dd z, ... , dD/ z; q)oo' Therefore, since .
ReSIdue
z
=
(
dqn
1
(d/ z; q)oo
)
=
(_I)ndq2n+G) , (q; q)n (q; q)oo
n
=
0,1,2, ... ,
(4.10.4)
it follows that
(4.10.5)
4.10 General basic contour integral formulas
127
> B, or if D = Band (4.10.2) holds. In addition, by considering the residues of p(z)zm-l outside of K or by just using the inversion z -+ Z-l and renaming the parameters, we obtain if D
1m= (blCl, ... ,bBcl,aI/cl, ... ,aAlcl;q)oo_m ~ (q, dlCl, ... , dDCl, c2/cl, ... , Cc ICl; q)oo
f
x
(dlCl, ... , dDCl, qcI/al, ... , qcI/aA; q)n n=O (q, blCl, ... , bBCl, qCI/C2, ... , qcI/cc; q)n
(n+l)/2)n(C-A) (a l ... aAq-m)n x ( -Clq Cl··· Cc + idem (Cl; C2, ... , cc)
if C
> A, or if C
=
A and l
In the special case when C in the form
(A
Im
(4.10.6)
=
al ... aAq-ml Cl··· Cc
< 1.
A we can use the
reps
(4.10.7) notation to write (4.10.5)
·A ) - (aldl, ... ,aAdl,bI/dl, ... ,bBldl;q)oo d m , B, , D - (q,cl dl , ... ,CAdl ,d2Idl, ... ,dD I ) 00 1 dl;q
cldl , ... , cAdI, qdI/bl , ... , qdI/bB . t( d )D-B] al d 1,···,aA d l,q did 1 2,···,q did 1 D ,q, q 1 + idem (d l ;d2, ... ,dD ) (4.10.8) rI,
[
XA+B'I'A+D-l
where t
= bl b2 ··· bBqm Id l ... d D , if D > B, or if D = Band (4.10.2) holds.
Similarly, from the D
1 (A B·C B) m
=
B case of (4.10.6) we have
(blCl, ... ,bBcl,aI/cl, ... ,aAlcl;q)oo -m (q,dlCl, ... ,dBCl,C2/cl, ... ,cclcl;q)ooCl
=
",
[dlCl, ... , dBCl, qCI/al, ... , qcI/aA. ( )C-A] I I ' q, U qCl blCl, ... ,bBcl,qcl C2,···,qCl Cc + idem (Cl; C2, ... , cc) (4.10.9) rI,
XA+B'I'B+C-l
> A, or if C = A and (4.10.7) holds, where u = al··· aAq-m ICI ... CC. Evaluations of Im which follow from these formulas will be considered in §4.11. From (4.10.8) and (4.10.9) it follows that if C = A and D = B, then we have the transformation formula
if C
(aldl , ... , aAdl , bI/dl , ... , bBldl ; q)oo dm (cldl , ... , cAdI, d2/dl, ... , dBI dl ; q)oo 1 rI,
XA+B'I'A+B-l
[Cldl, ... ,CAdl,qdI/bl, ... ,qdI/bB. bl ... bBqm ] al d 1,···aA d l,q did 1 2,···,qdid 1 B ,q, d 1··· d B
+ idem (d l ; d 2 , ... , dB) =
(blCl, ... , bBCl, aI/cl, ... , aAlcl; q)oo -m ~ (dlCl, ... , dBCl, c2/cl, ... , cAlcl; q)oo
128
Basic contour integrals
'" [dICI, ... ,dBCI,qcI/al, ... ,qcI/aA. al ... a Aq - m ] XA+B 'f'A+B-I bICI,···, bBCI,qCI / C2,···,qCI / CA' q, CI···CA + idem (CI; C2, ... , CA) (4.10.10) provided that Ibl ·· ·bBqml < Idl ·· ·dBI, lal·· ·aAq-ml < ICI·· ·cAI and m = 0,±1,±2, .... In some applications it is useful to have a variable Z in the argument of the series which is independent of the parameters in the series. This can be accomplished by replacing A by A + 1, B by B + 1 and setting bB + I = Z and aA+I = q/z in (4.10.10). More generally, doing this to the m = 0 case of (4.10.5) and of (4.10.6) gives the rather general transformation formula
(aldl , ... , aAdl , bI/dl , ... , bB/dl , z/dl , qdI/ z; q)(XJ (cld l , ... , CedI, d 2/d l , ... , dD/d l ; q)(XJ
x
f
(cldl , ... , CedI, qdI/bl , ... , qdI/bB ; q)n
n=O (q, aldl , ... , aAdl , qdI/d2, ... , qdI/dD; q)n
x (_d l q(n+I J/2)n(D-B-IJ (bl ... bBZ) n dl
··
·dD
+ idem (d l ; d2, ... , d D ) (bl CI, ... , bBCI, aI/ CI, ... , aA/ CI, CIZ, q/ CIZ; q)(XJ (dICI, ... , dDCI, C2/CI, ... , Ce /CI; q)(XJ
x
f
(dICI, ... , dDCI, qCI/al, .. . , qcI/aA; q)n n=O (q, bICI, ... , bBCI, qcI/c2, ... , qcI/ce; q)n
(_Clq(n+I J/2)n(e-A-IJ (a l ... aAq)n CI··· CeZ + idem (CI; C2, ... , ce), X
(4.10.11)
where, for convergence,
(i)
D >B
+ 1,
or D
bl ... bBz I
= B + 1 and I d l
...
dD
<1
and (ii)
C > A + 1, or C = A + 1 and l
al .. . aAql CI··· CeZ
<
l.
This is formula (5.2.20) in Slater [1966]. Observe that by replacing (4.10.11) by zqm and using the identity
Z
in
(4.10.12) we obtain from (4.10.11) the formula that would have been derived by using (4.10.5) and (4.10.6) with m an arbitrary integer.
4.11 Some additional extensions of the beta integral
129
4.11 Some additional extensions of the beta integral Askey and Roy [1986] used Ramanujan's summation formula (2.10.17) to show that 1 (ce iO /(3, qe iO / ca, cae- iO , q(3e- iO / c; q) 00 dB
j7r 21f -7r
(ae'O , be'o, ae-'o, (3e -,0; q) 00
(aba(3, c, q/c, cal (3, q(3/ca; q)oo (aa, a(3, ba, b(3, q; q)oo
where max(lql, lal, Ibl, lal, 1(31) contour integral form
r
~
21f21K
(4.11.1)
< 1 and ca(3 =I- 0; and they extended it to the
(cz/(3,qz/ca,ca/z,q(3/cz;q)oo dz (az, bz,a/z,(3/z; q)oo z
(aba(3, c, q/ c, cal (3, q(3/ ca; q)oo (aa, a(3, ba, b(3, q; q)oo
(4.11.2)
where aa, a(3, ba, b(3 =I- q-n, n = 0,1,2, ... ,ca(3 =I- 0, and K is a deformation of the unit circle as described in §4.9. These formulas can also be derived from the A = B = D = 2, m = 0 case of (4.10.8) by setting al = c/ (3, a2 = q/ca, bl = ca, b2 = q(3/c, CI = a, C2 = b, d l = a and d 2 = (3 and then using the summation formula (2.10.13) for the sum of the two 2¢1 series resulting on the right side. In Askey and Roy [1986] it is also shown how Barnes' beta integral (4.1.2) can be obtained as a limit case of (4.11.1). Analogously, application of the summation formula (2.10.11) to the A = 3, B = D = 2, m = 0 case of (4.10.8) gives ~
r
21f2 1 K
(8z,qzh,1'z/a(3,1'/z,qa(3hz;q)00 dz (az, bz, cz, a/ z, (3/ z; q)oo z
b /a, qah, 1'/(3, q(3h, 8/a, 8/b, 8/c; q)oo (aa, a(3, ba, b(3, ca, c(3, q; q)oo
where 8
=
(4.11.3)
abca(3, abca(31' =I- 0, and aa, a(3, ba, b(3, ca, c(3 =I- q-n,
n
= 0,1,2, ...
Note that (4.11.2) follows from the c -+ 0 case of (4.11.3). In addition, application of Bailey's summation formula (2.11.7) gives the more general formula
1
r (za!, -za!, qaz/b, qaz/c, qaz/d, qaz/ j; q) ( b z, cz, dz, j z; q ) qzal I -qza
21fi 1 K
2 ,
2 ,
00
00
(qzh, v/ a (3, 1'/ z, qa(3hz; q)oo dz (aaz,a(3z,a/z,(3/z;q)oo z (1'/a,qa/1',1'/(3,q(3/1',aq/cd,aq/bd,aq/bc,aq/bj,aq/cj,aq/dj ;q)oo (aa(3, ba, ca, da, ja, b(3, c(3, d(3, j(3, q; q)oo (4.11.4)
x
~~--~~--~~~~--~
130
Basic contour integrals
where aq = bedfa{3, bedfa{3, -=I- 0, aa{3, ba, ea, da, fa, b{3, e{3, d{3, f {3 -=I- q-n,
n
= 0,1,2, ... ,
and K is as described in §4.9; see Gasper [1989c]. 4.12 Sears' transformations of well-poised series Sears [1951d, (7.2)] used series manipulations of well-poised series to derive the transformation formula (qal/aM+2, ... , qal/a2M, q/aM+2, ... , q/a2M, af, -af, q/af, -q/af; q) 00
(al, ... , aM+l, a2/al, ... , aM+l/al; q)oo
~
X2M'f/2M-l
[ a l ,a2, ... ,a2M . ] / / ,q,-x qal a2,···,qal a2M
(qa2/aM+2, ... , qa2/a2M, qal/a2 aM+2, ... , qal/a2 a2M; q)oo
= a2(al/a2,a2,a3/a2, ... ,aM+l/a2,a§/al,a2a3/al, ... ,a2aM+l /al;q)oo x (af /a2, -af /a2, qa2/a f, -qa2/a f; q) 00
~
X2M'f/2M-l
[a§/al,a2,a2adal, ... ,a2a2M/al. ] / / / ,q,-x qa2 al,qa2 a3,···,qa2 a2M
+ idem (a2; a3,···, aM+l),
(4.12.1)
where x = (qad M /ala2··· a2M. Slater [1952c] observed that this formula could also be derived from (4.10.10) by taking A = B = M + 1, m = 1, choosing the parameters such that P(z) in (4.9.1) becomes (qa l z/aM+2, ... , qalz/a2M, qzaf, -qzaf; q) 00 (alz, ... , aM+lz; q)oo
x
(q/zaM+2, ... , q/za2M, l/zaf, -1/zaf; q) 00
(l/z,a2/ zal, ... ,aM+l/zal;q)00
'
(4.12.2)
and then using the fact that (a, -a, q/a, -q/a; q)oo - a 2(qa, -qa, l/a, -1/a; q)oo
= 2(a, -a, q/a, -q/a; q)oo
(4.12.3)
to combine the terms with the same 2M¢2M-l series. Similarly, taking A = B = M +2 and m = 1 in (4.10.10) and choosing the parameters such that P(z) in (4.9.1) becomes (qalz/aM+3, ... , qalz/a2M, qzaf, -qzaf, z(qad!, -Z(qal)!' q) 00 (alz, ... , aM+2 Z; q)oo (q/zaM+3, ... , q/za2M, l/zaf, -1/za f, q! /zaf, -q! /zaf; q) 00
x
~----------~~--~----------~--~----------~
(1/ z, a2/ zal, ... , aM+2/ zal; q)oo
(4.12.4)
4.12 Sears' transformations of well-poised series
131
we obtain
(qal/aM+3, ... , qal/a2M, q/aM+3, ... , q/a2M, q/al; q)= (a2' ... ' aM+2, adal, ... , aM+2/ a l; q)= [aI, a2, ... , a2M
A..
x 2M\f'2M-l qal / a2,···,qal / a2M ;q,x
= a2
]
(qa2/aM+3, ... , qa2/a2M, qal/a2aM+3, ... , qal/a2a2M; q)= (al/a2,a3/a2, ... ,aM+2/a2,a2a3/al, ... ,a2aM+2/al;q)=
(al/a§, qa§!al; q)= A.. [a§/al, a2, a2a3/al, ... , a2 a2M/al. ] (a2, a 2/ al; q)= 2M\f'2M-l qa2/ aI, qa2/ a3,···, qa2/ a2M ,q,x 2 + idem (a2; a3, ... , aM+2), (4.12.5) x
where x = (qat)M /al ... a2M, which is formula (7.3) in Sears [1951d]. Finally, if we take A = B = M +2 and m = 1 in (4.10.10) and choose the parameters such that P(z) in (4.9.1) becomes
(qalz/aM+3, ... , qalz/a2M+1, qzaf, -qzaf, ±q! zaf; q) = (alz, ... , aM+2z; q)= x
(q/zaM+3, ... , q/za2M+l, l/zaf, -l/zaf, ±q! /zaf ;q) = (/ / /) , (4.12.6) 1 z,a2 zal, ... ,aM+2 zal;q =
we obtain
(qal/aM+3, ... , qal/a2M+l, q/aM+3, ... , q/a2M+1; q)= (al' ... ' aM+2, a2/ a l, ... , aM+2/ a l; q)= x (af, -af, q/af, -q/af, ±(alq)!, ±(q/at)!; q) =
[al,a2, ... ,a2M+l ] / / ; q, ~y qal a2,···, qal a2M+l (qa2/aM+3, ... , qa2/a2M+l, qal/a2aM+3, ... , qal/a2a2M+l; q)= = a2(al/a2,a3/a2, ... ,aM+2/a2,a2,a§/al,a2a3/al, ... ,a2aM+2/al;q)= A..
x 2M+l\f'2M
x (af /a2, -af /a2, qa2/af, -qa2/af, ±a2(q/at)!, ±(qat)! /a2; q) =
a§/al,a2,a2a3/al, ... ,a2a2M+l/al x 2M+l¢2M [
;q,H]
qa2/al,qa2/a3, ... ,qa2/a2M+l (4.12.7) where y [1951d].
= (qal)M+! /al ... a2M+l, which are formulas (7.4) and (7.5) in Sears
132
Basic contour integrals Exercises
4.1 Let Re e > 0, Re d beta integral
_1_liOO 27ri
> 0, and Re(x + y) > 1. Show that Cauchy's [1825]
(1
-ioo
ds = r(x + y - 1)(1 + d/e)l- Y + C/d)l-x (1 + es)x(1- ds)Y (e + d)r(x)r(y)
has a q-analogue of the form
liOO
(-esqX,dsqY;q)oo ds (-es, ds; q)oo r q(x + y - 1) (_eqX /d, -dqY /e; q)oo r q(x)r q(Y) (e + d)( -eq/d, -dq/e; q)oo'
1 27ri
-ioo
°
where < q < 1. (Wilson [1985]) 4.2 Prove that
1 27ri
liOO
(q1+s, _qa+s, qa-b+1+s, _qa-b+1+s, qc+d+e-a+s; q) 00
(qc+s, qd+s, qe+s, _qa+1+s, _qa-2b+s; q)oo 7rqs ds x ~--~--~~~------~ sin 7r s sin 7r (e + d + e - a + s) (q,qc+d+e-a, q1+a-c-d-e,q1+a-b; q)oo =csc7r(e+d+e-a) (q1+ a, _q 1+ a, _qa -2b , qC; q ) 00 -ioo
(_q1+a-b, _qa, q1+a-c, q1+a-d, q1+a-e; q) 00 X
~~--~----~~------~--~--~=
XlO
(qd,qe,q1+a-c-d, q1+a-c-e, q1+a-d-e; q)oo
W g (qa; i q1+a/2, _i q1+a/2, l, _qb, qC, qd, qe; q, _q1+2a-2b-C-d-e) ,
where 1 + 2a - 2b
> e + d + e.
4.3 Show that
ae, be, ad ] 3cP2 [ b dh;q,gh a eg,ae (q, ae, be, ag, bg, eh; q)oo
133
Exercises 4.4 Prove that ~ 27r
j1l"
(Je iO , ke iO Id, qde- iO Ik, cke- iO , qe iO Ick, abcdghe iO I j; q) 00 dB (ae iO be iO ce- iO de- iO ge iO heiO . q)
-7r
" " "
00
(k,qlk,ckld,qdlck,cj,dj,acdg, bcdg, cdgh,abcdhlj; q)oo (q, ac, ad, bc, bd, cg, dg, ch, dh, cdjg; q)oo x 8 W 7 (cdjglq; cg, dg, j la, jib, j Ih; q, abcdhl1)·
4.5 Prove that
[
.,.f...
4'1-'3
q-n , abcdqn-l " ae iO ae- iO ab, ac, ad 2
(q;q)oo (q!, q!, ab, ac, bc; q)oo
1l" (q! eiHi I a, q! ae-i(}-i , aq! eiO - i , q! ei-iO I a, abcei I a; q) 00
1 X
27r { }
x
-7[
(ae i la bei la ce i la ae iO - i ae- iO - i. q) 00 " ' "
(dae-i,bc;q)n (~eird¢ (abce i la, ad; q) n a
and, more generally, (q,az,alz,bz, blz,cz, clz;q)oo (t;;, ql t;;, t;; Z 2, ql t;; Z 2, ab, ac, bc; q)oo 1 {1l" (t;;ze i la, qae- i It;;z, t;;aze- i , qei It;;az, abce i la; q)oo x 27r ) -1l" X
for n
(ae i I a, be i I a, cei I a, aze- i, ae- i I Z; q)oo
(daci,bc;q) (a .",)n n -e''I' d¢ i (abce la, ad; q)n a
= 0,1,2, ... , where
z
=
ei()
and
t;;
is an arbitrary parameter.
4.6 Prove that
(aqle,aqlj,aqlg,aqlh,qlae,qlaj,qlag,qlah;q)oo (qa2, ab, ac, ad, bla, cia, dla; q)oo x lOW9 (a 2; ab, ac, ad, ae, aj, ag, ah; q, q3labcdejgh)
-1
a
+ idem (a; b, c, d) = 0, where Iq31
< labcdejghl.
134
Basic contour integrals
4.7 Prove that _l(alq/bl,alq/b2, ... ,alq/br,q/albl, ... ,q/albr;q)OO (qai, ala2,···, alan a2/al, ... , ar/al; q)oo
al
X
2r+2 W 2r +l (ai; ala2, ... , alan alb l , ... , alb r ; q, qr-l /al ... arb l ... br )
+ idem (al; a2,···, ar) = 0, where r
1,2, ... , and Iqr-ll
=
< lal ... arb l ... brl.
4.8 Show that _ 1 lioo (_csqn+l ' bds , as·, q) oods 27ri -ioo (-CS, ds, basqn-l; q)oo (-a/ c, -bd/ c; q)oo (b, a/ d; q)n (C + d)( -dq/c, -ba/cq; q)oo (q, -cq/d; q)n'
°
where Re(c, d, ba) > and n = 0,1, .... Show that the q-Cauchy beta integral in Ex. 4.1 follows from this formula by letting n ----+ 00 and then setting b = qY, a = _cqx. 4.9 Extend the integral in Ex. 4.8 to 1 lioo (-acs, ac2s/a, ac 2s/ (3, ac2s/'Y, ac2S/6, ac2s/ A; q)oo 27ri -ioo (-cs, as, (3s, "Is, 6s, AS; q)oo
x
(1- a2~s2)
ds
(a/ q, -ac/ a, -ac/ (3, -ach, -ac/6, -ac/ A; q)oo c(q, -a/c, -(3/c, -'Y/c, -6/c, -A/C; q)oo (q2/a , ac2/a(3, ac2/a'Y, ac2/(3"1; q)n X ~~7---~~----~~~~~~~(-cq/a, -cq/ (3, -cqh, -a 2c3 /a(3'Yq; q)n'
where Re(c,a,(3,'Y,6,A) 1,2 ....
>
0, a 3 c5 = -a(3'Y6Aq2, ac = -Aqn+l, and n
4.10 Show that
r
~ (q2z/aa'Y,q2z/ba'Y,q2z/ca'Y,'Y/z;q)oo 1- qz2/a'Y dz 27r2 ) K (az, bz, cz, a/ z; q)oo z
(a'Y, b'Y, c'Y, aqh; q)oo (aa, ba, ca, q; q)oo '
where q2
=
abcaq2, I'Y/al
< 1, and the contour K is as defined in §4.9.
=
Notes 4.11 Show that ~
r (bqz,qzh,'"'(lz;q)oo
JK
27r~
135
(az, bz, al z; q)oo
dz x (qzhqm; q)m(alz; q)ml ... (arz; q)mr Z
b la, aqh, bqla; q)oo (aa, qlaa, ba; q)oo x (qlb'"'(qm; q)m(al/b; q)ml ... (arlb; q)mr (ba)m+ml +.+m r ,
provided
1'"'(1 a < 1, where 1
m, ml, ... ,mr are nonnegative integers, q =
a'"'(qm+ml +·+m r and K is as defined in §4.9. 4.12 Show that 1
-. 27r~
jiOO -ioo
(-acs, dqs; q)oo ( ) ( ) ( d .) al s; q ml . .. ars; q mr ds -cs, S, q 00
(-acld;q)oo
(I)
(I)
(c + d)( -cqld; q)oo al d; q ml ... ar d; q mr provided laq-(m1+·+mr)1 are nonnegative integers.
< 1, Re (c,d,al,···,a r ) > 0, and ml, ... ,mr
4.13 Show that
jiOO
(-acs,-ac 2slf,ac2sla,ac 2sl/3;q)00 27ri -ioo (-cs, - f s, as, /3s; q)oo 2 2 x (ac sh ,ac s15;q)00(1 - ac2 s 21 q) ds ('"'(s, 5s; q)oo (alq, acl f, -acla, -acl/3, -ach, -acI5; q)oo c(q, f Ic, -alc, -/3lc, -'"'( Ic, -5Ic; q)oo (q2 la, ac2la/3, ac2la,",(, ac21/3'"'(; q)n x ~~~~~~~-7~~~~~~~ (-cqla, -cql/3, -cqh, -a2c3 Ia/3'"'(q; q)n (ac 2I Pq, acl f, -ac2I fa, -ac2I f/3, -ac2I f'"'(, -ac2I j8; q)oo +~~~~~~~~~~~~~~~~~~~~~ f(q, cl f, -al f, -/31 f, -'"'(If, -51 f; q)oo (P q2lac 2, ac2/a/3, ac2/a'"'(, ac21/3'"'(; q)n x ~~~--~~~~~~~~~~~~ (- fqla, - fql /3, - fqh, -a2c4 I fa/3,",(q; q)n provided ac = f qn+l, a3 c5 = fa/3'"'(5q2, Re (c, f, a, /3, ,",(, 5) > 0, the integrand has only simple poles, and n = 0,1, .... 1
(For the formulas in Exercises 4.8 - 4.13, and related formulas, see Gasper [1989c].) Notes
§4.4 Kalnins and Miller [1988, 1989] exploited symmetry (recurrence relation) techniques similar to those used by Nikiforov and Suslov [1986], Nikiforov,
136
Basic contour integrals
Suslov and Uvarov [1991]' and Nikiforov and Uvarov [1988] to give another proof of (4.4.3) and of (4.11.1). §4.6 Contour integrals of the types considered in this section were used by Agarwal [1953c] to give simple proofs of the two-term and three-term transformation formulas for 8¢7 series. §4.12 Sears [1951b] also derived the hypergeometric limit cases of the transformation formulas in this section. Applications of (4.12.1) to some formulas in partition theory are given in M. Jackson [1949]. Exercises 4.1-4.5 For additional q-beta integrals, see Askey [1988a,b, 198ge], Gasper [1989c], and Rahman and Suslov [1994a,b, 1996a, 1998].
5 BILATERAL BASIC HYPERGEOMETRIC SERIES
5.1 Notations and definitions The general bilateral basic hypergeometric series in base q with r numerator and s denominator parameters is defined by
r1/Js(z)
== r1/Js [ab1,ab2""'bar;q,z]
f
1, 2,···,
s
(a1, a2,···, ar ; q)n (_I)(s-rlnq(s-rlG) zn. n=-oo (b 1, b2,···, bs; q)n
(5.1.1)
In (5.1.1) it is assumed that q, z and the parameters are such that each term of the series is well-defined (i.e., the denominator factors are never zero, q =I=- 0 if s < r, and z =I=- 0 if negative powers of z occur). Note that a bilateral basic hypergeometric series is a series L~=-oo Vn such that Vo = 1 and vn+dvn is a rational function of qn. By applying (1.2.28) to the terms with negative n, we obtain that
r1/Js(z) =
f
(a1,a2, ... ,ar ;q)n (_I)(s-r ln q(s-rlG)zn ... ,bs;q)n n=O (b1,b2,
+
f
(5.1.2)
(q/b1, q/b2, ... , q/bs;.q)n (b 1'" bs ) n n=l (q/a1, q/a2,"" q/ar , q)n a1 ... arz
Let R = Ib 1 ··· bs /a1'" arl. If s < rand Iql < 1, then the first series on the right side of (5.1.2) diverges for z =I=- 0; if s < rand Iql > 1, then the first series converges for Izl < R and the second series converges for all z =I=- O. When r < sand Iql < 1 the first series converges for all z, but the second series converges only when Izl > R. If r < sand Iql > 1, the second series diverges for all z =I=- O. If r = s, which is the most important case, and Iql < 1, the first series converges when Izl < 1 and the second when Izl > R; on the other hand, if Iql > 1 the second series converges when Izl > 1 and the first when Izl < R. We shall assume throughout this chapter that Iql < 1, so that the region of convergence of the bilateral series
(5.1.3)
137
138
Bilateral basic hypergeometric series
is the annulus (5.1.4) When bj = q for some j, the second series on the right sides of (5.1.2) and (5.1.3) vanish and the first series become basic hypergeometric series. If we replace the index of summation n in (5.1.1) by k + n, where k is an integer, then it follows that
= (al, ... ,ar;q)k z k [(_I)kq(~)]s-r (b l
X
, ... ,
r'l/Js [
bs ; qh
k(S-r J] al qk, ... , arqk k k ; q, zq . blq , ... ,bsq
(5.1.5)
When rand s are small we shall frequently use the single-line notation
r'l/Js(z) == r'l/Js(al,"" ar ; bl , ... , bs; q, z). An r'l/Jr series will be called well-poised if alb l = a2b2 = ... = arbn and verywell-poised if it is well-poised and al = -a2 = qb l = -qb 2. Corresponding to the definition of a VWP-balanced r+l¢r series in §2.1, we call a very-wellpoised r'l/Jr series very-well-poised-balanced (VWP-balanced) if (5.1.6) with either the plus or minus sign. The very-well-poised bilateral basic hypergeometric series in (5.3.1), (5.5.2), (5.5.3), and in §5.6 are VWP-balanced. Note 1
that if in (5.1.6) we replace r by r+l, al by qar, and then a3 by aI, then (5.1.6) reduces to the condition (2.1.12) for a r+l Wr series to be VWP-balanced. A well-poised r'l/Jr series will be called well-poised-balanced (WP-balanced) if
(ala2'" ar)z = -(±(alb1)br
(5.1.7)
with either the plus or minus sign. The well-poised bilateral basic hypergeometric series in (5.3.3), (5.3.4), (5.5.1), (5.5.4), and in (5.5.5) are WP-balanced. 5.2 Ramanujan's sum for l'l/Jl(a;b;q,z)
The bilateral summation formula oj,
1 '1-'1
('b' )_ (q,bja,az,qjaz;q)cxo a, ,q, z - ( b,qj a,z,b j az;q ) cxo ,
Ibjal <
Izl < 1,
(5.2.1)
which is an extension of the q-binomial formula (1.3.2), was first given by Ramanujan (see Hardy [1940]). In Chapter 2 we saw that this formula follows as a special case of Sears' 3¢2 summation formula (2.10.12). Andrews [1969, 1970a], Hahn [1949b]' M. Jackson [1950b], Ismail [1977] and Andrews and Askey [1978] published different proofs of (5.2.1). The proof given here is due to Andrews and Askey [1978].
139
5.2 Ramanujan's sum for l'l/Jl(a;b;q,z)
The first step is to regard 1 'l/Jl (a; b; q, z) as a function of b, say, f (b). Then
f(b)
l'l/Jl(a;b;q,z)
= =
f
f
(5.2.2)
(a; q)n zn + (q/b; q)n (b/azt n=O (b; q)n n=l (q/a; q)n
so that, by (5.1.4), the two series are convergent when Ib/al < Izl function of b, f(b) is clearly analytic for Ibl < min(l, lazl) when Izl
=
< 1. As a < 1. Since
f
(a; q)n+l zn n=-oo (b; q)n
= z-l(l- b/q)
f
(a;q)n+l zn+l n=-oo (b/q; q)n+l
= z-l(l- b/q) l'l/Jl(a; b/q; q, z), we get
f(bq) - z-l(l- b)f(b) = a l'l/Jl(a;bq;q,qz).
(5.2.3)
However,
a l'l/Jl(a;bq;q,qz) =a =
=
~ ~
(a;q)n ( )n qz n=-oo (bq; q)n
f
(5.2.4)
(a; q)n(1- bqn - 1) zn n=-oo (bq; q)n -ab- 1(1- b)f(b) + ab- 1f(bq). -ab-1
Combining (5.2.3) and (5.2.4) gives the functional equation
(1- ab-1)f(bq) = (1- b)(Z-l - ab-1)f(b), that is,
f(b)
=
1- b/a (1 _ b)(l _ b/az) f(bq).
(5.2.5)
Iterating (5.2.5) n - 1 times we get
f(b) = Since f(b) is analytic for Ibl
(b/a; q)n f(bqn). (b, b/az; q)n
< min(l, lazl), by letting n
f(b)
=
(b/a; q)oo f(O). (b, b/az; q)oo
(5.2.6) ----+ 00
we obtain (5.2.7)
140
Bilateral basic hypergeometric series
However, since
f(q) =
f=
(a; q)n zn = (az; q)oo n=O (q; q)n (z; q)oo
by (1.3.2), on setting b = q in (5.2.7) we find that
f(O) = (q, q/az; q)oo f(q) = (q, q/az, az; q)oo. (q/ a; q)oo (q/ a, z; q)oo Substituting this in (5.2.7) we obtain formula (5.2.1). Jacobi's triple-product identity (1.6.1) is a limit case of Ramanujan's sum. First replace a and z in (5.2.1) by a-I and az, respectively, to obtain
f=
(a-\q)n (az)n n=-oo (b; q)n
= (q,ab,z,q/z;q)oo ,
(5.2.8)
(b, aq, az, b/ z; q)oo
when Ibl < Izl < la-II. Now set b = 0, replace q by q2, a ---+ 0 to get (1.6.1).
Z
by zq, and then take
5.3 Bailey's sum of a very-well-poised 6'ljJ6 series
Bailey [1936] proved that 'ljJ [ 6
6
qa~,-qa~,b,c,d,e
. qa 2 ] a~, -a~, aq/b, aq/c, aq/d, aq/e ,q, bcde
(5.3.1)
(aq, aq/bc, aq/bd, aq/be, aq/cd, aq/ce,aq/de, q,q/a; q)oo (aq/b, aq/c, aq/d, aq/e, q/b, q/c, q/d, q/e,qa 2/bcde; q)oo ' provided Iqa 2/bcdel < 1. Since this VWP-balanced 6'ljJ6 reduces to a VWPbalanced 6¢5 series when one of the parameters b, c, d, e equals a, (5.3.1) can be regarded as an extension of the 6¢5 summation formula (2.7.1). There are several known proofs of (5.3.1). Bailey's proof depends crucially on the identity
(aq/d,aq/e,aq/f,q/ad,q/ae,q/af;q)oo a(qa 2, ab, ac, bfa, cia; q)oo x 8 W 7 (a 2; ab, ac, ad, ae, af; q, q2/abcdef)
(5.3.2)
+
idem (a; b, c)
=
0,
when Iq2/abcdefl < 1, which is easily proved by using the q-integral representation (2.10.19) of an 8¢7 series (see Exercise 2.15). If we set c = q/a, the first and third series in (5.3.2) combine to give, via (5.1.3),
(aq/d,aq/e,aq/f, q/ad,q/ae,q/af; q)oo a(qa 2,q/a2, q,ab, b/a; q)oo qa, -qa, ab, ad, ae, af q ] [ x 6'ljJ6 a, -a, aq / b, aq / d, aq / e, aq /f; q, bde f
'
5.4 A general transformation formula
141
while, by (2.7.1), the second series reduces to
6¢5
[
b2, bq, -bq, bd, be, bj q] b, -b, bqjd, bqje, bqj j; q, bdej
(qb 2, qjde,qjdj,qjej; q)oo (bqjd, bqje, bqj j, qjbdej; q)oo . This gives (5.3.1) after we replace a 2, ab, ad, ae, aj by a, b, e, d, e, respectively, and use the same square root of a everywhere. Slater and Lakin [1956] gave a proof of (5.3.1) via a Barnes type integral and a second proof via a q-difference operator. Andrews [1974a] gave a simpler proof and Askey [1984c] showed that it can be obtained from a simple difference equation. The simplest proof was given by Askey and Ismail [1979] who only used the 6¢5 sum (2.7.1) and an argument based on the properties of analytic functions. Setting e oj,
4 '/-'4
=
a!
[-a
in (5.3.1), we obtain
3]
1
-qa 2 , b, e, d . qa 2 ,q,2 , aqjb, aqje, aqjd bed 1
(aq,aqjbe,aqjbd,aqjed,qa~jb,qa~je,qa~jd,q,qja;q)oo
(aqjb, aqje, aqjd, qjb, qje, qjd, qa!, qa-~, qd jbed; q)oo provided Iqd jbedl
(5.3.3)
< 1. This is an extension of the q-Dixon formula (2.7.2). 1
1
If we set d = a 2 , e = -a 2 in (5.3.1) and simplify, we get the sum of a WP-balanced 2'l/J2 series
2'l/J2(b, e; aqjb, aqje; q, -aqjbe) (aqjbe; q)oo(aq2 jb2, aq2 je2, q2, aq, qja; q2)oo (aqjb, aqje, qjb, qje, -aqjbe; q)oo
I~~I < 1.
(5.3.4)
5.4 A general transformation formula for an r'l/Jr series In this section we shall derive a transformation formula for an r'l/Jr series from those for r+l¢r series in Chapter 4. First observe that (5.1.3) gives
(5.4.1)
142
Bilateral basic hypergeometric series
In (4.10.11) let us now make the following specialization of the parameters
D=B+1, A=B=r, C = A+ 1, cldl = C2 d2 = ... = CA+ldA+l = q, qdI/bl = 0:1, qdI/b2 = 0:2,"" qdI/bA = O:A, aldl = (31, a2 dl = (32, ... , aAdl = (3A, bl ... bAZ ----=x. d l ... dA+l
(5.4.2)
Then, combining the pairs ofthe resulting r+l¢r series in (4.10.11) via (5.4.1), simplifying the coefficients and relabelling the parameters, we obtain Slater's [1952b, (4)] transformation formula
(b l , b2, ... , br , q/al, q/a2,"" q/ar , dz, q/dz; q)oo r'¢r [aI, a2,···, ar ; q, z] (Cl,C2, ...
bl , b2, ... ,br
,Cr ,q/Cl,q/C2,'" , q/cr ;q)oo
q (Cl/al,Cl/a2"",Cl/ar,qbl/Cl,qb2/Cl, ... ,qbr/Cl,dclZ/q, q2/dclZ;q)oo Cl (Cl,q/Cl,Cl/C2,'" ,Cl/Cr,qC2/Cl," .,qCr/Cl;q)oo qaI/cl, qa2/Cl,"" qar/Cl ] ; q, Z qbl /Cl,qb2/Cl,'" ,qbr/Cl idem (Cl; C2, ... , Cr ),
r'¢r [
X
+
(5.4.3)
... br I < Izl < l. al'" ar Note that the c's are absent in the r'¢r series on the left side of (5.4.3). This gives us the freedom to choose the c's in any convenient way. For example, if we set Cj = qaj, where j is an integer between 1 and r, then the lh series on the right becomes an r¢r-l series. So if we set Cj = qaj, j = 1,2, ... , r, in (5.4.3), then we get an expansion of an r'¢r series in terms of r r¢r-l series:
where d
= ala2'" ar /clc2'" CTl
I bl
(b l , b2, ... , bTl q/al, q/a2,"" q/aTl z, q/z; q)oo '¢ [aI, a2,···, ar .
] (qal, qa2, ... , qar, l/al, 1/a2,"" l/a r ; q)oo r rbI, b2, ... , br ' q, Z a~-1(q,qal/a2, ... ,qal/ar,bl/al,b2/al, ... ,br/al,alz,q/alz;q)oo (qal,1/al,al/a2, ... ,al/ar,qa2/al, ... ,qar/al;q)oo
x
r¢r-l
[
qaI/bl, qaI/b2, ... , qaI/br bl ··· br ] ; q, - - - qal/a2, ... ,qal/ar al···arz
+ idem (al; a2,"" ar), (5.4.4) provided I bl ... br I < al ... ar
Izl < l.
On the other hand, if we set Cj
=
bj , j
=
1,2, ... , r in (5.4.3), then we
143
5.5 A transformation formula for a 2r'IjJ2r series obtain the expansion formula
(q/al,q/a2, ... ,q/an dz,q/dz;q)oo 'IjJ [aI, a2, ... ,ar . ] (q/b l ,q/b2, ... ,q/br ;q)oo r rbI, b2, ... ,br ,q,z q (q,bl/al,bl/a2, ... ,bl/ar,dbIZ/q,q2/dblz;q)oo bl (bl ,q/bl ,bl /b2, ... ,bl/br;q)oo x r¢r-l [ with d
(5.4.5)
qadbl, ... ,qar/bl ]. / / ; q, z + Idem (bl ; b2,···, br ), qb 2 bl, ... ,qbr bl
= ala2'" ar/b l b2 ··· br . 5.5 A general transformation formula for a very-well-poised 2r'IjJ2r series
Using (4.12.1) and (5.4.1) as in §5.4, we obtain Slater's [1952bj expansion of a WP-balanced 2r'IjJ2r series in terms of r other WP-balanced 2r'IjJ2r series: 1
1
1
1
(q/b l , ... , q/b 2n aq/h, ... , aq/b 2n a"2, -a"2, qa-"2, -qa-"2; q)oo
x (ar/a ,qa/ar,ala2/a , ... ,alar /a,qa/ala2, ... ,qa/alar ;q)oo albl/a, alb2/a, ... ,alb2r/a arqr ] [ X 2r 'ljJ2r / / / ; q, - b b alq bl , alq b2,···, alq b2r I'" 2r +idem(al;a2, ... ,ar ).
(5.5.1)
For the very-well-poised case when al = bl = qa!, a2 = b2 = -qa!, the first two terms on the right side vanish and we get
(q/b 3, ... , q/b 2r , aq/b3, ... , aq/b2r ; q)oo (aq,q/a,a3, ... ,ar ,q/a3, ... ,q/ar ;q)oo
x (a3/a, ... , aria, aq/a3, ... , aq/a r ; q)~l of,
X
2r'/'2r
[
qa !2 ,-qa !2 , b3,···, b2r ar-l qr-2] 1 ; q, - - - a"2, -a"2, aq/b 3, ... , aq/b 2r b3 ··· b2r 1
(a3q/b3, ... ,a3q/b2r,aq/a3b3, ... ,aq/a3b2r;q)oo (a3,q/a3,a3/a ,aq/a3,qa§/a,aq/a§;q)oo
x (a4/a3, ... ,ar/a3,qa3/a4, ... ,qa3/ar;q)~1 x (a3a4/a, ... , a3ar/a, aq/a3 a4,"" aq/a3 ar; q)~l
144
Bilateral basic hypergeometric series (5.5.2)
In particular, for r = 3 we have the following transformation formula for VWPbalanced 6 '¢6 series
(q/b 3, ... ,q/b6,aq/b3, ... ,aq/b6;q)oo (aq,q/a,a3,q/a3,a3/ a,aq/a3;q)oo qa!, -qa!, b3, b4, b5, b6 a2q ] 1 ; q, b b b b a"2, -a"2, aq/b 3, aq/b4, aq/b5, aq/b6 3 4 5 6 (a3q/b 3, ... ,a3q/b6,aq/a3b3, ... ,aq/a3b6;q)oo .J,
x
[
6 '/-"6
1
(a3,q/a3,a3/a,aq/a3,qa~/a,aq/a~;q)oo
x
6'¢6
[
1
a2q qa3a-!, -qa3a-!, a3 b3/a, a3 b4/a, a3 b5/a, a3 b6/a 1 1 ; q, b b b b . a3a-"2, -a3a-"2, qadb3, qa3/b4, qa3/b5, qa3/b6 3 4 5 6 (5.5.3)
If we now set a3 sum
= b6, then the
6'¢6
series on the right side becomes a
6¢5
with
(qb~/a,aq/b3b4,aq/b3b5,aq/b4b5;q)oo
(qb 6/b 3, qb6/b 4, qb6/b 5, qa 2/b 3b4b5b6; q)oo . This provides another derivation of the 6'¢6 sum (5.3.1); see M. Jackson [1950a]. As in Slater [1952b], Sears' formulas (4.12.5) and (4.12.7) can be used to obtain the transformation formulas
(q/bl, ... ,q/b2r,aq/bl, ... ,aq/b2r;q)oo
(a2/al, ... ,ar+l/al,qal/a2, ... ,qal/ar+l,aq/aIa2, ... ,aq/aIar+l;q)oo x (al,q/al,aq/al,al/a,aIa2/a, ... ,aIar+l/a;q)~1 X
aIbda, aIb2/a, ... , aIb2r /a qra r ] 2r '¢2r [ aIq / bl , aIq / b2,···, aIq /b2r; q, bI b2'" b2r (5.5.4)
and
(q/b l , ... , q/b 2r - l , aq/b l , ... , aq/b2r - l ; q)oo (a,al, ... ,ar,q/a,q/al, ... ,q/ar;q)oo x
(a!, -a!, q/a!, -q/a!, ±(aq)!, ±(q/a)!; q)oo (al/a, ... ,ar/a,aq/al, ... ,aq/ar;q)oo
~~--~~--~~--~~~--~~~~-
5.6 Transformation formulas for 8'¢8 and 1O'¢1O series
145
al (al q/bl , ... ,al q/b2r- l , aq/ albl , ... ,aq/ al b2r- l ; q)oo (aq/ai,aq/ala2, ... ,aq/alar ,a2/a l, ... ,ar /al;q)oo (a! /al, -a! /al, alq/a!, -alq/a!; q)oo x (a/al,q/al,ai/a,ala2/ a, ... ,alar /a;q)oo (±(aq)! /al, ±al(q/a)!; q)oo (qal/a,al,qal/a2, ... ,qal/ar ;q)oo
x-,-----'--:-..:........::"----'---=.:,---=...:~--'----..:,.=...:....::....,-
X 2r-I'¢2r-1 [
al bI/a, ... ,al b2r_I/a qaI/b l , ... , qaI/b 2r - 1
=fqr-!a r -!] ;q, -----=--bl b2 ··· b2r- 1
+ idem (al; a2, ... , ar ). (5.5.5)
5.6 Transformation formulas for very-well-poised s'¢s and 1O'¢1O series In this section we consider two special cases of (5.5.2) that may be regarded as extensions of the transformation formulas for very-well-poised S¢7 and 1O¢9 series derived in Chapter 2. First, set r = 4 in (5.5.2) and replace b3, b4 , b5 , b6 , b7 , bs by b, e, d, e, j, g, respectively, choose a3 = j, a4 = 9 and simplify to get (aq/b,aq/e, aq/d, aq/e, q/b, q/e,q/d, q/e; q)oo (j,g,j/a,g/a,aq,q/a;q)oo of, [ qa!,-qa!,b,e,d,e,j,g . a3q2 ] x s~s 1 1 'q'------b d j a"2, -a"2, aq/b, aq/e, aq/d, aq/e, aq/ j, aq/g e e 9 (q,aq/bj,aq/ej,aq/dj,aq/ej,jq/b,jq/e, jq/d,jq/e; q)oo (j,q/j,aq/j,j/a,g/j,jg/a,qj2/a ;q)oo
x
S¢7
[
p la, qja-!, -qja-!, jb/a, je/a, jd/a, je/a, jg/a. a 3q2 ] 1 1 ,q, ------b d j ja-"2, - ja-"2, jq/b, jq/e, jq/d, jq/e,Jq/g e e 9
+ idem (f;g), where I
b:;:;9
I
<
(5.6.1) 1.
Replacing a, b, e, d, e, j, 9 by a 2, ba, ea, da, ea, j a, ga, respectively, we may rewrite (5.6.1) as
(aq/b, aq/e, aq/d, aq/e, q/ab, q/ae, q/ad, q/ae; q)oo (ja,ga, j/a,g/a, qa 2,q/a2; q)oo qa, -qa, ba, ea, da, ea, ja, ga q2 ] [ X s'¢s a, -a, aq/b, aq/e, aq/d, aq/e, aq/ j, aq/g ; q, bedejg (q,q/bj,q/ej,q/dj,q/ej,qj/b,qj/e, qj/d,qj/e; q)oo (ja, q/ja, aq/j,j/a,g/j,jg,qj2;q)oo ¢ [ X 8 7
p,qj,-qj,jb,je,jd,je,jg. q2] j, _ j,Jq/b, jq/e, jq/d, jq/e, jq/g' q, bedejg
146
Bilateral basic hypergeometric series + idem (f; g),
(5.6.2)
provided q2/bcdefgl < 1. Note that no a's appear in the 8¢7 series on the right side of (5.6.2). This is essentially the same as eq. (2.2) in M. Jackson [1950a]. For the next special case of (5.5.2) we take r = 5 and replace b3 , ... ,b lO by b, c, d, e, f, g, h, k, respectively, choose a3 = g, a4 = h, a5 = k, and finally, replace a, b, ... , k by a 2 , ba, ... , ka and simplify. This gives I
(aq/b, aq/c, aq/d, aq/e, aq/f,q/ab, q/ac, q/ad, q/ae,q/af; q)oo (ag,ah,ak,g/a,h/a, k/a, qa 2 ,q/a2; q)oo [ X 1O'¢1O
qa, -qa, ba, ca, da, ea, fa, ga, ha, ka
q3
]
a, -a, aq/b, aq/c, aq/d, aq/e, aq/ f, aq/g, aq/h, aq/k ; q, bcdefghk
(q, q/bg, q/cg, q/dg, q/eg, q/fg, qg/b, qg/c, qg/d, qg/e,qg/f;q )00 (gh,gk,k/g,h/g, ag, q/ag, g/a, aq/g, qg2; q)oo
x
[ 1O¢9
g2,qg,-qg,gb,gc,gd,ge,gf,gh,gk q3] g, _g, qg/b, qg/c, qg/d, qg/e, qg/ f, qg/h, qg/k ; q, bcdefghk
+ idem (g; h, k), (5.6.3) where q3/bcdefghkl < 1. Notice that all of the very-well-poised series in this section are VWP-balanced. I
Exercises 5.1 Show that
5.2 Letting c ----+
00
in (5.3.4) and setting a
= 1, b = -1, show that
1 +4 00 (_1)n qn(n+1)/2 1 + qn
~
5.3 In (5.3.1) set b = a, c
= [ (q;q)oo
]2
(-q; q)oo
= d = e = -1 and then let a ----+ 1 to show that
1 + 8 ~ (_q)n = [ (q; q)oo ] 4 ~ (1 + qn)2 (-q; q)oo See section 8.11 for applications of Exercises 5.1-5.3 to Number Theory. 5.4 Set b = c
= d = e = -1 and then let
a ----+ 1 in (5.3.1) to obtain
00 q2n(4 _ qn _ q-n)
1+16~
(1+qn)4
= [(q;:)00]8 (-q, q)oo
Exercises
147
5.5 Show that 00
L
q4n 2 z2n(1
+ zq4n+1) =
(q2, -zq, -q/ z; q2)00,
z yf O.
n=-oo
5.6 Prove the quintuple product identity 00
n=-CXJ
= (q, -Z, -q/ Z; q)00(qZ2, q/ Z2; q2)00,
z yf O.
See the Notes for this exercise. 5.7 Show that
q, 3 6 q,
q3 5 5 ] 6; q ,q . q
+
q5n+4} (1 - q5n+4 )2
Deduce that
{q5n+1 q5n+2 q5n+3 ~ (1 - q5n+1)2 - (1 - q5n+2)2 - (1 - q5n+3)2 00
= q
(q5; q5)~ ,Iql < l. (q; q)oo
See Andrews [1974a] for the above formulas. 5.8
Deduce (5.4.4) directly from (4.5.2).
5.9
Deduce (5.3.1) from (5.4.5) by using (2.7.1).
5.10 Show that
q (e/a, e/b, e/ab, qc/e, q2/ e , q2 I/e; q)oo ni, [e/c, e/q . ] (/ / /) 2 0/2 , q, q e,q e,e I,ql e;q 00 e/a,e/b (q,q/a,q/b, c/a,c/b, c/el,qel/c; q)oo (e, I, q/e, q/ I, c/ab; q)oo
-
e
'd
+1
(. I) em e,
5.11 Show that
1/J [ 8
8
qa~, -qa~, c, d, e, I, aq-n, q-n a2q2n+2] a~, -a ~ ,aq/ c, aq/ d, aq/ e, aq/ I, qn+l, aqn+1 ; q, cdel
(aq, q/a,aq/cd,aq/el; q)n (q/c, q/d, aq/e, aq/I; q)n X
] e, I, aqn+1 lcd, q-n 41/J4 [ aq / c,aq / d,qn +1 ,el / aqn ; q, q ,
n
= 0,1, ... ,
148
Bilateral basic hypergeometric series and deduce the limit cases [
2'lP2
e,j ; q, aq/ e, aq/ d
aq ] ej
(q/e,q/d, aq/e,aq/j; q)= (aq,q/a,aq/ed,aq/ej;q)=
f
X
(1- aq 2n)(e,d,e,j;q)n (qa 3 )n n 2 n=-= (1 - a)(aq/e, aq/d, aq/e, aq/ j; q)n edej q
and
which reduces to the first and second Rogers-Ramanujan identities when a = 1 and a = q, respectively. (Bailey [1950a] and Garrett, Ismail and Stanton [1999]) 5.12 Using (5.6.2) and (2.11.7), show that s'l/Js
[
qa,-qa,ab,ae,ad,ae,aj,ag ] a, -a, aq/b, aq/ e, aq/ d, aq/ e, aq/ j, aq/ 9 ; q, q
(q, qa 2, q/a 2, ag, g/a, q/be, q/bd,q/be,q/bj,q/ed,q/ee, q/ej; q)= (bg,eg,dg,eg,jg,aq/b,aq/e,aq/d,aq/e,aq/j,q/ab,q/ae;q)= (q/de, q/dj, q/ej; q)= x (q/ad, q/ae, q/aj; q)='
provided bedejg
= q and
(bj, q/bj, ej, q/ej, dj, q/dj, ej, q/ej, ag, q/ag, g/a, aq/g; q)= = 1. (bg,q/bg,eg, q/eg,dg, q/dg, eg, q/eg,aj,q/aj,j/a,aq/j; q)= Following Gosper [1988b], we may call this the bilateral Jackson formula. 5.13 Deduce from Ex. 5.12 the bilateral q-Saalschiitz formula
f
(a, b, e; q)n qn n=-= (d, e, j; q)n (q,d/a,d/b,d/e,e/a,e/b,e/e,q/j;q)= (d, e, aq/ j, bq/ j, eq/ j, q/ a, q/b, q/ e; q)= ' provided dej
= abeq2 and (e/a, aq/e,e/b, bq/e, e/e, eq/e, j,q/j; q)=
= (f la, aq/ j, j /b, bq/ j, j /e, eq/ j, e, q/e; q)=.
Exercises 5.14 Show that
l
149
d
(-qt/c, qt/d: q)oo dqt (-at/c, bt/d, q)oo d(1 - q) = 1- b l'l/h(q/a;bq;q,-ad/c) cd(1 - q)(q, ab, -c/d, -d/c; q)oo (c + d)(a, b, -bc/d, -ad/c; q)oo -c
when labl < lad/cl < l. (Andrews and Askey [1981]) 5.15 Show that
1
00 (ct, -dt: q)oo dqt (at, -bt, q)oo 2(1 - q)(c/a, d/b, -c/b, -d/a, ab, q/ab; q)00(q2; q2)~ (cd/abq,q;q)00(a 2,q2/a2,b 2,q2/b2;q2)00
-00
(Askey [1981]) 5.16 Show that
roo (aat, aft, abt, bit, act, cit, adt, d/t; q)oo dqt
10
(aqt 2, q/at 2; q)oo (1 - q)(aa, a, ab, b, ac, c, ad, d; q)oo (aq,q/a;q)oo
t
[q..ja,-q..ja,q/a,q/b,q/C,q/d. a 2abcd] t;;, t;;, , q, - - 3Y a, -y a, aa, ab, ac, ad q (1 - q)(q, aab/q, aac/q, aad/q, abc/q, abd/q, acd/q; q)oo (a 2abcd/q3; q)oo
x
.1.
6'/-'6
when la 2abcd/q 3
1
<
l.
5.17 Show that
roo (alt, ... ,art,bdt, ... ,bs/t;q)OOt'Y-ld t
10
(CIt, ... , crt, ddt, ... , ds/t; q)oo q (1- q)(al, ... , an bl , ... , bs ; q)oo (CI"'" cr , dl , ... , ds; q)oo CI, ... ,Cnq/bl, ... ,q/bs bl ... bs ] [ X r+s'l/Jr+s / / ; q, d d q'Y aI, ... , ar , q d l ,···, q ds I . .. s
when
5.18 Derive Bailey's [1950b] summation formulas:
(i)
'l/J [ 3
3
-.!L] _
b, c, d . (q, q/bc, q/bd, q/cd; q)oo q/b, q/c, q/d' q, bcd - (q/b, q/c, q/d, q/bcd; q)oo'
150
Bilateral basic hypergeometric series (ii)
L] _
'¢ [ 3
3
b,e,d . (q,q2/be,q2/bd,q2/ed;q)00 q2/b, q2/ e , q2 /d' q, bed - (q2/b, q2/e, q2/d, q2/bed; q)oo' b, e, d, e, q-n
]
(iii) 5'¢5 [ /b / /d / n+l ; q, q q ,q e,q ,q e,q
= (q, q/be, q/bd, q/ed; q)n
W
h
bd en e e
(q/b,q/e,q/d,q/bed;q)n b, e, d, e, q-n ] (iv) 5'¢5 [ 2/b 2/ 2/d 2/ n+2 ; q, q ,q e,q q ,q e,q (1 _q)(q2,q2/be,q2/bd,q2/ed;q)n (q2/b,q2/e,q2/d,q2/bed;q)n when bede = qn+3, where n = 0,1, ....
5.19 Show that
(i) and
(ii)
t
k=-n
t k=-n-l
for n
= 0,1, ....
(_1)k [ 2n n+k
= qn+l ,
]3 qk(3k+l)/2 = q
(_1)k [ 2n+ 1 ]3 l (3k+l)/2 n+k+1 q
(q;qhn (q,q,q;q)n
=
(q;qhn+l (q,q,q;q)n
(Bailey [1950b])
5.20 Derive the 2'¢2 transformation formulas
(i)
2
'¢ [a, b. ] _ (az, d/a, e/b, dq/abz; q)oo 2 e, d' q, z (z, d, q/b, ed/abz; q)oo X 2'¢2 [
(ii)
n/,
2 '1-'2
a, abz/d. d] ,q, aZ,e a
,
[a, b. ] _ (az, bz, eq/abz, dq/abz; q)oo (/ e, d , q, Z q a, q/b ,e, d; q) 00
abz/e, abz/d. Cd] x 2'¢2 [ b ' q, -b . az, z
a z
(Bailey [1950a]) 5.21 Verify that Ex. 2.16 is equivalent to
(b/a, aq/b, dj/a,aq/dj,ej/a, aq/ej,bde/a, aq/bde; q)oo
= (f la, aq/ j, bd/a, aq/bd, be/a, aq/be, dej la, aq/dej; q)oo b 2 2 - -(d,q/d,e,q/e,j/b,qb/j,bdej/a ,a q/bdej;q)oo' a
(See Bailey [1936])
Exercises
151
5.22 Extend the above identity to a
b (be aq bd aq be aq b f aq g aq !!.. aq g hq. ) a'b' c a 'bd' a' b e'a 'bf" a g " ah ' h ' g,q oo _ a
b (Ch aq dh aq eh aq f h aq ~ aq g aq g bq. ) h ' a , dh' a'e h ' a , fh' a 'b' ag , 'b ' g,o q o a'c
_ (c g aq dg aq eg aq f 9 aq ~ aq !!.. aq ~ hq. ) - ag a , cg , a 'd' 9 a , eg , a ' f " 9 a b' a 'h' h' b ,q - bh
where a 3 q2
(
q
c,
q
q
~' d, d' e, ~' f,
= bcdefgh.
00
qbhq9hq9bq) h' b' h' b' q 00'
7'
g'
g;
(See Slater [1954a])
5.23 More generally, show that it follows from the general formula for sigma functions in Whittaker and Watson [1965, p. 451, Example 3] and Tannery and Molk [1898, §400], and also from (5.4.3) that
~
(ak/ bl,ak/ b2, ... ,ak/ bn;q)00 a a k=l (ak/ l, ak/ 2,"" ak/ a k-l, ak/ a k+l,"" ak/an; q)oo
x
(qb1/ak,qb2/ak, ... ,qbn /ak;q)00 = 0, (qal/ak,qa2/ak, ... ,qak-l/ak,qak+l/ak, ... ,qan/ak;q)00
where ala2 ... an = b1b2 ··· bn · (See Slater 1954a]) 5.24 Extend the summation formula (1.9.6) to oj,
r+2'f/r+2
=
[a,b,blq'ml, ... ,brq'mr -ll-n] db b b ;q,a q , q, 1,"" r
(q, q, bq/a, d/b; q)oo (bdb; q)'ml ... (br/b; q)'mr bn , (bq, q/ a, q/b, d; q)oo (b 1; q)'ml ... (b r ; q)'mr
where ml, ... , mr are nonnegative integers, n is an integer, and Iq/al Iqnl
<
Iq'm1+'+'m r /dl.
<
(W. Chu [1994a]) 5.25 More generally, extend the transformation formula in Ex. 1.34(ii) to oj,
r+2'f/r+2
=
[a, b, b1q'ml, ... , brq'mr -1 1-n] d , bcq ,1, b .. ·, br ;q,a q
(q, cq, bq/a, d/b; q)oo (bdb; q)'ml ... (br/b; q)'mr bn (bcq, q/ a, q/b, d; q)oo (b 1; q)'ml ... (b r ; q)'mr A.. [ c- 1 , bq/d, bq/b1, ... , bq/br . d n-('ml +'+'mr)] x r+2'f/r+l bq/a, bql-'m1 /b 1, ... , bql-'mr /b r ' q, C q ,
where ml, ... , mr are nonnegative integers, n is an integer, and Iq/al Iqnl < Iq'ml +'+'m r /cdl. (W. Chu [1994a])
<
152
Bilateral basic hypergeometric series
5.26 Extend the summation formula in Ex. 2.33(i) to n
1
I-N]
aqn1 +1 k+ qa 1/2 ,-qa1/2 ,b,a / b,c,d,el, ... , e k , - - , " aq "e e1 k ' q aq 1/2 1/2 aq b aq ad aq aq -n -nk' '--da ,-a 'b' q'c'd'e-;""'ek,elq 1, ••• ,ekq c (q,q,aq, q/a,aq/bc, aq/bd, bq/c, bq/d; q)oo (aq/b,aq/c, aq/d,bq, bq/a,q/b,q/c, q/d; q)oo
'¢ 6+2k 6+2k
[
rr
x k (aq/bej,bq/ej;q)n'J j=1 (aq/ej, q/ej; q)nj , where nl,"" nk are nonnegative integers, N 1 when the series does not terminate. (W. Chu [1998a])
= nl +-. +nk, and laql-N /cdl <
5.27 Prove that
S(a- 2,bc, bd, cd) S(b/a, cia, d/a, abcd)
+
. ( ) Idem a; b, c, d = 2,
where S is defined in Ex. 2.16. See Askey and Wilson [1985, pp. 10, 11], where it is used to evaluate the integral in (6.1.1).
Notes §5.2 Andrews [1979c] used Ramanujan's sum (5.2.1) to prove a continued fraction identity that appeared in Ramanujan's [1988] "lost" notebook. Formal Laurent series and Ramanujan's sum are considered in Askey [1987]. A probabilistic proof of (5.2.1) can be found in Kadell [1987b]. Milne [1986, 1988a, 1989] derived multidimensional U(n) generalizations of (5.2.1). §5.3 Gustafson [1987b, 1989, 1990] derived a multilateral generalization of (5.2.1), (5.3.1) and related formulas by employing contour integration and Milne's [1985d, 1987, 1993, 1994a,b, 1997] work on U(n) generalizations of the q-Gauss, q-Saalschiitz, and very-well-poised 6¢5 summation formulas. §5.4 M. Jackson [1954] employed (5.4.3) to derive transformation formulas for 3'¢3 series. §5.6 A transformation formula between certain 4¢3 and s'¢s series was found by Jain [1980b], along with transformation formulas for particular 7'¢7 series, and then used to deduce identities of Rogers-Ramanujan type with moduli 5, 6, 8, 12, 16, 20 and 24. Some recent results on bilateral basic hypergeometric series are given in Schlosser [2003a,b,c]. Ex.5.6 Watson [1929b] derived this identity in an equivalent form. For various proofs of the quintuple product identity (and of its equivalent forms) and applications to number theory, Lie algebras, etc., see Adiga, Berndt, Bhargava and Watson [1985], Andrews [1974a], Atkin and Swinnerton-Dyer [1954], Bailey [1951]' Carlitz and Subbarao [1972]' Gordon [1961]' Hirschhorn [1988], Kac [1978, 1985], Sears [1952]' and Subbarao and Vidyasagar [1970]. Exercises 5.12 and 5.13 See Rahman and Suslov [1994a, 1998] for more general formulas.
Notes
153
For integrals of Ramanujan-type that correExercises 5.15 and 5.16 spond to the summation formulas of basic bilateral series, see Rahman and Suslov [1994b, 1998] and Ismail and Rahman [1995]. Ex.5.18 Using (i) and (ii) one can show that the formula in Ex. 2.31 holds even when aI, a2, a3 are not nonnegative integers, provided that Iql < 1 and Iqa 1 +a 2 +a3 +11 < l. Exercises 5.21-5.23 Additional identities connecting sums of infinite products are given in Slater [1951, 1954b, 1966] and Watson [1929b].
6 THE ASKEY-WILSON q-BETA INTEGRAL AND SOME ASSOCIATED FORMULAS
6.1 The Askey-Wilson q-extension of the beta integral It should be clear by now that the beta integral and extensions of it that can be evaluated compactly are important. A significant extension of the beta integral was found by Askey and Wilson [1985]. Since it has five degrees of freedom, four free parameters and the parameter q from basic hypergeometric functions, it has enough flexibility to be useful in many situations. This integral is
j
1
1
1
h(x;I,-I,q"2,-q"2) dx -1 h(x; a, b, e, d) viI - x 2 2n( abed; q)oo (q, ab, ae, ad, be, bd, ed; q)oo'
(6.1.1)
where
h(x; aI, a2,···, am)
= h(x; aI, a2,···, am; q) =
h(x; adh(x; a2) ... h(x; am),
=
II (1 -
00
h(x; a)
=
h(x; a; q)
2axqn + a 2q2n)
n=O
(6.1.2) and max (Ial, Ibl,
lei, Idl, Iql) < 1.
(6.1.3)
As in (6.1.1), we shall use the h notation without the base q displayed when the base is q. Askey and Wilson deduced (6.1.1) from the contour integral
(Z2,z-2;q)00 dz l l 2ni ) K (az, az-l, bz, bz-l, ez, ez- , dz, dz- ; q)oo z 2(abed; q)oo (q, ab, ae, ad, be, bd, ed; q)oo' 1
{
(6.1.4)
where the contour K is as defined in §4.9 and the parameters a, b, e, d are no longer restricted by (6.1.3), but by the milder restriction that their pairwise products are not of the form q-j, j = 0, 1,2, .... Askey and Wilson's original proof of (6.1.4) required a number of interim assumptions that had to be removed by continuity and analytic continuation arguments. In their paper 154
6.1 The Askey-Wilson q-extension of the beta integral
155
they also provided a direct evaluation of the reduced integral
1
1 h(x; 1, -1)
-1 h(x; a, b)
dx
VI -
x2 27r( -abq; q)oo
1
1
1
(6.1.5)
1
(q, -q, aq2, -aq2, bq2, -bq2, ab; q)oo
by using summation formulas for 1 'l/J1 and 4'l/J4 series. Simpler proofs of (6.1.1) were subsequently found by Rahman [1984] and Ismail and Stanton [1988]. In the following section we shall give Rahman's proof since it only uses formulas that we have already proved, whereas the Ismail and Stanton proof uses some results for certain orthogonal polynomials which will not be covered until Chapter 7. We shall conclude this section by showing that the beta integral
1 1
-1
(1 _ x)<>(1 + x)f3dx = 2<>+f3+1 r(a + l)r(fi + 1) r(a+fi+2)
(6.1.6)
is a limit case of (6.1.5). Let 0 < q < 1, a = q<>+~, b = _qf3+~ and use the notation (z; q)oo () Z; q <> = (zq<>; q)oo
(6.1. 7)
and the definition (1.10.1) of the q-gamma function to express the right side of (6.1.5) in the form 22<>+2f3+ 2 r q(a + l)r q(fi + 1) _7r_ (-q; q)<>+f3( -q~; q)<>+~ (-q~; rq(a+fi+2) q(~) 22<>+2f3+ 1
q)f3+~
By (1.10.3) this tends to 22<>+2f3+ 1 r(a + 1) r(fi + 1)/r(a + fi + 2) as q ---+ 1-, since r (~) = ..fif. For the integrand in (6.1.5) we have h(x;I,-I) h( x,. a, b) =
(i()) (-i()) (i()) (-i()) e ;q <>+1 e ;q <>+1 -e ;q f3+1 -e ;q f3+1 2
2
2
2
and hence h(x;I,-I) lim q--+1- h(x; q<>+~, _q!3+~)
= [(1- ei ()) (1- e-i())r+~ [(1 +
ei ()) (1 +
e-i())lf3+~
= 2<>+f3+ 1(1 - cos e)<>+~ (1 + cos e)!3+~, which shows that (6.1.6) is a limit of (6.1.5). Formula (6.1.1) is substantially more general than (6.1.5) since it contains two more parameters. It is the freedom provided by these extra parameters which will enable us to prove a number of important results in this and the subsequent chapters.
156
The Askey-Wilson q-beta integral 6.2 Proof of formula (6.1.1)
Denote the integral in (6.1.1) by J (a, b, c, d). Since x of B, one can write
J(a, b, c, d) = ~ 2
cos B is an even function
=
j7r h(x; 1, -1, y'ij, -y'ij) dB. -7r h(x; a, b, c, d)
(6.2.1)
Let us assume, for the moment, that a, b, c, d and their pairwise products and quotients are not of the form q-j, j = 0,1,2, .... It is easy to check that, by (2.10.18),
h(x; l)/h(x; a, b) -:-:-:-_--:('-;-a_---'l,-;cb:---:-'\....::q;.-)00----:----:_ b(l - q) (q, alb, bq/ a, ab; q)oo h(x; -l)/h(x; c, d) (-c-l,-d-\q)oo d(l-q)(q,c/d,dq/c,cd;q)oo
lb a
(6.2.2)
(qu / a, qu / b, u; q) 00 ~ (u/ ab; q)oo h(x; u)
Id (qv/c,qv/d:-v;q)oo (-v/cd,q)oo
c
dqv h(x;v) ,
,
(6.2.3)
and
h
1
1
1
1
1
q2(-q 2U- ,-q 2v- ;q)oo (X;-q2)/ (X;U,V) = v(l-q)(q,U/V,vq/U,uv;q)oo 1
h
(6.2.4) Also,
j7r h(x; q!) dB 2 -7r h(x; tq!) = ~ j7r (q! eiO , q! e- iO ; q)oo dB 2 -7r (tq!eiO,tq!e-iO;q)oo = ~ j7r f f (r\ qh(r\ q)R (tq!) kH ei(k-R)OdB 2 -7rk=O R=O (q;q)k(q;q)R = ~ f f (r\ qh(r\ q)R (tq!) kH j7r-7r ei(k-R)OdB 2 k=O R=O (q; qh(q; q)R
~
=
7r
f
k=O
(r 1, rl; q)k (qt2)k = (q, q; q)k
7r
(qt, qt; q)oo (q, qt 2; q)oo
(6.2.5)
for Itq! I < 1, by (1.5.1). Since
(qt 2; q)oo = (qt 2, q2t 2; q2)00 = (tq! , -tq! , qt, -qt; q)oo, we have
j7r h(x; q!) dB = 2 -7r h(x; tq!)
~
qt; q)oo (q, tq! , -tq! , -tq; q)oo . 7r(
(6.2.6)
(6.2.7)
6.3 Integral representations for very-well-poised
series
81>7
157
Thus
I(a,b,e,d) bd(1 - q)3(q; q)'!xo(a/b, bq/a, ab, e/d, dq/e, cd; q)oo X
I
b
d qU
a
(qu/a, qu/b, u; q)oo (u/ab;q)oo
Id dqV (qv/e,v(-v/ed,vq/u,u/v,uv;q)oo qv/d, -v, -q! /u, -q! Iv; q)oo c
1
X
lv:~'2 dqt (t'!~ /u, ~q~ lv, qt; q)oo uq
(tq'2, -tq'2, -qt/uv; q)oo n(a- 1 , b- 1 , -e- 1 , -d- 1 ; q)oo bd(1 - q)2(q; q)'!xo(a/b, bq/a, ab, e/d, dq/e, cd; q)oo X
I
a
b
2
Id dqV (qv(v,uv,-v/ed;q)oo / qv / d, -uv; q)oo n( a-l, b-l, q! , -q! , q)oo Ib d (qu/ a, qu/b, edu; q)oo
d
(qu/ a, qu/b; q)oo qU (-u,u/ab;q)oo
e,
c
-1;
=
b(1 - q)(q; q)~(a/b, bq/a, ab, e, d, cd; q)oo
a
qU (eu, du, u/ab; q)oo
n( -1, q! , -q!, abed; q)oo (q, ab, ae, ad, be, bd, cd; q)oo'
(6.2.8)
by repeated applications of (2.10.18). Since (-I,q!,-q!;q)oo = 2(q!,-q!,-q;q)oo = 2, which follows from (6.2.6) by setting t = 1, we get (6.1.1). By analytic continuation, the restrictions on a, b, e, d mentioned above may be removed.
6.3 Integral representations for very-well-poised
81>7
series
Formulas (2.10.18) and (2.10.19) enable us to use the Askey-Wilson q-beta integral (6.1.1) to derive Riemann integral representations for very-well-poised 81>7 series. Let us first set 1
w(x; a, b, e, d)
=
and
J(a, b, e, d, f, g) = where max(lal, Ibl, lei, Idl,
1 1
-1
h(x;g) w(x; a, b, e, d) h(x; f) dx,
If I, Iql) < 1 and 9 is arbitrary.
h(x;g) h(x;d,f)
1
_!h(x;I,-I,q'2,-q'2) (1 - x) 2 h(. b d) x,a, ,e, 2
(6.3.1)
(6.3.2)
Since, by (2.10.18),
(g/d,g/f;q)oo f(1 - q)(q, d/ f, qf /d, fd; q)oo X
i
d
f
d
(qu/ d, qui f, gu; q)oo qU (gu/df; q)ooh(x; u) ,
(6.3.3)
158
The Askey-Wilson q-beta integral
we have
J(
b d f ) (g/d, g/ f; q)oo a, ,e, , ,g = f(l-q)(q,d/f,qf/d,df;q)oo
{f d
Jd
X
qU
(qu/d, qui f, gu; q)oo (gu/df; q)oo
11 ( -1
)
w x; a, b, e, u dx.
(6.3.4)
By (6.1.1),
1 1
-1
W
b )d _ x, a, ,e, u x -
(.
27f(abeu; q)oo . (q, ab, ae, be; q)oo(au, bu, eu; q)oo
(6.3.5)
Substituting this into (6.3.4), we obtain
J(
=
b d f ) a, ,e, , , 9
i
x d
f d qU
27f(g/d, g/ f; q)oo f( 1 - q) (q,. q)200 (d/f ,q f/d ,J, dff ab,ae, b·) e, q 00
(qu/d, qui f, gu, abeu; q)oo ( ). au,bu,eu,gu / df;q 00
(6.3.6)
The parameters in this q-integral are such that (2.10.19) can be applied to obtain
J(
b d f ) a, ,e, "g
=
27f(g/f,fg,abef,bedf,edaf,dabf;q)00 (q,ad,bd,ed,af, bf,ef,df,ab,ae, be, abedf2; q)oo
x 8W7(abedf2q-\af,bf,ef,df,abedfg-\q,g/f),
(6.3.7)
provided Ig/ fl < 1, if the series does not terminate. By virtue of the transformation formula (2.10.1) many different forms of (6.3.7) can be written down. Two particularly useful ones are
J( a, b,e, d , f ,g )
=
(
27f(ag,bg,eg,abed,abef;q)00 ) q, ab, ae, ad, af, be, bd, bf, cd, ef, abeg; q 00
x 8 W 7(abegq-\ ab, ae, be, g/d, g/ f; q, df),
(6.3.8)
which was derived in Nassrallah and Rahman [1985], and
J(
b d f ) a, ,e, "g
=
27f(ag, bg,eg, dg, fg,abedf/g; q)oo (q,ab,ae,ad,af,be,bd,bf,ed,ef,df,g2;q)00 x 8 W 7(lq-\ g/a, g/b, g/e, g/d, g/ f; q, abedfg-1). (6.3.9)
If the series in (6.3.9) does not terminate, then we must impose the condition labedfg-11 < 1 so that it converges. Note that, if in (6.3.8) we let 0 < q < 1, a = -b = q!, e = qa+!, d = z, f = _qf3+!, 9 = zq' with Re(a, (3) > -~ and then take the limit q --+ 1-, we obtain, after some simplification,
r
1 a . . _ r(a + (3 + 2) f3 -, 2F1(--y,a+1,a+(3+2,z)-r(a+1)r((3+1)Jo X (I-x) (l-xz) dx.
(6.3.10) This shows that (6.3.8) is a q-analogue of Euler's integral representation (1.11.10).
6.4 Integral representations for very-well-poised
1O¢9
series
159
Another limiting case of (6.3.8) was pointed out in Rahman [1986b]. To derive it, replace a,b,e,d,f,g in (6.3.8) by qa,qb,qc,qd,qf,qg, respectively. Also, replace x in the integral (6.3.2) by cos(t log q), which corresponds to replacing ei () by qit. Now let q ---+ 1- and use (1.10.1) and (1.10.3) to get the formula 00 ~ r(a + it)r(a - it)r(b + it)r(b - it)r(e + it)r(e - it) 47f -00 r(2it)r( -2it)
1
x r(d + it)r(d - it)r(j + it)r(j - it) dt r(g + it)r(g - it) r(a + b)r(a + e)r(a + d)r(a + f)r(b + e)r(b + d) r(a + g)r(b + g)r(e + g) x r(b + f)r(e + d)r(e + f)r(a + b + e + g) r(a + b + e + d)r(a + b + e + f) ~--~~--~~~~~----~~
a+b+e+g-l,~(a+b+e+g+l),a+b,a+e, b+e x 7 F.6 1 2(a + b + e + 9 - 1), e + g, b + g, a + g,
[
9 - j,
9 - d, a
+ b + e + d, a + b + e + j
.,
1] ,
(6.3.11)
> O.
where Re (a, b, e, d, f)
6.4 Integral representations for very-well-poised
1O¢9
series
If we set 9 = abedj in (6.3.7), then the 8 W 7 series collapses to one term with value 1 and so we have the formula
1
h(x;I,-I,q!,-q!,abedf) dx h(x; a, b, e, d, f) VI - x 2 27f(abed,abej,bedj,abdj,aedj;q)00 (q, ab, ae, ad, aj, be, bd, bj, cd, ej, dj; q)oo = go(a, b, e, d, f), say, 1
-1
where (max lal, Ibl, lei, Idl, Ijl, Iql)
10 1 x a- 1(1 -
(6.4.1)
< 1. This is a q-analogue of the formula
x)b-1(1 - tx)-a-b dx
=
~~~~(:? (1 -
t)-a, Re(a, b)
> o.
Replace j by jqn in (6.4.1), where n is a nonnegative integer, to get
1
(je i () , je- i (); q)n v(x;a,b,e,d,f)( b dj -1 ace -i() ,abedj ei().,q ) n dx 1
= go
(
(aj, bj, ej, dj; q)n ) a,b,e,d,j (bedj,aedj,abdj,abej;q)n'
(6.4.2)
where
. b d j) v (x, a, ,e, ,
=
(1-
x
2)-! h(x; 1, -1, q!, -q!, abedf) h(. x,a, b,e, d , j)
.
(6.4.3)
160
The Askey-Wilson q-beta integral
Let u
abedj. If
=
1 1 -1
=
Izl < 1, then
(6.4.2) gives the formula
v(x;a,b,e,d,f) r+sWr+4(Ujq-\al, ... ,ar,jei!J,je-i!J;q,z) dx
go(a, b, e, d, f) r+7Wr+6(u jq-\ al, ... , ar, aj, bj, ej, dj; q, z).
(6.4.4)
In particular, for r = 3 and z = u2/ala2a3, we have the formula
1 1 -1
v(x;a,b,e,d,j) 8W7(ujq-\al,a2,a3,jei!J,je-i!J;q,u2/ala2a3) dx
= go(a, b, e, d, f) lOW9(U jq-\ aI, a2, a3, aj, bj, ej, dj; q, u2/ala2a3), (6.4.5) where lu2/ala2a31 < 1, if the series do not terminate. Let us assume that (6.4.6) which ensures that the very-well-poised series on either side of (6.4.5) are balanced. Then, by (2.11.7) 8W 7(u jq-l; aI, a2, a3, jei!J, je-i!J; q, q) (u j, qaI/u j, a2, qaI/a2, a3, qaI/a3; q)ooh(x; j, qaI/ f) (u j /qal, u j /a2, u j /a3, qala2/u j, qala3/u j, q2 aI/u j; q)ooh(x; u, qaI/u)
+~~--~~~~~~~~~~~~~~~~~~~----~
x
8 W 7 (qai!u j;
aI, qala2/u j, qala3/uj, qalei!J /u, qale-i!J /u; q, q) (u j, u j /a2a3, qaI/u j, u / j; q)ooh(x; u/a2, u/a3)
(6.4.7)
and hence
1 1
-1
h(x;I,-I,q!,-q!,u/a2,u/a3) h(x;a,b,e,d,j,u/a2a3)
= go(a, b, e, d, f)
{
dx
viI -x 2
(~j /a2, ~ j /a3, uj ja 2; u / j a3; ~)oo Uj, Uj a2a3, u j, u ja2a3; q
00
x lOW9(U jq-l; aI, a2, a3, aj, bj, ej, dj; q, q)
+
(aj, bj, ej, dj, a2, a3, qaI/a2, qaI/a3; q)oo (u/a, u/b, u/e, u/d, qaal/u, qbal/U, qeal/u,qdal/U; q)oo (qal/aj,qal/bj,qal/ej,qal/dj,qal/uj;q)oo x (u / j, u f!qal, u j /a2a3, u/ ja2a3, q2 aI/u j; q)oo x lOWg(qai!uj; aI, u/ ja2, u/ ja3, qaaI/u, qbaI/u, qeaI/u, qdaI/u; q, q) },
(6.4.8) where u = abedj and ala2 a3q = u 2. Since the integrand on the left side of (6.4.8) is symmetric in a, b, e, d and j, the expression on the right side must have the same property. This provides an alternate proof of Bailey's four-term transformation formula (2.12.9) for VWP-balanced 1O
6.4 Integral representations for very-well-poised
1O¢9
series
161
If we set a3 = q-n, n = 0,1,2, ... , then the coefficient of the second on the right side of (6.4.8) vanishes and we obtain
1 1
-1
h(x;I,-I,q!,-q!,a) (aeiIJ/g,ae-iIJ/g;q)n h(x; a, b, e, d, f) (aeiIJ, ae- iIJ ; q)n
= go(a, b, e, d, f) (at
7'
10 Wg
dx
VI -
x2
a!fg;)q)n
a ,a
,q
n
¢ [v' qy'v, -qy'v, g, aj, bj, ej, dj, a 2 qn-l/g, q-n ] y'v, _y'v, af/g, a/a, a/b,a/e,a/d, jgql-n/ a ,ajqn;q,q ,
X 10 9
(6.4.9) where v = a j /q. By applying the iteration of the transformation formula (2.9.1) given in Exercise 2.19, this can be written in the more symmetric form
1 1
-1
=
1
1
h(x;I,-I,q2,-q2,aqn,T) h(x; a, b, e, d, j, Tqn)
dx
vff=X2
( b d j) (Ta,Tb,Te,Td,Tj,a/T;q)n go a, ,e" (a/ a, a /b ,a / e, a /d , a /j ,T; 2 q) n
x
1O¢9 [
T2 q-1 , Tq 12 , -Tq 12 ,T / a, T /b ,T / e, T /d , T /j , aTq n-l , q-n Tq
_1 2,
-Tq
_1 2,
Ta, Tb, Te, Td, T j, Tq
I-n
2 n
/ a, T q
1
;q,q,
(6.4.10) where a = abedj and T is arbitrary. Similarly, by applying (2.12.9) twice one can rewrite (6.4.8) in the form
1 1
-1
h(x;I,-I,q!,-q!,A,JL) h(x; a, b, e, d, j, g)
dx
VI -
x2
2~(AJL/aj,AJL/bj,AJL/ej,AJL/dj,Ag,JLg,A/g,JL/g;q)oo
(q, ab, ae, ad, ag, be, bd, bg, cd, eg, dg, jg, j /g, AJLg/ j; q)oo ¢ [VI, qy'Vl, -qy'Vl, ag, bg, eg, dg, AI j, JL/ j, AJL/q ] XlO 9 y'Vl, _y'Vl, AJL/aj, AJL/bj, AJL/ej, AJL/dj, JLg, Ag, gq/j;q,q 2~(AJL/ag,AJL/bg,AJL/eg,AJL/dg,Aj,JLj,A/j,JL/j;q)oo
+~~~~~~~~~~~~~~~~~~-=~
(q, ab, ae, ad, aj, be, bd, bj, cd, ej, dj, gj, g/ j, AJLj /g; q)oo ¢ [V2' qyVi, -qyVi, aj, bj, ej, dj, A/g, JL/g, AJL/q ] XlO 9 yVi, _yVi, AJL/ag, AJL/bg, AJL/eg, AJL/dg, JLj, Aj, jq/g;q,q (6.4.11) where VI = AJLg/ jq, V2 = AJLj /gq, AJL = abedjg and max(lal, Ibl, lei, Idl, Ijl, Igl, Iql) < 1. For these and other results see Rahman [1986b]. If A or JL equals a, b, e or d, then the right side of (6.4.11) is summable by (2.11.7), while the integral on the left side gives the sum by (6.4.1). For this reason the integral in (6.4.1) may be considered as an integral analogue of Bailey's formula (2.11.7) for the sum of two balanced very-well-poised S¢7 series. Likewise, the integral in (6.4.2) is an integral analogue of Jackson's sum (2.6.2). Indeed, a basic integral may be called well-poised if it can be written in the form
162
The Askey-Wilson q-beta integral n
_eiO' aJ_e- iO . q) I1 (aJ
1 I1 11"
,00
d().
j:,l
-7r
(6.4.12)
iO ' b-e-iO'q) (b-e J J ,00
j=l
The integrals in (6.1.1), (6.3.2), (6.4.1) and (6.4.2) are all well-poised.
6.5 A quadratic transformation formula for very-well-poised balanced 1O¢9 series
In (6.4.11) let us set J1 = -A, 9 = - f and b = -a so that A2 the expression on the right side of (6.4.11) becomes
= -a 2edj2. Then
2n(A2jef,A2jdf;q)00(A4ja2f2,A2f2,A2jf2;q2)00 (q, -1, -a 2, - j2, A2; q)00(a 2e2, a 2d2, a 2j2; q2)00(ed, ef, df; q)oo x 10 W g (A2q-1; _A 2q-1, af, -af, Aj f, -Aj f, ef, df; q, q) + 2n( _A 2jef, _A 2jdf; q)00(A4 ja 2j2, A2 j2, A2 j j2; q2)00 (q, -1, -a 2, - j2, A2; q)00(a 2e2, a 2d2, a 2j2; q2)00(ed, -ef, -df; q)oo x 10 W g (A2q-1; _A 2q-1, af, -af, Aj f, -Aj f, -ef, -df; q, q). (6.5.1) We now turn to the integral on the left side of (6.4.11). Observing that (6.5.2) where x
= cos() and ~ = cos2() = 2x2 -1, it follows from (2.10.18) that h(X;A- A) h(~;A2;q2) h(x; a, -a, f, - f) h(~; a2, j2; q2) (A2ja 2, A2jf2; q2)00 j2(1- q2)(q2, a 2j j2, q2 j2 ja 2, a2j2; q2)00 X
If2 a2
(q 2uj a2, q2uj j2, A2U; q2)00 d (A2uja2 j2; q2)ooh(~; u; q2) q2U.
(6.5.3)
Hence the integral on the left side of (6.4.11) can be expressed as
(6.5.4) by (6.1.1). Since A2
=
-a 2edj2, the q-integral on the right side of (6.5.4)
6.6 The Askey-Wilson integral when max (Ial, Ibl, lei, Idl)
~
1
163
reduces to
l
(q 2u/ a 2, q2u/ p, -edqu, -a 2edpu; q2)00 d q 2U ( e 2 u, d 2 u, -u, -uq,. q2) 00 a2 p(1- q2)(q2, a 2/ p, q2 P /a 2; q2)00 (a 2e2, a 2d 2, e2 j2, d 2j2; q2)00 X (_a 2e2 p, _a 2d 2 p, a 2e2d 2p, -edqp, -a 2edj4; q2)00 (-a 2, - j2, -qj2, -a 2e2d 2j4; q2)00 2 X 8 W 7 ( _a e2d 2 j4q-2; e2 j2, d 2j2, _ j2, cd, eda 2 j2q-\ q2, _qa 2 ), i
2
(6.5.5) by (2.10.19). Using this in (6.5.4) and equating with (6.5.1) we obtain the desired quadratic transformation formula
(-a 2ej, -a 2dj; q)oo 2 -1 2-1 (dj j. ) lOW9 ('\ q ;-,\ q ,aj,-aj,A/j,-'\/j,ej,dj;q,q) ,e ,q 00 (a 2ej, a 2dj; q)oo 2 -1 2-1 + (-dj,-ej;q)oo lOW9 ('\ q ;-,\ q ,aj,-aj,A/j,-'\/j,-ej,-dj;q,q) =
(_qa 2 _a 2e2p _a2d2 p. q2) (2 jd d 2j2 _'2 2d2 j4. '2) 00 e , ,ae ,q 00 2 2 2 2 2 2 8 W 7 ( _a e d j4 q-2; e j2, d j2, - j2, cd, eda j2q-\ q2, _qa 2 ),
(-1, -edj2, -eda 2 j2; q)oo X
(6.5.6)
with ,\2 = -eda 2p. This gives a nonterminating extension of the transformation formula (3.10.3) when the bibasic q, series is a balanced 1O¢9 . For an extension of (3.10.3) when the q, series is not balanced, see Nassrallah and Rahman [1986]. 6.6 The Askey-Wilson integral when max (Ial, Ibl, lei, Idl) ~ 1
Our aim in this section is to extend the Askey-Wilson formula (6.1.1) to cases in which Iql < 1 and the absolute value of at least one of the parameters is greater than or equal to 1. Since the integral in (6.1.1), which we have already denoted by I(a, b, e, d), is symmetric in a, b, e, d, without loss of generality we may assume that lal ~ 1. Let us first consider lal ~ 1 > max(lbl, lei, Idl)·
(6.6.1)
If a = ±1, then the functions h(x; ±1) and h(x; a) in the integrand in (6.1.1) cancel and by continuity it follows that
I( ±1, b,e, d) =
27f(±bed; q)oo
(q, ±b, ±e, ±d, be, bd, cd; q)
(6.6.2)
00
However, if lal = 1 and a -I=- ±1, then h(x; a) = 0 for some x in the interval (-1,1) and so the integral in (6.1.1) does not converge. Similarly, this integral does not converge if laqnl = 1 and aqn -I=- ±1 for some positive integer n.
164
The Askey-Wilson q-beta integral If there is a nonnegative integer m such that
(6.6.3) and if ab, ac and ad are not of the form q-n for any nonnegative integer n, then the integral in (6.1.1) converges and we can evaluate it by the following technique. Observe that, since h(x;a)
=
(ae i !:l,ae- i !:l;q)m+1 h(x; aqm+1)
=
a 2m +2qm+m2 h(x; aqm+1, q-m la)lh(x; qla),
(6.6.4)
we have
J( a, b,c, d) 1
=
a -2m-2 q -m-m 1
2
1
x j h(x;1,-1,q2-q2,qla) -1 h(x; b, c, d, aqm+1, q-m fa)
dx
V1- x 2 '
(6.6.5)
where the parameters b, c, d, aqm+1, q-m I a in the denominator of the integrand are now all less than 1 in absolute value. By (6.3.8),
J(
b d) a, ,c,
=
2n(qbla,qcla,qdla,abcdqm+l,bcdq- m l a;q)00 (q, bc, bd, cd, abqm+1, acqm+l, adqm+l, bcdqla; q)oo a- 2m- 2q-m-m 2 x ~--~----~~~--~~-(bq-m la, cq-m la, dq-m la; q)oo
x 8 W 7(bcda-\ bc, bd, cd, q-m a-2, qm+\ q, q).
(6.6.6)
The series in (6.6.6) is balanced and so we can apply Bailey's summation formula (2.11.7). After some simplification we find that 2n(abcd;q)00 ) q,ab,ac,ad,bc,bd,cd;q 00
J( a, b,c, d) = (
+ L m (·b a, ,c, d) ,
(6.6.7)
where L m
. b d _ 2n( aql d, bql d, cql d, qm+1, a 2bcqm+1, q-m I a 2 , bcq-m; q)oo (a, ,c, ) - (q, ab, ac, bc, abqm+1, acqm+1, adqm+l, aqm+l I d, abcql d; q)oo a-2m-1d-1q-m-m2
x
~--~------~~~~----~~~
(bq-m I a, cq-m I a, dq-m I a, q-m I ad; q)oo
x 8 W 7(abcd-\ bc, q-m lad, aqm+1 Id, ab, ac; q, q).
(6.6.8)
By (2.10.1), 8 W 7 (abcd-\
bc, q-m lad, aqm+1Id, ab, ac; q, q)
(abcqld,qlad,abq, acq; q)oo (bqld,cqld,q, a 2bcq; q)oo x 8 W7(a 2 bc; abcd, ab, ac, a 2 qm+1, q-m; q, qlad).
(6.6.9)
6.6 The Askey-Wilson integral when max (Ial, Ibl, Icl, Idl) ~ 1
165
Since m is a nonnegative integer, the series on the right side of (6.6.9) terminates and hence, by Watson's formula (2.5.1), 8
2 yy 7 a 2b c,. a bcd ,ab,ac, a 2 qm+1 ,q -m., q, q I a d) -_ ((a bbcq, q; q)m ) a q,acq;q m ¢ [q-m, ab, ac, q-m I ad . ] _ (a 2bcq, q, aqlb, aql c; q)m X43 bcq-m, aqld, q-m,q,q - (qa2,abq,acq,qlbc;q)m q] a 2 , qa, -qa, ab, ac, ad, a 2qm+1, q-m [ X 8¢7 a, -a, aq Ib ,aq I c, aq Id , q-m ,a 2 qm+1; q, ab cd
TXT (
(a 2bcq,q,aqlb,aqlc;q)m (qa 2 ,abq,acq,qlbc;q)m
f
X
(a 2;q)k(l-a 2q2k)(ab,ac,ad;q)k ( q )k (q; qh(1 - a 2 )(aqlb, aqlc, aqld; q)k abcd .
k=O
(6.6.10)
Using (6.6.10) and (6.6.9) in (6.6.8), we obtain L m
2n(a- 2; q)= ( . b d) = _ a, ,c, (q,ab,ac,ad,b I a,c I a,d I a;q )=
f
X
k=O
(a 2; q)k(1 - a 2q2k)(ab, ac, ad; q)k ( q )k (q; qh(1 - a 2 )(aqlb, aqlc, aqld; q)k abcd .
(6.6.11)
Hence
1
1 h(x;I,-I,q!,-q!)
-1
X
f
h(x; a, b, c, d)
k=O
2n(a- 2;q)=
dx
VI -
x2
+
(q, ab, ac, ad, bla, cia, dla; q)=
(a 2;q)k(l-a 2q2k)(ab,ac,ad;q)k ( q )k (q; q)k(1 - a 2)(aqlb, aqlc, aqld; qh abcd
2n(abcd; q)= (q, ab, ac, ad, bc, bd, cd; q)= '
(6.6.12)
where max(lbl, Icl, Idl, Iql) < 1, laqm+ll < 1 < laqml for some nonnegative integer m, and the products ab, ac, ad are not of the form q-n, n = 1,2, .... Askey and Wilson [1985] proved this formula by using contour integration. By continuity, formula (6.6.12) also holds if the restriction (6.6.3) is replaced by aqm = ±l. Note that, if one of the products ab, ac or ad is of the form q-n for some nonnegative integer n, the integral in (6.6.12) converges even though the denominator on the right side of (6.6.12) equals zero as does the denominator in the coefficient of the sum in (6.6.12). If we let ab tend to q-n then, since Ibl < 1 and laqm+11 < 1 < laqml, we must have n:S; m. We may then multiply (6.6.12) by 1 - abqn and take the limit ab ~ q-n. The result is a terminating 6¢5 series on the left side and its sum on the right, giving the summation formula (2.4.2). If max(lcl, Idl, Iql) < 1 and there are nonnegative integers m and r such that (6.6.13)
166
The Askey-Wilson q-beta integral
then the above technique can be extended to evaluate I(a, b, e, d) provided the products ab, ae, ad, be, bd are not of the form q-n and alb i=- q±n for any nonnegative integer n. Splitting h(x; b) in the same way as in (6.6.4) and using (6.3.4) we get
I(a,b,e,d) = b- 2r - 2q-r-r 2 J(a, bqr+1, q-r /b, e, d, q/b) - 2r - 2 -r-r2 q
=b
j
X
d
c
d
qU
(q/be, q/bd; q)oo d(l - q)(q, e/d, dq/e, cd; q)oo
(qu/e, quid, qu/b; q)oo (/b d. ) qu e, q 00
11 (. -1
W
b r+1 -rib ) d x, a, q , q ,u x. (6.6.14)
However, by (6.6.12),
1 w(x; q, bqr+l, q-r /b, u) dx r 1-1
27r( aqu; q)oo (q, q, abqr+1, aq-r /b; q)oo(au, buqr+l, uq-r /b; q)oo 27r(a- 2; q)oo (q, abqr+1, q-r lab, bqr+1/a; q)oo(au, u/a; q)oo X
~ (a 2; q)k(l - a 2q2k)(au; q)k ( )-k ~ (q; qh(l _ a2)(aq/u; q)k au .
(6.6.15)
Since by (2.10.18),
j
d
c
dqu (qu/e, quid, qu/b; q)oo (au; qh (qu)-k (qu/bed, au, u/a; q)oo (aq/u; q)k
= (_l)k -2k -k(k+1)/2
jd
(qu/e, quid, qu/b; q)oo ,uq- k/ a;q ) 00 d(l - q)(q, e/d, dq/e, cd, q/ab, aq/b; q)oo (q/be,q/bd, ae, ad, e/a, d/a; q)oo a
q
c
(ab,ae, ad; qh X (aq/b,aq/e,aq/d;q)k
d
qU ( qu /b e d ,auq k
(q)k abed '
(6.6.16)
we find that
I(a, b, e, d)
+
= Lm(a; b, e, d)
27r(adq, edq, dq/b, qr+2, q1-r /b 2; q)oob- 2r - 2q-r-r 2 (q,q,ad,ed,dq2/b,abqr+1,beqr+1,bdqr+1,aq-r/b,eq-r/b,dq-r/b;q)oo
x 8 W 7(dq/b; bdqr+1, q, dq-r /b, q/be, q/ab; q, ae),
(6.6.17)
where Lm (a; b, e, d) is as defined in (6.6.11). The reduction of this 8 W 7 will be done in two stages. First we use (2.10.1) twice to reduce it to a balanced 8 W 7 and then apply (2.11.7) to obtain 8 W 7(dq/b;bdqr+1,q,dq-r/b,q/be,q/ab;q,ae)
6.6 The Askey-Wilson integral when max (Ial, Ibl, Icl, Idl) ~ 1
167
(dq2/b,q/bd,abcdqr+1,acdq-r/b;q)00 (q1-r/b 2,qr+2,ac,qacd2;q)00 X 8 W7( acd 2; abcd, bdqr+1, dq-r /b, ad, cd; q, q/bd) (dq2/b,aq/b,cq/b,q, abcdqr+1,acdq-r/b; q)oo (ac,adq, cdq, acdq/b,qr+2, q1-r/b 2; q)oo x 8 W 7 (acdb-\ cd, ad, ac, qr+1, q-r /b 2; q, q) (dq2/b, aq/b, bfa, q, q/ab, abcd, bcqr+1, bdqr+1, cq-r /b, dq-r /b; q)oo (ac,adq, cdq,dq/b, bd, bc,qr+2, bqr+1/a, q-r/ab, q1-r/b 2; q)oo + ~ (dq2/b, q, bq/a, cq/a, dq/a, ad, qr+1, b2cdqr+1, cdq-r, q-r /b 2; q)oo a (bcdq/a, dq/b, bc, bd, adq, cdq, qr+2, bqr+1 la, q-r lab, q1-r /b 2; q)oo X 8 W 7 (bcda-\
bc, bd, cd, bqr+1 la, q-r lab; q, q).
(6.6.18)
Substituting (6.6.18) into (6.6.17) and simplifying the coefficients we get
27r(abcd; q)oo I(a, b, c, d) - Lm(a; b, c, d) - ( b d b bd d ) q,a ,ac,a ,c, ,c;q 00 27r(bq/a, cq/a, dq/a, qr+1, b2cdqr+1, cdq-r, q-r/b 2; q)oo (q,bc,bd,cd,abqr+1,bcqr+1,bqr+1/ a,aq-r/b,cq-r/b,dq-r;q)00 b-2r-1 -1 -r-r2 X ( -r / b : dq/ .) 8 W 7 (bcda-\ bc, bd, cd, bqr+1 la, q1-r lab; q, q). a, c q a, q 00 q (6.6.19) The expression on the right side of (6.6.19) is the same as that in (6.6.8) with a, b, c, d, m replaced by b, d, c, a and r, respectively, and so has the value
27r(b- 2; q)oo (q,ba, bc,bd,a/b,c/b,d/b; q)oo (b 2; q)k(l- b2q2k)(ba, bc, bd; q)k ( q )k X (q; q)k(l - b2 )(bq/a, bq/c, bq/d; q)k abcd
t
k=O
= Lr(b; c, d, a),
(6.6.20)
by (6.6.11). So we find that
I(a, b, c, d) - Lm(a; b, c, d) - Lr(b; c, d, a) 27r(abcd; q)oo (q, ab, ac, ad, bc, bd, cd; q)oo'
(6.6.21)
where the parameters satisfy the conditions stated earlier. It is now clear that we can handle the cases of three or all four of the parameters a, b, c, d exceeding 1 in absolute value in exactly the same way. For example, in the extreme case when min(lal, Ibl, lei, Idl) > 1 > Iql with
laqm+11 < 1 < laqml, Icq s+11
< 1 < IcqSI,
Ibqr+11 < 1 < Ibqrl, Idqt+11 < 1 < Idqtl,
(6.6.22)
for some nonnegative integers m, r, s, t such that the pairwise products of a, b, c, d are not of the form q-n and the pairwise ratios of a, b, c, d are not
168
The Askey-Wilson q-beta integral
of the form q±n for n
=
0,1,2, ... , we have the formula
I(a, b, c, d) - Lm(a; b, c, d) - Lr(b; c, d, a) - Ls(c; d, a, b) - Lt(d; a, b, c) 27r( abcd; q)oo (6.6.23) (q, ab, ac, ad, bc, bd, cd; q)oo' where Ls(c; d, a, b) and Lt(d; a, b, c) are the same type of finite series as those in (6.6.11) and (6.6.20), and can be written down by obvious replacement of the parameters.
Exercises
6.1 Prove that
17r
sin 2 () d()
IT
(1- 2aj cos() + a;) j=1 when max(la11, la21, la31, la41) o
2
7r(1 - a1 a2a3a4) I1 (1 - ajak) 1<;j
< 1.
6.2 Use the q-binomial theorem and an appropriate transformation formula for the 2cP1 series to show that
1
1 h(x; 1, -1) -1 h(x; a, -a)
dx
VI -
x2 27r(qa 2; q)oo
1. .!. 1. 1. (q, -q, aq2, -aq2, aq2, -aq2, -a2.,q)oo '
6.3 Prove that
lal
< 1.
r
10 =
(e 2iO , e- 2iO ; q2)00 d() (qe 2iO , qe- 2iO ; q2)00 1 - 2z cos () + z2 27r(q, qz2,q3 z 2;q2)00 oo l-q ( 2)n qz (q2,q2,q2 z 2;q2)00 n=O 1- q2n+1 '
L
where 0 < q < 1 and Izq!1 < 1. 6.4 If 0 < q < 1 and max(lal, Ibl, Icl, Idl)
1
1 h(x;I,-I,q!,j) -1 h(x; a, b, c, d)
< 1, show that
dx
VI -
x2
27r(aj, bj, cj, -abcq!, abcd; q)oo 1
1
1
(q, ab, ac, ad, bc, bd, cd, -aq2, -bq2, -Cq2, abcj; q)oo
x 8 W 7 (abcjq-\ ab, ac, bc, - jq-!, Jld; q, -dq!). 6.5 If max(lal, Ibl, Icl, Idl, Iql)
1
< 1, show that
1 h(X;I,~I,q!,-q!,-q/c) dx -1 h(x, a, -a, b, -b, c) VI - x 2 27r( -q; q)oo( _a 2b2c2, _q2 /c 2; q2)00 (q; q)oo( -a 2, -b 2, a 2b2, a 2c2, b2c2; q2)00·
Exercises
169
6.6 If b = a-I, Icl < 1, Idl < 1 and laqm+11 < 1 < laqml for some nonnegative integer m such that ac, ad are not of the form q-n, n = 0,1,2, ... , show that
I(
d)
-1
a, a X
, c,
27f(qa 2)-1
+ (q,q,acq,adq,c/ aq,d / aq;q )=
m-1 (1 - a2q2k+2) (acq, adq; q)k(q/cd)k { ; (1 - qk+l )(1 - a2qk+l )(aq2 /c, aq2/d; q)k 27f
(q,q,ac,ad,c/a,d/a;q)= x
~ {a2 ~ qr + (Cd)-~ _ qr -
ac_ l1 _ qr - ad_ l1 _ qr } qr,
where I(a, b, c, d) is as defined in (6.2.1). If m side is to be interpreted as zero.
=
0, the series on the left
6.7 Applying (2.12.9) twice deduce from(6.4.8) that
r
h(cos();I,-I,q!,-q!,A,J-L)d() h(cos();al,a2,a3,a4,as,a6)
Jo
= A (J-L 2;?)=
IT
(Aa j ;?)= (J-L/ A, q)= j=l (Ajaj, q)=
x
10 W g (A2q-1;
+ A (A2; .q)= (A/ J-L, q)= x
AJ-L/q, Ajal, A/a2, Aja3, Aja4, A/as, Aja6; q, q)
IT
j=l
10 W g (J-L2 q-l;
(J-Laj;.q)= (J-L/aj, q)=
AJ-L/ q, J-L/ aI, J-L/ a2, J-L/ a3, J-L/ a4, J-L/ as, J-L/ a6; q, q),
6
where AJ-L
= I1 aj, j=l
max(lql, lall,···, la61)
< 1, and
(Rahman [1986b, 1988b]) 6.8 Prove that 8W 7
(abczq-\ az, bz, cz, abcdflq, 1/ J; q, dqJ
(q,az, a/z,bz, b/z, cz, abcz, abc/Az, AJ-L2q/ abdz,J-L2, ab/J-L2; q)= 27f(ab, ac, bc, q/dz, zq/ Jd, abczJ, qJ-L2/dz, aq/ JdJ-L2; q)=
170
The Askey-Wilson q-beta integral
Aq a2be .) x ( - , - 2 ,bef,q bd AJ-L 00 x
SW7
XSW7
11
h(x;l,-l,q!,-q!,I!j,/aJ-L) A -1 h(x· Q 12. HZ I!:. abe ~) , J-L' J-L' "" , z' AJ-L ' abd ab Aq ) a aei!) ae- iO q ( dfJ-L2; ---;;-' -J-L-' bdf' fAJ-L2' abedi q, bef ) dx J-L 2 J-LeiO J-Le- iO q ab AJ-L 2q ( -d ; - ' - - ' - d "'-bd;q,ez ~' z z z Z /\ a e V 1 - x2
where Izl = 1 andmax(lal, Ibl, lei, Idl, If I, IJ-LI, la/J-LI, Ib/J-LI, labe/AJ-LI, IAJ-Lq/abdl, Iq/dzl, Iql) < 1. 6.9 Choosing A = abedq-1 and f
where z
= cos'lj;
= qn deduce from Ex. 6.8 that
and max(lal, Ibl, IJ-LI, Iql)
< 1.
6.10 Show that
roo (iax, -ia/x, ibx, -ib/x, iex, -ie/x, idx, -id/x; q)oo dx (-qx 2,_q/x2;q)00 x (log q-1 )(ab/q, ae/q, ad/q, be/q, bd/q, ed/q, q; q)oo (abed/ q3; q)oo
Jo
when labed/q 31 < 1 and Iql (Askey [1988b])
<
1.
Iql
< 1,
6.11 Show that
when Re x > 0, Re y (Askey [1980b])
> 0,
and k
= 0, 1, ....
171
Exercises 6.12 Show that
1o00 100 t 0
x - 1t x - 1 1
,00
(-at 1 qx+y+2k ,-at qx+y+2k. q) 2 ( - at1,-at2;q)00
2
x ti k (t2q1-k It1; qhk d q t 1d q t2
rq(x)rq(X + k)rq(y)rq(y + k)rq(k + l)rq(2k + 1) rq(x + y + k)rq(x + y + 2k)rq(k + l)rq(k + 1) x (-aqX, _ aq x+2k, _q1-x I a, _q1-x-2k I a; q)oo (-a, -a, -qla, -qla; q)oo when Re x > 0, Re y (Askey [1980b])
> 0, Iql < 1, and k = 0, 1, ....
6.13 With w(x; a, b, c, d) defined as in (6.3.1), prove that
1111 -1
1
1
1
1
w(x; a, aq2, b, bq2) w(y; a, aq2, b, bq2)
-1
x l(q!ei(lIH),q!ei(lI-1»;qhI2dxdy 2n
2
=
J1
[(1 - ab)(q, abq(j-1)k+1/2, abq(j-1)k+1; q)ooJ2 2
x
II
(a 2b2qj k+1,qk+l,q;q)00 (a2q(j-1)k+1/2 b2q(j-1)k+1/2 qjk+1. q)
j=l
'
where x = cosO, y = coscp, (Rahman [1986a])
Iql < 1, and k =
"
00 ,
0,1, ....
6.14 Let C.T. f(x) denote the constant term in the Laurent expansion of a function f(x). Prove that if j and k are nonnegative integers, then C.T. (qx, x-\ q)j(qx 2, qx- 2; q2)k
= 2~ 17r(qe2i!:1,e-2ill;q)j(qe4ill,qe-4ill;q2)k dO (q; qhj(q2; q2hk(q2j+\ q2h (q; q)j(q2; q2h(q; q)j+2k (Askey [1982b])
I:
6.15 Verify Ramanujan's identities
(i)
e-x2+2mX(_aqe2kX,_bqe-2kX;q)00 dx
vfif( abq; q)ooem2 (ae2mk..fil, be- 2mk ..fil; q)oo '
1-00 00
(ii)
e-x2+2mxdx
(ae 2ikx ..fil, be- 2ikx ..fil; q)oo vfif em2 (_aqe 2imk , _bqe- 2imk ; q)oo (abq; q)oo
172
The Askey-Wilson q-beta integral where q = e- 2k2 and Iql < l. (See Ramanujan [1988] and Askey [1982a])
6.16 Derive the q-beta integral formulas
(i)
{OO Jo
(-tqb,-qa+~/t;q)oo
(-t,-qlt,q)oo
dqt = rq(a)rq(b) t rq(a+b)
and (ii)
{OO (_tqb, _qa+l It; q)oo dt = -log q r q(a)r q(b) ,
Jo
(-t,-qlt;q)oo
t
where 0 < q < 1, Re a > 0 and Re b > (Askey and Roy [1986], Gasper [1987])
1-q rq(a+b)
o.
6.17 Extend the above q- beta integral formulas to
(i)
{OO t c - 1 (-tqb, _qa+l It; q)oo d t
Jo
(-t, -qlt; q)oo q (_qC, _ql-c; q)oo r q(a + c)r q(b - c) (-l,-q;q)oo rq(a+b)
and
(ii)
{OO t c - 1 (_tqb, _qa+~ It; q)oo dt
Jo
(-t, -qlt, q)oo r(c)r(l- c)rq(a + c)rq(b - c) rq(c)rq(l- c)rq(a + b)
where 0 < q < 1, Re (a + c) > 0 and Re (b - c) > o. (See Ramanujan [1915]' Askey and Roy [1986], Gasper [1987], Askey [1988b], and Koornwinder [2003]) 6.18 Use Ex. 2.16(ii) to show that, for y
= cos ¢;,
if x> y, if x < y, if x = y. (Ismail and Rahman [2002b]) 6.19 Denoting S(al' ... ' am) Ex. 2.16(i) to prove that
= S(al' ... ' am; q) =
n:=l (ak, qlak; q)oo,
use
Notes
173
Deduce that n
r
7r
io
rr rr
(2W -2iIJ.) h(cosf); Ak, q/Ak) e ,e ,q 00 k=1 df) h(cos f); a, b) n + l ( f ) /) h cos ;am,q am m=1
n+l n .) S(Ak/aj, Akaj) q, q, q 00 j=1 k=1
21f =
(1 _ ab)(
L II
n+l X
II [S(araj, ar/aj)(aaj, baj, aq/aj, bq/aj; q)oo]-I. r=l r=l-j
(Ismail and Rahman [2002b]) 6.20 Show that
1
1 h(x; 1, -1, ql/2, _ql/2, aql/2, ql/2/ a )
-1
h(x;al,a2, ... ,a6)
dx
viI -x 2
rr (aajql/2, ajql/2 / a; q)oo j=1 (q, q, qa 2, q/ a 2; q)oo rr (ajak; q)oo l::;j
(al a2a3a4a5a6/ q; q)oo
X
8'l/J8 [
/ ] qa, -qa, aql/2 / aI, ... , aql/2 / a6 a a a a a, -a, aalq 1/2 , ... , aa6q 1 / 2 ; q, al a2 3 4 5 6 q ,
provided a i=- q±n, n = 0,1,2, ... , lala2a3a4a5a6/ql k = 1, ... ,6. (See Gustafson [1987b] and Rahman [1996a])
< 1, and lakl < 1,
6.21 Prove that
sin V2 cosh VI (q, _qe 2V1 , _qe- 2V1 , tl t2t3t4/ q3; q)oo I(qe 2iv2 ; q)oo 12 ' where V = VI + iV2, 0 < V2 Itlt2t3t4/q31 < l. (Ismail and Masson [1994])
<
1f/2 or V2
= 1f/2 and VI :::; 0, Itjl < 1,
Notes Setting al = -a2 = a, a3 = -a4 aq~, a5 = b, a6 = C, A = a 2 c and using Ex. 2.25, Rahman [1988b] evaluated this integral in closed form and found the corresponding systems of biorthogonal rational functions. Ex.6.7
a 2 bq, JL
=
174
The Askey-Wilson q-beta integral
Ex. 6.8 This may be regarded as a q-extension of the terminating case of Erdelyi's [1953, 2.4(3)] formula
r
1 r( c) t lL - 1 (1 _ t)c- IL -1 (1 _ tx ).. -a-b r(JL)r(c - JL) 10 x F(A - a, A - b; JL; tx) F(a + b - A, A - JL; c - JL; (1 - t)x/(1 - tx)) dt,
F( a, b; c; x) =
where Ixl < 1, 0 < Re JL < Re c. Also see Gasper [1975c, 2000]. Ex.6.10 Askey [1988b] found the 4¢3 polynomials which are orthogonal with respect to the weight function associated with this integral. Exercises 6.11- 6.13 The double integrals in these exercises are q-analogues of the n = 2 case of Selberg's [1944] important multivariable extension of the beta integral: 2z
=
IT j=l
r(x + (j - l)z)r(y + (j - l)z)r(jz + 1) r(x+y+(n+j-2)z)r(z+l) ,
where Re x > 0, Re y > 0, Re z > - min(l/n, Re x/(n - 1), Re y/(n - 1)). Aomoto [1987] considered a generalization of Selberg's integral and utilized the extra freedom that he had in his integral to give a short elegant proof of it. Habsieger [1988] and Kadell [1988b] proved a q-analogue of Selberg's integral that was conjectured in Askey [1980b]. For conjectured multivariable extensions of the integrals in Exercises 6.11-6.13, other conjectured q-analogues of Selberg's integral, and related constant term identities that come from root systems associated with Lie algebras, see Andrews [1986, 1988], Askey [1980b, 1982b, 1985a, 1989f, 1990], Evans, Ismail and Stanton [1982]' Garvan [1990], Garvan and Gonnet [1992]' Habsieger [1988], Kadell [1988a-1994]' Macdonald [1972-1995]' Milne [1985a, 1989], Morris [1982]' Rahman [1986a], Stanton [1986b, 1989], and Zeilberger [1987, 1988, 1990a]. Also see the extension of Ex. 6.11, the q-Selberg integrals, and the identities in Aomoto [1998], Evans [1994]' Ito [2002]' Kaneko [1996-1998], Lassalle and Schlosser [2003], Macdonald [1998a,b], Rains [2003a], Stokman [2002] and Tarasov and Varchenko [1997].
7 APPLICATIONS TO ORTHOGONAL POLYNOMIALS
7.1 Orthogonality Let a(x) be a non-constant, non-decreasing, real-valued bounded function defined on (-00,00) such that its moments /Ln =
I:
xnda(x),
n = 0,1,2, ... ,
(7.1.1)
are finite. A finite or infinite sequence Po (x), Pl (x), ... of polynomials, where Pn(x) is of degree n in x, is said to be orthogonal with respect to the measure da(x) and called an orthogonal system of polynomials if (7.1.2) In view of the definition of a(x) the integrals in (7.1.1) and (7.1.2) exist in the Lebesgue-Stieltjes sense. If a(x) is absolutely continuous and da(x) = w(x)dx, then the orthogonality relation reduces to
I:
Pm(X)Pn(X)W(X) dx = 0,
m -=I- n,
(7.1.3)
and the sequence {Pn (x)} is said to be orthogonal with respect to the weight function w(x). If a( x) is a step function (usually taken to be right-continuous) with jumps Wj at x = Xj,j = 0,1,2, ... , then (7.1.2) reduces to LPm(Xj)Pn(Xj)Wj = 0,
m -=I- n.
(7.1.4)
j
In this case the polynomials are said to be orthogonal with respect to a jump function and are usually referred to as orthogonal polynomials of a discrete variable. Every orthogonal system of real valued polynomials {Pn(x)} satisfies a three-term recurrence relation of the form
(7.1.5) with p-l(X) == O,Po(x) == 1, where An, B n , C n are real and AnCn+ l > 0. Conversely, if (7.1.5) holds for a sequence of polynomials {Pn(x)} such that P-l(X) == O,Po(x) == 1 and An, B n , C n are real with AnCn+ l > 0, then there exists a positive measure da(x) such that (7.1.6) 175
176
Applications to orthogonal polynomials
where
_ rrn A k - l Vn C' k
k=l
Vo = 1.
(7.1.7)
°
If {Pn(X)} = {Pn(x)}~=o and AnCn+1 > for n = 0,1,2, ... , then the measure has infinitely many points of support, (7.1.5) holds for n = 0,1,2, ... , and (7.1.6) holds for m, n = 0,1,2, .... If {Pn(x)} = {Pn(x)};;=o and AnCn+1 > for n = 0,1,2, ... , N -1, where N is a fixed positive integer, then the measure can be taken to have support on N + 1 points xo, Xl,"" XN, (7.1.5) holds for n = 0,1, ... , N - 1, and (7.1.6) holds for m, n = 0,1,2, ... , N. This characterization theorem of orthogonal polynomials is usually attributed to Favard [1935], but it appeared earlier in published works of Perron [1929]' Wintner [1929] and Stone [1932]. For a detailed discussion of this theorem see, for example, Atkinson [1964], Chihara [1978], Freud [1971] and Szego [1975]. In the finite discrete case the recurrence relation (7.1.5) is a discrete analogue of a Sturm-Liouville two-point boundary-value problem with boundary conditions p-I(X) = O,PN+I(X) = 0. If Xo, Xl, ... , XN are the zeros of PN+l(X), which can be easily proved to be real and distinct (see e.g., Atkinson [1964] for a complete proof), then the orthogonality relation (7.1.6) can be written in the form
°
N
N
LPm(Xj)Pn(Xj)Wj = V;;:-l L Wj bm,n, (7.1.8) j=O j=O m, n = 0,1, ... , N, where Wj is the positive jump at Xj and Vn is as defined in (7.1.7). The dual orthogonality relation N
N
LPn(Xj)Pn(Xk)Vn = wjl L Wn bj,k, n=O n=O
(7.1.9)
j, k = 0,1, ... , N, follows from the fact that a matrix that is orthogonal by rows is also orthogonal by columns. It can be shown that
Wj = [ANvNPN(Xj )P~+l (Xj)
r
N
lL
Wn,
j
= 0, 1, ... ,N,
(7.1.10)
n=O
where the prime indicates the first derivative. In general, the measure in (7.1.6) is not unique and, given a recurrence relation, it may not be possible to find an explicit formula for o:(x). Even though the classical orthogonal polynomials, which include the Jacobi polynomials
P~Q,(3)(x) =
(0:
:!I)n
2FI (
-n, n
+ 0: + (3 + 1; 0: + 1; 1; X)
,
(7.1.11)
and the ultraspherical polynomials
C~(x)
((2>')t) p2-~')..-~)(x) >'+"2n n _ L -'(>'h(>')n-k -----'---:----'----'------:-:- ei(n-2k)e
=
-
k=O
k!(n-k)!
'
X
= cos e,
(7.1.12)
7.2 The finite discrete case
177
are orthogonal with respect to unique measures (see Szego [1975]), it is not easy to discover these measures from the corresponding recurrence relations (see e.g., Askey and Ismail [1984]). However, for a wide class of discrete orthogonal polynomials it is possible to use the recurrence relation (7.1.5) and the formulas (7.1.8)-(7.1.10) to compute the jumps Wj and hence the measure. We shall illustrate this in the next section by considering the q-Racah polynomials (Askey and Wilson [1979]).
7.2 The finite discrete case: the q-Racah polynomials and some special cases Suppose {Pn(x)} is a finite discrete orthogonal polynomial sequence which satisfies a three-term recurrence relation of the form (7.1.5) and the orthogonality relations (7.1.8) and (7.1.9) with the weights Wj and the normalization constants Vn given by (7.1.10) and (7.1.7), respectively. We shall now assume, without any loss of generality, that Pn(XO) = 1 for n = 0,1, ... , N. This enables us to rewrite (7.1.5) in the form
(x - XO)Pn(x) where n
=
An [Pn+l(X) - Pn(x)]- en [Pn(x) - Pn-l(X)] ,
= 0,1, ... ,N.
Setting j
= k = 0 in (7.1.9)
N
L Vn =
(7.2.1)
we find that
N
L
(7.2.2) W n· n=O n=O It is clear that in order to obtain solutions of (7.2.1) which are representable W;;l
N
in terms of basic series it would be helpful if Vn and quotients of products of q-shifted factorials. (2.4.2) in mind, let us take
Vn
L
Vn were equal to n=O Therefore, with the 6¢5 sum
(abq; q)n (1 - abq2n+l) (aq, eq, bdq; q)n ( )-n (q; q)n(1- abq) (bq, abq/e, aq/d; q)n edq
=
n
II
=
k=l
(1 _ abqk) (1 - abq2k+l) (1 - aqk)(l - eqk)(l - bdqk) (1- qk) (1- abq2k-l) (1- bqk)(l- abqk/e)(l- aqk/d)edq' (7.2.3)
where bdq
=
q-N, 0 < q < 1, so that
~
[ab q, q(abq)~, -q(abq)~, aq, eq, q-N .
_
~ Vn -
n=O
6¢5
1
1
N
(abq)'i, -(abq)'i, bq, abq/e, abq +
2'
(abq2, b/e; q)N (bq, abq/e; q)N' where it is assumed that a, b, e, d are such that Vn > 0 for n view of (7.1. 7) we can take A
q,
bqN ]
(7.2.4)
= 0,1, ... , N.
_ (1 - abqk) (1 - aqk) (1 - eqk) (1 - bdqk) (1 _ abq2k-l) rk,
k-l -
e
In
(7.2.5)
178
Applications to orthogonal polynomials
(7.2.6) where {rdf=l is an arbitrary sequence with rk =I- 0,1 ::::; k ::::; N. Since Co = 0 and Ao = (1 - aq)(l - eq)(l - bdq)rl' we have from the n = 0 case of (7.2.1) that -1 (1- q-l)(Xj - xo)qr 11 Pl ( X· ) (7.2.7) J (1 - q)(l - aq)(l - eq)(l - bdq) This suggests that we should look for a basic series representation of Pn (x j ) whose (k + l)-th term has (q, aq, eq, bdq; qh as its denominator, which in turn suggests considering a terminating 4¢3 series. In view of the product ( -n) k (n+ a+,8 + l)k in the numerator ofthe (k+ l)-th term in the hypergeometric series representation of p~<x,(3)(x) in (7.1.11) and the dual orthogonality relations (7.1.8) and (7.1.9) required for Pn(Xj) it is natural to look for a 4¢3 series whose (k + l)-th term has the numerator
Replacing q<X, qf3, q'Y, q8 by a, b, e, d, respectively, we are then led to consider a 4¢3 series of the form 4¢3
[
q-n, abqn+l, q-j, edqj+l . ] b d ' q, q . aq, eq, q
(7.2.8)
Observing that (q-j, edqJ+\ q) k is a polynomial of degree k in the variable q-j + edqj+1, we find that if we take
(7.2.9) then Xj - Xo = -(1- q-j)(l- edqJ+l) , and so (7.2.7) is satisfied with
(7.2.10) Then AkCk+1 > 0 for 0 ::::; k ::::; N - 1 if, for example, a, b, e are real, d b-1q-N-1, be < 0, max(laql, Ibql, leql, lablel) < 1 and 0 < q < 1. We shall now verify that _ Pn(Xj) satisfies (7.2.1) with x
4¢3
[q-n,abqn+l,q-j,edqj+l. ] b d ' q, q aq, eq, q
=
(7.2.11)
= Xj. A straightforward calculation gives
(7.2.12)
7.2 The finite discrete case
o< n
179
::::: N. So we need to verify that
q-n, abqn+l, q-j, edqj+l ] b d ; q, q aq, eq, q Anq-n (1 - abq2n+2) [q-n, abqn+2, ql- j , edqj+2 . _ ] - (1 - aq) (1 - eq) (1 - bdq) 4¢3 aq2, eq2, bdq2 ' q, q
4¢3
[
C nq l-n (1- abq2n) - (l-aq)(l-eq)(l-bdq)
4¢3
[ql-n, abqn+l, ql- j , edqj+2 . ] aq2,eq2,bdq2 ,q,q,
(7.2.13) where An and C n are given by (7.2.5) and (7.2.6) with Tk as defined in (7.2.10). Use of (2.10.4) on both sides of (7.2.13) reduces the problem to verifying that [
4¢3
q-n, abqn+l, bq-j Ie, bdqi+l ] bq, a bq I e, bdq ; q, q
(1- abqn+l)
(1- bdqn+l)
(1 - abq2n+l) (1 - bdq) x
4¢3
[
q-n, abqn+2, bq-j Ie, bdqi+l ] bq, a bq I e, bd q2 ; q, q
(1 - qn) (1 - aqn Id) (1 - abq2n+l) (1 _ bdq) bdq X4¢3
[
ql-n, abqn+l, bq-j Ie, bdqi+l ] bq, a blbd2 ;q,q, q e, q
(7.2.14)
which follows immediately from the fact that (q-n; qh (abqn+l; q) k+l
(1 -
bdqn+l)
(1 - abq2n+l) (1 - bdqk+l)
(1-
+
(q-n;q)k+l (abqn+\q)k af) bdqn+l (1 - abq2n+l) (1 _ bdqk+l) = (q-n, abqn+1; q) k .
The verification that (7.2.11) satisfies the boundary condition PN+l(Xj) = ::::: N is left as an exercise (Ex. 7.3). Note that the 4¢3 series in (7.2.11) remains unchanged if we switch n, a, b, respectively, with j, e, d. This implies that the polynomials Pn(Xj) are self-dual in the sense that they are of degree n in Xj = q-j + edqi+l and of degree j in Yn = q-n + abqn+l and that the weights Wj are obtained from the Vn in (7.2.3) by replacing n, a, b, e, d, by j, e, d, a, b, respectively, i.e.,
o for 0 ::::: j
W· J
=
(edq; q)j (1 - edq2i+l) (eq, aq, bdq; q)j ( )_j abq , (q; q)j(l - edq) (dq, edqla, eqlb; q)j
(7.2.15)
o : : : j ::::: N. Thus {Pn(Xj)};;=o is orthogonal with respect to the weights Wj, while {Pj (Yn)}f=o is orthogonal with respect to the weights Vn (see Ex. 7.4). The calculations have so far been done with the assumption that bdq = q-N, but the same results will hold if we assume that one of aq, eq and bdq is q- N.
180
Applications to orthogonal polynomials
The polynomials Pn(Xj) were discovered by Askey and Wilson [1979] as q-analogues of the Racah polynomials
Pn
F [-n,n+a+ f3 +1,-j,j+1'+6"+1.1]
( ,\.) = 4
J
a
3
+ 1, l' + 1, f3 + 6" + 1
"
(7.2.16)
where ,\j = j(j + l' + 6" + 1), which were named after the physicist Racah, who worked out their orthogonality without apparently being aware of the polynomial orthogonality. In the physicist's language the Pn ('\j) 's are known as Racah coefficients or 6j-symbols. The connection between the 6j-symbols and the 4F3 polynomials (7.2.16) was first made in Wilson's [1978] thesis. The q-analogues in (7.2.11) are called the q-Racah polynomials. We shall now point out some important special cases of the q-Racah polynomials. First, let us set d = b-1q-N-l in (7.2.11) and replace e by be to rewrite the q- Racah polynomials in the more standard notation
Wn(x;a,b,e,N;q)
=
4¢3
[
] q-n,abqn+1, q-X, eqx-N -N ;q,q . aq,q ,beq
(7.2.17)
Then, by (7.1.8), (7.2.3) and (7.2.15), N
LP(x;q)Wm(X;q)Wn(x;q) x=o
=
~m(,n),
(7.2.18)
n q
where
Wn(X; q) p(x; q)
== Wn(x; a, b, e, N; q), == p(x; a, b, e, N; q) (eq-N. q) (1 eq2X-N) (aq beq q-N. q)
=
'x ' , ' x (abq)-X (7.2.19) (q; q)x(l - eq-N)(ea-1q-N, b-1q-N, cq; q)x
and
hn(q)
== hn(a, b, e, N; q) = (bq, aqle; q)N (abq; q)n (1 - abq2n+l) (aq, beq, q-N; q)n (qN ler . (abq2, lie; q)N
(q; q)n(1- abq) (bq, aqle, abqN+2; q)n (7.2.20)
Setting e = 0 in (7.2.17) gives the q-Hahn polynomials
Qn(x)
_
=
. . _ Qn(x, a, b, N, q) -
3¢2
[q-n, abqn+l, q-X . ] aq, q-N ,q, q ,
(7.2.21 )
which were introduced by Hahn [1949a] and, by (7.2.18), satisfy the orthogonality relation N
L Qm(x)Qn(x) (aq; q)x(bq; q)N-x (aq)-X x=o (q; q)x(q; q)N-x 2 (abq2; q)N(aq)-N -----'--(q_;q--'-,)_n('-:-1_-_a_bq--'::),.:-(b.,. . "q,-:-a,-qb_N_+----:;c:-'q)---,n_(_ )n (n)-Nn.r -:N aq q 2 Um n (q; q)N (abq; q)n(1- abq2n+l )(aq, q- ; q)n ' (7.2.22)
7.3 The infinite discrete case
181
for m, n = 0,1, ... , N. Setting a = 0 in (7.2.17) we obtain the dual q-Hahn polynomials (Hahn [1949b])
Rn(P,(x))
== Rn(P,(x); b, e, N; q) _ -
where p,(x)
=
3¢2
[q-n, q-X, eqx-N . ] q-N, beq , q, q ,
(7.2.23)
+ eqx-N, which satisfy the orthogonality relation
q-X
N (eq-N. q) (1 eq2X-N) (beq q-N. q) x ' " Rm(P,(x))Rn(P,(x)) 'x " x qNX-(2)(_beq)-X (q; q)x(1- eq-N)(b-1q-N, eq; q)x
f='o =
(!t
e; ~)N (b (q, b~;;)n) (eq-Nt bm,n, q;q N eq,q ;q n
(7.2.24)
m, n = 0,1, ... , N. For some applications of q-Hahn, dual q-Hahn polynomials, and their limit cases, see Delsarte [1976a,b, 1978], Delsarte and Goethals [1975], Dunkl [1977-1980] and Stanton [1977-1986c].
7.3 The infinite discrete case: the little and big q-Jacobi polynomials As a q-analogue of the Jacobi polynomials (7.1.11), Hahn [1949a] (also see Andrews and Askey [1977]) introduced the polynomials
Pn(x;a,b;q)
= 2¢1(q-n,abqn+\aq;q,xq).
(7.3.1)
It can be easily verified that
. (1-
hm Pn
q--->l
x. a (3. ) _ p~a,(3)(x) - - , q ,q ,q ((3) .
2
Pn a,
(7.3.2)
(1)
He proved that
~ (x. b.) (x. b.) (bq; q)x ( )X _ bm,n ~ Pm q ,a, ,q Pn q ,a, ,q ( .) aq - h ( b. )'
(7.3.3)
< q,aq < 1 and h ( b. ) = (abq; q)n(1- abq2n+l )(aq; q)n(aq; q)= ( )-n
(7.3.4)
x=o
q, q x
n a, ,q
where 0
n a, ,q
(q,. q) n (1 - abq ) (bq,. q)n (abq 2.) ,q =
aq.
Observe that (7.3.3) and (7.3.4) also follow from (7.2.22) when we replace x by N - x and then let N ---+ 00. To prove (7.3.3), assume, as we may, that o :::; m :::; n, and observe that
f
(bq; q)x (aq)xqxkPm(qx; a, b; q) x=o (q; q)x m (q-m abqm+1. q) . ' J qj l¢o(bq;-;q,aq1+k+ j ) j=O (q,aq;q)j
=2:
'
182
Applications to orthogonal polynomials
(7.3.5) by the q-binomial and q-Saalschiitz formulas. Then the orthogonality relation (7.3.3) follows immediately by using (1.5.2) and (7.3.5). It is easy to verify that Pn(x; a, b; q) satisfies the three-term recurrence relation (7.3.6) for n 2: 0, where A
_ (1- abqn+l) (1- aqn+l) _ n n - (1 _ abq2n+l) (1 _ abq2n+2) ( q ),
(7.3.7)
(l-qn)(l-bqn) n C n = (1 _ abq2n) (1 _ abq2n+l) (-aq ),
(7.3.8)
so the condition that AnCn +l > 0 for n = 0,1, ... , is satisfied if 0 < q, aq < 1 and bq < 1. When b < 0 the polynomials Pn(x; a, b; q) give a q-analogue of the Laguerre polynomials L~(1) (x) since liml Pn ((1 - q)x; qOl, _q(3; q)
q-+
= L~OI) (x) / L~OI) (0).
(7.3.9)
Andrews and Askey [1985] introduced a second q-analogue of the Jacobi polynomials, (7.3.10) which has the property that •
•
01
(3
,.
_
hm Pn(x, q ,q ,-q ,q) -
q-+l
p~OI,(3)(X) ((3) , Pn 01, (1)
(7.3.11)
where'Y is real. In view of the third free parameter in (7.3.10) they called the Pn(x; a, b, c; q) the big q-Jacobi and the Pn(x; a, b; q) the little q-Jacobi polynomials. We shall now prove that the big q-Jacobi polynomials satisfy the orthogonality relation
l
aq
cq
( ) ( ) (x/a, x/c; q)oo Pm x; a, b, c; q Pn x; a, b, c; q ( b / . ) dqx x,xc,qoo
(7.3.12)
hn(a, b, c; q)'
where h ( b . ) n a, ,c, q
= M-l(abq;q)n(1_abq2n+l)(aq,cq;q)n(_ ( q; ) q n ( 1 - abq ) (bq, abq /c;) q n
2)-n acq
q
-G) (7.3.13)
7.3 The infinite discrete case
183
and M =
ja
q (x/a, x/c; q)oo dqx cq (x, bx/c; q)oo
aq(1 - q)(q, cia, aq/c, abq2; q)oo (aq,bq,cq, abq/c; q)oo
(7.3.14)
by (2.10.20). Since
j _ja
(x/a, X/C; q)oo (bx/c;q)j(X;q)k ( b / .) dqx cq x, x c, q 00 q (x/a, x/c; q)oo d - cq (k xq, bxql0/ C; q)00 qX aq
_ M(bq,abq/c;q)j(aq,cq;q)k (abq2; q)j+k ' the left side of (7.3.12) becomes M (bq, abq/c; q)m (c/b)m
(aq, cq; q)m m n (q-m abqm+1. q) (q-n abqn+1. q) '"' '"' ' '1 ' ' k j+k x ~ ~ (q,. q)j (q,. q) k (abq 2.) q, j=O k=O ,q j+k 0
(7.3.15)
where we used (7.3.10) and the observation that, by (3.2.2) and (3.2.5),
Pm(x; a, b, c; q) =
(bq,abq/c;q)m( /b)m C ( aq,cq;q ) m
r/,
3'1'2
[q-m,abqm+1,bX/C. ] bq,a bq/ c ' q, q . (7.3.16)
Assume that 0 ::; m ::; n. Since, by (1.5.3)
the double sum in (7.3.15) equals
n-m (qm-n abqm+n+1. q) (q -n abqn+1. q) , ' m ( b m+2) m ' " ' ' , k k (abq2; qhm a q ~ (q, abq2m+2; qh q _
-
(q -n abqn+1. q) ' ' n ( b n+2)n J: (b 2. ) a q Um,n' a q ,q 2n
Substituting this into (7.3.15), we obtain (7.3.12).
(7.3.17)
184
Applications to orthogonal polynomials
7.4 An absolutely continuous measure: the continuous q- ultraspherical polynomials In this and the following section we shall give two important examples of orthogonal polynomials which are orthogonal with respect to an absolutely continuous measure da(x) = w(x)dx. In his work of the 1890's, in which he discovered the now-famous RogersRamanujan identities, Rogers [1893b, 1894, 1895] introduced a set of orthogonal polynomials that are representable in terms of basic hypergeometric series and have the ultraspherical polynomials (7.1.12) as limits when q ---+ 1. Following Askey and Ismail [1983], we shall call these polynomials the continuous q-ultraspherical polynomials and define them by the generating function
((3te iO , (3te- iO ; q) 00 (·0 "0) te' , te-'· q '00
n
00
= "Cn(x; (3lq)t , ~
where x = cos 0, 0 ::; 0 ::; 7f and max(lql, Itl) theorem, it follows from (7.4.1) that
Cn(x; (3lq)
=
(7.4.1)
n=O
< 1. Using the q-binomial
t
((3; qh((3; q)n-k ei(n-2k)O k=O (q; qh(q; q)n-k
-- ((3; q)n inO ( ) e q;q
n
rI-. ( -n (3. 2'1-'1 q "
(3-1 q1-n.,q, q(3-1 e -2iO) .
(742) . .
Note that 1· C ( . AI ) - ~ (A)k(A)n-k i(n-2k)O 1m n x, q q - ~ k'( k)'. e • n k=O = C~(x).
q--->l
(7.4.3)
Before considering the orthogonality relation for Cn(x; (3lq), we shall first derive some important formulas for these polynomials. For 0 < 0 < 7f, 1(31 < 1, set a = q(3-1 e2iO, b = q(3-1, C = qe 2iO and z = (32 qn in (3.3.5) to obtain
2cP1 (q-n, (3; (3-1 q1-n; q, q(3-1 e -2iO) =
" " ((3qn, (3e- 2iO ; q) 00 rI-. (q(3-1 q(3-1 e 2,O. qe 2,o. q (32qn) n+1 -2iO.) 2'1-'1 , " , (q ,e ,qoo ((3, q(3-1, (3q n e2iO , (3-1 q1-n e -2iO; q) 00
+ (qn+1, e 2iO , q(3-1 e -2iO, (3-1 q1-n; q)oo X
2cP1 (q(3-l,q(3-1e-2iO;qe-2iO;q,(32qn).
(7.4.4)
Then, application of the transformation formula (1.4.3) to the two 2cP1 series on the right side of (7.4.4) gives
7.4 An absolutely continuous measure
185
(7.4.5) where
x = cos B.
(7.4.6)
Rewriting the right side of (7.4.5) as a q-integral we obtain the formula
Cn (x· ;3lq) ,
=
2i sin B (;3,;3; q)oo (;32; q) n (l-q)W{3(xlq) (q,;32;q)00 (q;q)n e - iO (queiO,que-iO;q)oo n X e iO (;3ueiO , ;3ue-iO ; q)oo u dqu,
l
(7.4.7)
which was found by Rahman and Verma [1986a]. Now use (1.4.1) to obtain from (7.4.5) that
Cn (x;;3lq) = (;3,;3;;q) ooWi 1(xlq) (;32;q)n (q,;3 ; q)oo (;3q; q)n 2iO inO X {(I - e ) e 2¢1 (q;3-1, qn+\ ;3qn+\ q, ;3e 2iO ) + (1 - e- 2iO ) e- inO 2¢1 (q;3-1, qn+\ ;3qn+l; q, ;3e- 2iO )} 00
= 4sinB Wil(xlq)
L b(k, n;;3) sin(n + 2k + l)B,
(7.4.8)
k=O
where 0
< B < 7[, 1;31 < 1 and b(k,n;;3) = (;3,;3;;q)oo (;32;q)n(q;3-\q)k(q;q)n+k ;3k. (q,;3 ; q)oo (q; q)n(q; q)d;3q; q)n+k
(7.4.9)
The series on the right side of (7.4.8) is absolutely convergent if 1;31 < 1. For Ixl < 1, Iql < 1 and large n it is clear from (7.4.5) that the leading term in the asymptotic expansion of Cn(cosB; ;3lq) is given by
C ( B·;3I) n cos, q
rv
(;3;q)oo {(;3e 2iO ;q)00 -inO + (;3e- 2i O;q)00 ino} (q;q)oo (e 2iO ;q)00 e (e- 2iO ;q)00 e
=2~;3;q?00 q;q
IA(eiO)lcos(nB-a),
(7.4.10)
00
where (7.4.11) and, as elsewhere, f(n) rv g(n) means that lim n--+ oo f(n)jg(n) = 1. For further results on the asymptotics of Cn(x; ;3lq), see Askey and Ismail [1980] and Rahman and Verma [1986a].
186
Applications to orthogonal polynomials
If we use (3.5.4) to express 2¢1 (q-n, (3; (3-1 q1-n; q, q(3-1 e-2iO) as a terminating VWP-balanced S¢7 series in base q~ and then apply (2.5.1), we obtain
,/-. ( -n (3. (3-1 1-n. (3-1 e -2iO) 2'/"1 q " q ,q,q
=
(q ~-n e -2iO.,q) n (_ q(1-n)/2(3-1.,q ~) n ((3 - 1 q21-n ; q21) n
-n/2 ,(31 2 e -iO ,- (31 2 e -iO , -q -n/2 1 1] (3 1_ n /2 -iO 1_ n /2 -iO ; q2, q2 - ,q4 e ,-q4 e (3 n/2 , (312 e iO , (312 e -iO . ~ ~ ] _ ((3 2.'q) n (3-n/2 -inO,/-. [ q-n/2 ,q - ((3 ) e 4 '/"3 1 1 1 1 ,q, q ;q n -(3, (32 q4, -(32 q4 X
4¢3
[q
(7.4.12)
by (2.10.4). However, by (3.10.13),
(7.4.13) and hence, from (7.4.2), (7.4.12) and (7.4.13), we have Cn(cosB; (3lq)
_ ((32;q)n -n/2 [q-n,(32qn,(3~eiO,(3~e-iO. ] - (q,q . ) n (3 4¢3 (3 q2,1 (3 ,- (3 q21 , q, q .
(7.4.14)
Since W,a(cosBlq) = IA(e iO )I- 2 for real (3, it follows from Theorem 40 in Nevai [1979] and the asymptotic formula (7.4.10) that the polynomials C n (cos B; (3lq) are orthogonal on [O,n] with respect to the measure W,a( cos Blq)dB, -1 < (3 < 1. One can also guess the weight function by setting (3 = q>' and comparing the generating function (7.4.1) and the expansion (7.4.8) with the q ---+ 1 limit cases and the weight function (1 - e2iO )>. (1 - e- 2iO )>' for the ultraspherical polynomials C~ (cos B). We shall now give a direct proof of the orthogonality relation
1" where Iql
Cm (cos B; (3lq)Cn (cos B; (3lq)W,a( cos Blq) dB
=
h:(~lq)'
(7.4.15)
< 1,1(31 < 1 and h n ((3lq)
=
(q, (32; ~)oo (q; ;!n(1- (3qn). 2n((3, (3q, q)oo ((3 ,q)n(1- (3)
(7.4.16)
As we shall see in the next section, (7.4.15) can be proved by using (7.4.14) and the Askey-Wilson q- beta integral (6.1.1); but here we shall give a direct proof by using (7.4.2), (1.9.10) and (1.9.11), as in Gasper [1981b]' to evaluate the integral. Since the integrand in (7.4.15) is even in B, it suffices to prove that (7.4.17)
7.4 An absolutely continuous measure
187
when 0 ::; m ::; n. We first show that, for any integer k,
j
7r -7r
e
ike
( I) { 0, if k is odd, W/3 cos () q d() = Ck/2 ((3I)·f . even, q, 1 k IS
(7.4.18)
where
(7.4.19)
I:
By the q-binomial theorem,
=
eike W/3( cos ()Iq)
ff r=Os=O
d()
((3-\ q)r((3-\ q)s (3r+s (q;q)r(q;q)s
r
J-7r
ei(k+2r-2s)e d(),
which equals zero when k is odd and equals
2n
f
s=o
((3-1: q)s((3.-\ q)s+j (31+2s (q, q)s(q, q)s+j
= 2n(~-.\~)j (3j 2¢1 ((3-1, (3-1 qj ; q1+\ q,(32) q, q
when k
=
2j, j
= 0,1, ....
J
By (1.4.5), the above 2¢1 series equals
((3, (3q1+\ q)oo rI-. ( -1 (3-1. (3. (3 j+1) 2\f'1 q , "q, q ((3 2 ,qj+1.) ,q 00 =
((3,(3,;!q)oo (q;.q)j (1+qj). (q,(3 ,q)oo ((3q,q)j
From this and the fact that W /3 ( cos () q) is symmetric in (), so that we can handle negative k's, we get (7.4.18). Hence, from (7.4.2), 1
(7.4.20) equals zero when n - k is odd and equals
(7.4.21 ) when n - k = 2j is even. From (1.9.11) it follows that this 4¢3 series and hence the integral (7.4.20) are equal to zero when n > Ikl. Hence (7.4.17) holds when m -I- n. If k = ±n -I- 0, then (1.9.10) gives
I:
eikOCn(cos();(3lq)W/3(cos()lq) d()
2n((3, (3q; q)00((32; q)n (q, (32; q)oo((3q; q)n
(7.4.22)
188
Applications to orthogonal polynomials
from which it follows that (7.4.17) also holds when m
=
n.
7.5 The Askey-Wilson polynomials In view of the 4¢3 series representation (7.4.14) for the continuous q-ultraspherical polynomials it is natural to consider the more general polynomials
(7.5.1) which are of degree n in x = cosO, and to try to determine the values of cv, /3, ,,(, 8, E for which these polynomials are orthogonal. Because terminating balanced 4¢3 series can be transformed to other balanced 4¢3 series and to VWP-balanced 8¢7 series which satisfy three-term transformation formulas (see, e.g., (7.2.13), (2.11.1), Exercise 2.15 and the three-term recurrence relation for the q-Racah polynomials), one is led to consider balanced 4¢3 series. From Sears' transformation formula (2.10.4) it follows that if we set cv = abcdq-l, /3 = a, "( = ab, 8 = ac and E = ad, then the polynomials
=
(ab, ac, ad; q)na-n
4¢3
[
q-n abcdqn-l ae iIJ ae- iIJ ] ' b 'd' ; q, q a ,ac,a
(7.5.2) are symmetric in a, b, c, d. In addition, for real 0 these polynomials are analytic functions of a, b, c, d and are, in view of the coefficient (ab, ac, ad; q)na-n, realvalued when a, b, c, d are real or, if complex, occur in conjugate pairs. Askey and Wilson [1985] introduced these polynomials as q-analogues of the 4F3 polynomials of Wilson [1978, 1980]. Since they derived the orthogonality relation, three-term recurrence relation, difference equation and other properties of Pn(x; a, b, c, dlq), these polynomials are now called the AskeyWilson polynomials. Since the three-term recurrence relation (7.2.1) for the q-Racah polynomials continues to hold without the restriction bdq = q-N, by translating it into the notation for Pn(x; a, b, c, dlq) we find, as in Askey and Wilson [1985], that the recurrence relation for these polynomials can be written in the form 2xPn(x) = AnPn+l(X)
+ BnPn(x) + CnPn-l(X),
n 2 0,
(7.5.3)
with P-l(X) = O,Po(x) = 1, where An =
1 - abcdqn-l
~------~~~------~~
(1 - abcdq2n-l) (1 - abcdq2n) ' (1 - qn)(1 - abqn-l )(1 - acqn-l )(1 - adqn-l) (1 - abcdq2n-2) (1 - abcdq2n-l) X (1- bcqn-l)(I- bdqn-l)(I- cdqn-l),
(7.5.4)
(7.5.5)
7.4 An absolutely continuous measure
189
and
= a + a-I
Bn
-
A n a- l (l- abqn)(l - aeqn)(l - adqn)
- Gna/ (1 - abqn-I) (1 - aeqn-I) (1 - adqn-I).
(7.5.6)
It is clear that An, B n , G n are real if a, b, e, d are real or, if complex, occur in conjugate pairs. Also AnGn+1 > 0, n = 0,1, ... , if the pairwise products of a, b, e, d are less than 1 in absolute value. So by Favard's theorem, there exists a measure da(x) with respect to which Pn(x; a, b, e, dlq) are orthogonal. In order to determine this measure let us assume that max(lal, Ibl, lei, Idl, Iql) < 1. Then, by (2.5.1), Pn( cos B; a, b, e, dlq) (ab, ae, be, de- iO , ; q)
=
"0
netn
(abee iO ; q)n S W7 (abee iO q-\ ae iO , be iO , ee iO , abedqn-I, q-n; q, qd-Ie- iO ) , (7.5.7)
X
and, by (2.11.1), S
W (abeeiOq-l. ae iO beiO ee iO abedqn-I q-n. qd-Ie- iO ) 7
""
"
q) 0 ( abee iO , bedeiOqn , beiOqn+1 , eeiOqn+1 , ae- iO , be- iO , ee- iO , de-iOqn.' 0 2iO 2iO ab ae be qn+1 bdqn edqn beqn+Ie e. q) ( ,
X S W7
"
,
"
"
CXJ
(beq n e 2iO ; beqn, beiO , ee iO , qa-Ie iO , qd-Ie iO ; q, adqn)
(abeeiO, abee- iO qn, bede- iO qn, be- iO qn+l, ee- iO qn+l; q) 00 +~--~~~~--~--~~~~--~~~~~--~
X
(qn+l, bdqn, edqn, abeeiOqn, beqn+Ie- 2iO ; q)oo q) 0 -2tnO " ( ae iO , beiO , ee iO , de iO '. 0 ( ab, ae, ad, e2"0) e t ; q 00
X SW 7
where
(beqn e -2iO; beqn, be- iO , ee- iO , qa-Ie- iO , qd-Ie- iO ; q, adqn) , (7.5.8)
°< B <
7L
Hence
Pn(cosB; a, b, e, dlq) =
(be, bd, cd; q)n { Qn (e iO ; a, b, e, dlq)
+ Qn (e- iO ; a, b, e, dlq) },
(7.5.9)
where Qn(z; a, b, e, dlq)
=
(abezqn,bedzqn,bzqn+l,ezqn+l,a/z,b/z,e/z,d/z;q)oo n z ( be bd cd abqn aeqn qn+1 bez 2qn+1 z-2. q) ,
X SW7
"
,
"
"
00
(beqn z 2; beqn, bz,ez, qa-Iz,qd-Iz; q,adqn) .
(7.5.10)
It is clear from (7.5.10) that Qn(z; a, b, e, dlq) '" zn B(Z-I )/(be, bd, cd; q)oo,
(7.5.11)
where (7.5.12)
190
Applications to orthogonal polynomials
as n ----; 00, uniformly for z, a, b, c, d in compact sets avoiding the poles z2 k = 0,1, .... Using (7.5.9) we find that
=
q-k,
Pn (cos e; a, b, e, dlq) '" ein () B(e- i()) + e- in () B(e i())
= 2IB(ei ())1 cos(ne - (3), where (3
(7.5.13)
= arg B(ei ()) and 0 < e < 7f (see Rahman [1986c]). Then IB(ei ())1 2 = [sine w(cose;a,b,e,dlq)r 1 ,
(7.5.14)
where, in order to be consistent with the Pn(x; a, b, e, dlq) notation, we have used w(x; a, b, e, dlq) to denote the weight function w(x; a, b, e, d) defined in (6.3.1). It follows from Theorem 40 in Nevai [1979] that the polynomials Pn(x;a,b,e,dlq) are orthogonal on [-1,1] with respect to the measure w(x; a, b, e, dlq)dx when max(lal, Ibl, lei, Idl, Iql) < 1. We shall now give a direct proof of the orthogonality relation (7.5.15) where
w(x) == w(x; a, b, e, dlq) hn == hn(a, b, e, dlq) -1 =/i
and
(abedq-1; q) n (1 - abedq2n-1) (a,b,e,dlq)(.) 1 ) ( ab, ae, ad, be, bd, cd,. q)n , q, q n ( 1 --abedq (7.5.16)
with
/i(a, b, e, dlq) = [11 w(x; a, b, e, dlq) dx 27f( abed; q)oo (q, ab, ae, ad, be, bd, cd; q)oo'
(7.5.17)
by (6.1.1). First observe that, by (7.5.17),
1
1 (aei(),ae-i();q)j (bei(),be-i();q)k w(x;a,b,e,dlq) dx
-1
=
11
w(x; aqj, bqk, e, dlq) dx
-1
=
.
k
/i(aqJ, bq ,e, dlq).
(7.5.18)
By using (7.5.2) and the fact that Pn(x; a, b, e, dlq) that the left side of (7.5.15) equals
=
Pn(x; b, a, e, dlq) we find
/i(a, b, e, dlq)(ab, ae, ad; q)m(ba, be, bd; q)na-mb-n
~ (q-m,abedqm-\q)j j ( b d.) q j=O q, a e ,q j
x ~
3¢2
[q-n,abedqn-1,abqj. ] abedqj ab ,q, q . (7.5.19) ,
7.4 An absolutely continuous measure
191
Assuming that 0 ::::: n ::::: m and using the q-Saalschiitz formula to sum the series, the sum over j in (7.5.19) gives
3¢2
( cd abcdqm-1 q-m. q) , " n n q ( q 1-nj a b.,q) n ( a bcd.) ,q 2n
A.
2'/-'1
(n-m b d n+m-1 b d 2n ) q ,a c q ; a c q ; q, q
(cd, abcdqm-1, q-m; q)n (q1+n-m; q)m-n (q1-n jab; q)n (abed; q)2n (abcdq2n; q)m-n (q,cd;q)n (1- abcdq-1) ()n m (abcdq- 1 ,ab; q)n (1 - abcdq 2 n- 1 ) ab 15 ,n·
(7.5.20)
Combining (7.5.19) and (7.5.20) completes the proof of (7.5.15). Askey and Wilson proved a more general orthogonality relation by using contour integration. They showed that if Iql < 1 and the pairwise products and quotients of a, b, c, d are not of the form q-k, k = 0,1, ... , then [11
+ 27r ~Pm(Xk)Pn(Xk)Wk
Pm (X)Pn(X)W(X) dx
(7.5.21 )
hn(a, b, c, dlq)'
where Xk are the points ~ (Jqk + j-1 q-k) with j equal to any of the parameters a, b, c, d whose absolute value is greater than 1, the sum is over the k with Ijqkl > 1, and Wk
== wk(a, b, c, dlq) (a- 2 ; q)oo (q, ab, ac, ad, bja, cja, dja; q)oo (a 2 ;qh(l-a2q2k)(ab,ac,ad;q)k X
(q; q)k(1 - a 2 )(aqjb, aqjc, aqjd; qh
(
q )k abed
(7.5.22)
when Xk = ~ (aqk + a- 1q-k). For a proof and complete discussion, see Askey and Wilson [1985]. Also, see Ex. 7.3l. In order to get a q-analogue of Jacobi polynomials, Askey and Wilson set a
=
q(2a+1)/4, b = q(2a+3)/4, C
=
_q(2,6+1)/4, d
=
_q(2 f3+ 3 )/4
(7.5.23)
and defined the continuous q-Jacobi polynomials by p(a,,6)(xlq) = (qa+1; q)n n (q;q)n q-n, qn+a+,6+1, q(2a+1)/4 e iO, q(2a+1)/4 e -iO . ] [ X 4¢3 qa+1, _q(a+,6+1)/2, _q(a+,6+2)/2 ,q, q.
(7.5.24)
On the other hand, Rahman [1981] found it convenient to work with an apparently different q-analogue, namely, p~a,,6)(x; q) =
,6+1.) ( a+1 q ,-q ,q n (q, -q; q)n
(7.5.25)
192
Applications to orthogonal polynomials
However, as Askey and Wilson pointed out, these two q-analogues are not really different since, by the quadratic transformation (3.10.13),
=
(-q; q)n qan p(a,(3) (X' q) (_qa+(3+I; q)n n ,.
(7.5.26)
limI p~a,(3)(xlq)
= limI p~a,(3)(x; q) = p~a,(3)(x).
(7.5.27)
p(a,(3)(Xlq2) n
Note that q--+
q--+
The orthogonality relations for these q-analogues are (7.5.28) and (7.5.29) where 0
< q < 1,
a
2 -~, jJ 2 -~, 2
(7.5.30)
an (a, jJlq)
=
(q, qa+l, q(3+I, _q(a+(3+I)/2, _q(a+(3+2)/2; q) 00 27rq ((a+(3+2)/2 ,q (a+(3+3)/2.,qoo )
x
( 1 - q2n+a+(3+I) (q qa+(3+I _q(a+(3+I)/2. q)
'" n -n(2a+I)/2 (1 - qa+(3+I) (qa+l, q(3+I, _q(a+(3+3)/2; q) n q (7.5.31 )
and
From (7.4.14), (7.5.24) and (7.5.25) it is obvious that
Cn (cos 0; qAlq) (q2\q)n (qA+2;1) q
q n
-nA/2p,(A-!,A-!)( n
01)
cos q
(7.5.33)
7.4 An absolutely continuous measure
193
It can also be shown that
C
( . AI ) -
(q\ -q; q)n (1q2,-q2,q 1.)
2n x, q q -
q
-n/2 ~(A-~'-~)(2 2 - l' ) n x, q ,
(7.5.35)
n
C
2n+ 1
(x' qAlq) = x (q\ -1; q)n+l q-n/2 ~(A-~'~)(2x2 - l' q) , n , , q2,-q2;q n+l
(1 1)
(7.5.36) which are q-analogues of the quadratic transformations
C A (x) = (A)n p,(A-~'-~)(2x2 - 1) (~)n
2n
and A () C2n+ 1 X
(A)n+l n(A-~'~)(2 2 rn X '2 n+l
=
(7.5.37)
n
X ( 1)
_
1)
(7.5.38)
,
respectively. To prove (7.5.35), observe that from (7.4.2)
C 2n (cos e; qAlq) _ (q\ q)2n 2inO - ( ) e q; q 2n
A,
2'/-'1
(-2n A. l-A-2n. I-A -2iO) q ,q, q , q, q e ,
and hence, by the Sears-Carlitz formula (Ex. 2.26),
C 2n(cose; qAlq)
=
(q\ q)2n ( q21 e -2iO ,q21-n e -2iO ; q) e2inO (q;q)2n n
x
A,
4 '/-'3
Reversing the
4¢>3
[
q-n, _q-n, _q~-n, q~-A-n. ] ql-A-2n, q~ -n e 2iO, q~ -n e -2iO , q, q .
(7.5.39)
series, we obtain
C2n (cos e; qAlq) _ -
(q A qA+~. q) ' ' n -n/2 ( 1 ) q q, q2; q
1
4¢>3
.
1
.
[q-n,qn+A,q2e2.0,q2e-2.0. ] A+~ _ ~ _ ' q, q . q ,q, q
n
(7.5.40) This, together with (7.5.25), yields (7.5.35). The proof of (7.5.36) is left as an exercise. Following Askey and Wilson [1985] we shall obtain what are now called the continuous q-Hahn polynomials. First note that the orthogonality relation (7.5.15) can be written in the form
1:
Pm(cose; a, b, c, dlq)Pn(cose; a, b, c, dlq)w(cose; a, b, c, dlq) sine de
hn(a, b, c, dlq)'
(7.5.41)
194
Applications to orthogonal polynomials
Replace B by B + ¢, a by ae i¢, b by ae- i¢ and then set c = be i¢, d = be- i ¢ to find by periodicity that if -1 < a, b < 1 or if b = a and lal < 1, then
[ : Pm(cos(B + ¢); a, blq)Pn(cos(B + ¢); a, blq)W(B) dB (7.5.42) where
(7.5.43) (7.5.44) and
x (1 _ a2b2q-l) (q , a2 , b2 , ab " ab abe 2i ¢ , abe- 2i ¢.'q) n
(7.5.45) The recurrence relation for these polynomials is 2 cos(B + ¢ )Pn(cos(B + ¢); a, blq)
= AnPn+l (cos(B + ¢); a, blq) + BnPn(cos(B + ¢); a, blq)
+ CnPn-l(cos(B + ¢); a, blq)
(7.5.46)
for n = 0,1, ... , where P-l (x; a, blq) = 0,
A -
1 - a 2 b2 qn-l
(7.5.4 7)
-,---..".-:-;:,......."..----:-..,...-7-----,,-:-:::-;:--:-
n - (1 _ a 2b2q2n-l) (1 _ a 2b2q2n) '
and
Bn
=
ae i¢ + a-1e- i¢ - Ana-1e-i¢(1 - a 2qn)(1 - abqn)(l - abe 2i ¢qn) - Cnae i¢ /(1- a 2qn-l)(1_ abe 2i ¢qn-l)(1_ abqn-l). (7.5.49)
If we set a obtain
= qO<, b = qf3, B = x log q in (7.5.42) and then let q
[ : Pm(x; 0:, (3)Pn(x;
0:,
(3)lr(o: + ix)r((3 + ix)1 2dx = 0,
m -=I- n,
---->
1 we
(7.5.50)
7.6 Connection coefficients
195
where . (3) _ · n Pn ( x, a, - 2
D
3£2
[-n, n+ 2aa++(32(3, 2-a 1, a- ix ,.1].
(7.5.51)
For further details and an extension to one more parameter, see Askey [1985b], Askey and Wilson [1982, 1985], Atakishiyev and Suslov [1985], and Suslov [1982, 1987].
7.6 Connection coefficients Suppose fn(x) and gn(x), n = 0,1, ... , are polynomials of exact degree n in x. Sometimes it is of interest to express one of these sequences as a linear combination of the polynomials in the other sequence, say, n
gn(X)
=
L ck,nfk(X).
(7.6.1)
k=O
The numbers Ck,n are called the connection coefficients. If the polynomials fn(x) happen to be orthogonal on an interval I with respect to a measure da(x), then Ck,n is the k-th Fourier coefficient of gn(x) with respect to the orthogonal polynomials fk(X) and hence can be expressed as a multiple of the integral II fk(X)gn(x)da(x). A particularly interesting problem is to determine the conditions under which the connection coefficients are nonnegative for particular systems of orthogonal polynomials. Formula (7.6.1) is sometimes called a projection formula when all of the coefficients are nonnegative. See the applications to positive definite functions, isometric embeddings of metric spaces, and inequalities in Askey [1970, 1975], Askey and Gasper [1971]' Gangolli [1967] and Gasper [1975a]. As an illustration we shall consider the coefficients Ck,n in the relation n
Pn(x; a, (3, 1', dlq) =
L Ck,nPk(X; a, b, c, dlq)·
(7.6.2)
k=O
Askey and Wilson [1985] showed that
Ck,n
=
(ad,(3d,1'd,q;q)n (a(31'd qn-\q)k k2-nkdk-n kq (ad, (3d, 'I'd, q, abcdq 1; qh (q; q)n-k
x S¢4
[qk-n, a(31'dqn+k-1, adqk, bdqk, cdqk . ] abcdq2k, adqk, (3dqk, 1'dqk ,q, q .
To prove (7.6.3), temporarily assume that max observe that, by orthogonality,
bk,j =
11
w(x;a,b,c,dlq)Pk(x;a,b,c,dlq)(deiO,de-iO;q)j dx
-1
vanishes if j
<
(Ial, Ibl, Icl, Idl, Iql) <
k, and that
bk,j = /i(a, b, c, dlq)(ab, ac, ad; q)ka-k
(7.6.3) 1, and
(7.6.4)
196
Applications to orthogonal polynomials x (ad,bd,cd;q)j
(abcd;q)j
[q-k,abedqk-1,adqj. ] abedqj,ad ,q,q
A-,
3'1'2
= A:(a, b, c, dlq)(ab, ae, bc, q-j; q)k(ad, bd, cd; q)j(dqj)k /(abed; q)j+k (7.6.5) if j 2 k. Since
Pn(x; a, (3, 'Y, dlq) _ . -n n (q-n,a(3'Ydqn-l,deiIJ,de-iIJ;q)j j - (ad, (3d, 'Y d , q)n d ( d (3d d' ) q , j=O q, a, , 'Y , q J
L
(7.6.6)
0
we find that
Ck,n
=
hk(a, b, e, dlq)
111 W(X; a, b, e, dlq)Pk(x; a, b, e, dlq)Pn(x; a, (3, 'Y, dlq) dx
_ . -n n (q-n,a(3'Ydqn-\q)j j - hk(a, b, e, dlq)(ad, (3d, 'Y d , q)n d d (3d d' ) q bk,J j=O q, a, , 'Y , q J
L (
0
0
(7.6.7) and hence (7.6.3) follows from (7.5.16), (7.5.17) and (7.6.5). The 5rP4 series in (7.6.3) is balanced but, in general, cannot be transformed in a simple way, so one cannot hope to say much about the nonnegativity of ek,n unless the parameters are related in some way. One of the simplest cases is when the 5rP4 series reduces to a 3rP2 , which can be summed by the q-Saalschiitz formula. Thus for (3 = band 'Y = e we get n
Pn(X; a, b, c, dlq) =
L ek,npdx; a, b, e, dlq)
(7.6.8)
k=O
with
ek,n
=
(a/a; q)n-k (abedqn-\ qh(q, be, bd, cd; q)nan-k (q, be, bd, cd; q)k(abcdqk-1; q)k(q, abedq2k; q)n-k'
(7.6.9)
It is clear that Ck,n > 0 when 0 < a < a < 1, 0 < q < 1 and max (be, bd, cd, abed) < 1. Another simple case is when the 5rP4 series in (7.6.3) reduces to a summable 4rP3 series. For example, this happens when we set d = q!, c = -q! = 'Y, b = -a and (3 = -a. The 5rP4 series then reduces to
(7.6.10) which equals ( a2qn+k+1 qk-n+1 a2qk+n+2 a2qk-n+2/a2.q2) '
,
,
(q,a2q2k+1,a2q2k+2,a2q2/a2;q2)00
,
00
(
()/ n+k) n-k 2 a q 2
(
)
7.6.11
by Andrews' formula (Ex. 2.8). Clearly, this vanishes when n - k is an odd integer and equals
(qa 2, q2a 2; q2)n_2 o(q, 00 2 /a 2; q2)Jo (qa 2, q2a 2; q2)n_j
J -----;-----;c----';;:~'____;;co__------"(a 2q2n+ 1-4 j )j
(7.6.12)
7.7 A difference equation and a Rodrigues-type formula when n - k
197
2j,j = 0, 1, ... , [~]. Since by (3.10.12) and (7.4.14)
=
.
Pn (X, a,
__ ~ a,
~I) _
(q2,qa 2 ;q2)n (. 21 2) (2.) On X, a q , a ,q n
q ,q q -
(7.6.13)
we obtain, after some simplification, Rogers' formula
o (. n
X, "(
1) q
= [~l ~
k=O
(3k("( j (3; qh(,,(; q)n-k(l - (3qn-2k) 0 (. (31 ) ( . ) ((3 .) (1 _ (3) n-2k X, q q, q k q, q n-k
(7.6.14)
after replacing 0;2, a 2 , q2 by ,,(, (3 and q, respectively. It is left as an exercise (Ex. 7.15) to show that formulas (7.6.2) and (7.6.14) are special cases of the q-analogue ofthe Fields-Wimp formula given in (3.7.9). For other applications of the connection coefficient formula (7.6.2), see Askey and Wilson [1985].
7.7 A difference equation and a Rodrigues-type formula for the Askey-Wilson polynomials
The polynomials Pn(x; a, b, c, dlq), unlike the Jacobi polynomials, do not satisfy a differential equation; but, as Askey and Wilson [1985] showed, they satisfy a second-order difference equation. Define Eif(e iO )
6qf (e iO )
= f (q±~eiO), (Et - E;;) f (e iO )
=
Dqf(x) = 6q!(X) , uqx
and
(7.7.1)
X = cose.
(7.7.2) (7.7.3)
Clearly,
~ (q~
6q( cos e) =
and 6q (ae iO , ae- iO ; q) n
q-~)
-
(e iO - e- iO ) = -iq- ~ (1 - q) sin e,
= 2aiq- ~ (1 -
qn) sin
e (aq~ eiO , aq~ e- iO ; q)n-l,
(7.7.4) (7.7.5)
so that "0
D q ( ae'
"0)
,
ae-'·'qn
=-
2a(1-qn)
(1 _ q)
1"0
1
"0
(aq2 e' , aq2 e-' ,. q) n-l,
(7.7.6)
which implies that the divided difference operator Dq plays the same role for (aeiO,ae-iO;q)n as djdx does for xn. When q ---+ 1, formula (7.7.6) becomes
d~ (1- 2ax + a 2)n = -2an(1- 2ax + a 2t- 1 . Generally, for a differentiable function .
hm Dqf(x)
q--+l
f (x) =
we have
d
-d f(x). X
198
Applications to orthogonal polynomials
Following Askey and Wilson [1985], we shall now use the operator Dq and the recurrence relation (7.5.3) to derive a Rodrigues-type formula for Pn(x; a, b, c, dlq). First note that by (7.5.2) and (7.7.5)
8q Pn(x; a, b, c, dlq) 1 1 1 1 1 n sine (1 - qn)(l - qn- 1 abcd)Pn-l(X; aq2, = -2iq2bq2, cq2, dq2Iq). (7.7.7) If we define
(7.7.8) and
r (eiO ) = n
4'" 'f'3
[q-n,abCdqn-l,aeiO,ae-iO. ] ab , ac , ad ' q, q ,
(7.7.9)
then the recurrence relation (7.5.3) can be written as
q-n(1- qn)(l_ abcdqn-l)rn(e iO ) = A( -e) [rn(q-1e iO ) - rn(e iO )] + A(e) [rn(qe iO ) - rn(e iO )].
(7.7.10)
Also, setting (7.7.11) (7.7.12) we find that (7.7.10) can be expressed in the form
q-n(1- qn)(l_ abcdqn-l)V(eiO)V(e-iO)Pn(x) = 8q [{E:V(e iO )} {E';-V(e- iO )} {8 qPn(xH]·
(7.7.13)
Combining (7.7.7) and (7.7.13) we have
- q-n/2V(e iO)V(e- iO)Pn(X; a, b, c, dlq) = 8q [{ E:V(e iO )} {E.;-V(e- iO )} (e iO _ e- iO ) x Pn-l (X; aq! , bq! , cq! , dq! Iq)] . (7.7.14) Since
(e iO _ e- iO ) E:V(eiO)E';-V(e- iO ) (e iO -
c iO ) (qe 2iO , qe- 2iO ; q)
00
h( cos 61; aq! )h( cos 61; bq! )h( cos 61; cq! )h( cos 61; dq!) W (eiO;aq!,bq!,cq!,dq!lq) eiO
-
e- iO
199
Exercises (7.7.14) can be written in a slightly better form
q-n/2W(eiO ; a, b, c, dlq)Pn(x; a, b, c, dlq)
= liq [(e iO
- e-iO)-l W(e iO ; aq~, 1
1
1
1
bq~, cq~ ,dq~ Iq) ]
Pn-l (x; aq2 ,bq2 ,cq2, dq2lq) .
X
(7.7.15) Observing that, by (6.3.1),
Vl- x 2w(x; a, b, c, dlq) =
W(e iO ; a, b, c, dlq)
we find by iterating (7.7.15) that
w(x; a, b, c, dlq)Pn(x; a, b, c, dlq) =
(_I)k
(1 ;
q) k qnk/2-k(k+l)/4
k k k k k k k k ] x Dqk[ w(x; aq2, bq2, cq2, dq2Iq)Pn_k(X; aq'i, bq'i, cq'i, dq'ilq)
=
(-It
(1 ;
q) n q(n 2 -n)/4 D~ [w(x; aqn/2, bqn/2, cqn/2, dqn/2Iq)] . (7.7.16)
This gives a Rodrigues-type formula for the Askey-Wilson polynomials. By combining (7.7.7) and (7.7.15) it can be easily seen that the polynomials Pn(x) = Pn(x; a, b, c, dlq) satisfy the second-order difference equation 1
1
1
1
Dq [ w(x; aq2, bq2, cq2, dq 2 Iq)D qPn(x)
]
+ AnW(X; a, b, c, dlq)Pn(x) = 0,
(7.7.17)
where (7.7.18)
Exercises 7.1 If {Pn(x)} is an orthogonal system of polynomials on (-00, (0) with respect to a positive measure da(x) that has infinitely many points of support, prove that they satisfy a three-term recurrence relation of the form
XPn(X)
=
AnPn+l(X)
+ BnPn(x) + CnPn-l(X), n;::: 0,
with p-l(X) = 0, po(x) = 1, where An, B n , Cn are real and AnCn+1 > for n ;::: 0. 7.2 Let PO(X),Pl(X), ... ,PN(X) be a system of polynomials that satisfies a three-term recurrence relation
°
200
Applications to orthogonal polynomials n = 0,1, ... , N, where Darboux formula
P-l (x)
= O,Po(x) = 1. Prove the Christoffel-
n
(x - Y) LPj(x)pj(Y)Vj = Anvn [Pn+1 (x)Pn (y) - Pn(X)Pn+l(Y)] ' j=O :S n :S N, where Vo = 1 and vnCn = vn-1An- 1, 1 :S n :S N. Deduce
°
that
n
LP](x)Vj = Anvn [p~+1(X)Pn(X) - P~(X)Pn+l(X)l j=O and hence
N
LP](Xk)Vj j=O
=
ANVNPN(Xk)p~+l(Xk)
if x k is a zero of PN +1 (x) .
= N, the recurrence relation (7.2.1) reduces to (1 - q-j) (1 - cdqHl) PN(Xj) = CN [PN(Xj) - PN-l (Xj)] ,
7.3 Show that when n
where Pn(Xj) is given by (7.2.11), Xj by (7.2.9), and An and Cn by (7.2.5) and (7.2.6). Hence show that (7.2.1) holds with x = Xj,j = O,l, ... ,N. (Askey and Wilson [1979]) 7.4 If Pn(Xj), Vn and Wj are defined by (7.2.11), (7.2.3) and (7.2.15), respectively, prove directly (i.e. without the use of Favard's theorem) that N
N
LPm(Xj)Pn(Xj)Wj = j=O and
V;;:-l
N
L Wj Dm,n j=O N
LPn(Xj)Pn(Xk)Vn = wjl L Wn Dj,k. n=O n=O [Hint: First transform one of the polynomials, say Pn(x), to be a multiple of the
4¢3
series on the left side of (7.2.14)].
7.5 Let one of a, b, c, d be a nonnegative integer power of q-l and let
¢(a, b) =
4¢3 [
a~~~/~d; q, q] ,
where efg = abcdq. Prove the Askey and Wilson [1979] contiguous relation
A¢(aq-l, bq) where
+ B¢(a, b) + C¢(aq, bq-l) = 0,
A = b(l - b)(aq - b)(a - e)(a - f)(a - g), B = ab(a - bq)(b - a)(aq - b)(l - c)(l - d) - b(l - b)(aq - b)(a - e)(a - f)(a - g) + a(l - a)(a - bq)(e - b)(f - b)(g - b), C = -a(l - a)(a - bq)(e - b)(f - b)(g - b),
Exercises
201
and derive the following limit case which was found independently by Lassalle [1999]: ( (c-b)+ ab(1-c) _ aC(l-b)) y x _ (l-c)(y-a)(y-b) y(l - y) -
3
(l-b)(x-a)(x-c) x(1- x)
3(P2
(a,b,C;q, x y ) x,y abc
7.6 Determine the conditions that a, b, c, d must satisfy so that AnCn+l > 0 for 0 ::; n ::; N - 1, where An and C n are as defined in (7.2.5) and (7.2.6) and one of aq, cq, bdq is q-N, N a nonnegative integer. 7.7 Prove (7.2.22) directly by using the appropriate transformation and summation formulas derived in Chapters 1-3. Verify that
n
for x, y
=
0,1, ... ,N, which is the dual of (7.2.18).
7.8 (i) Prove that the q-Kmwtchouk polynomials Kn(x;a,N;q)
=
3
satisfy the orthogonality relation
(-N )
N
" Km (q -x., a, N')K ; q x ( -a )X ~ , q n (q -x., a, N') , q q( . ) x=o q, q x N _(N+l) (q;q)n(1+a- 1)(-a- 1qN+\q)n = (-qa -1) ;q a q 2 N (-a-I; q)n(1 + a- 1q2n)(q-N; q)n X (_ aq N+l )-n qn(n+l)bm ,n,
and find their three-term recurrence relation. (Stanton [1980b]) (ii) Let Kn(x;a,Nlq) = 2
be another family of q-K mwtchouk polynomials. Prove that they satisfy the orthogonality relation
~
~Km(q x=o
-x
;a,Nlq) Kn(q
-x
;a,Nlq)
(aq; q)N-x( _l)N-X q G) (.) ( . ) q,q x q,q N-x
_ (q, aq; q)n(q; q)N-n (_l)n N (N+2n+l)_n(n+l);: () a q U m n' q,q;q N '
-
7.9
Prove that
202
Applications to orthogonal polynomials where Pn(x)
An =
Pn(x; a, b; q) are the little q-Jacobi polynomials and
=
_qn (1- aqn+1)) (1- abqn+1) Cn (1 - abq2n+1) (1 - abq2n+2) ,
(1- qn) (1- bqn) (_aqn) (1 - abq2n) (1 - abq2n+1) .
= -:----=-:--:-:C-.,.......,..,.......=.--'--:,:..-..".---=-:-:-
7.10 Prove that
where Pn(x)
=
Pn(x; a, b, c; q) are the big q-Jacobi polynomials and (1- aqn+1) (1- cqn+1) (1- abqn+1) n (1 _ abq2n+1) (1 _ abq2n+2) ,
A -
--'----------:,.,----------':~---,--:;--,----;-,------''------:----;o---:-=-----'--
Cn =
(1 - qn) (1 - bqn) (1 - abc-1qn) n+1 (1 - abq 2 n ) ( 1 - abq 2 n+ 1) acq .
7.11 The affine q-Krawtchouk polynomials are defined by
K~ff (x; a, N; q) =
q
3¢2 [
-n
,
0]
x, aq,
q
-N ; q, q , 0
< aq <
l.
Prove that they satisfy the orthogonality relation
~ KAff (q-x' a
~m x=o
N.q)
'"
qN-1)X
x ( --a
KAff n
(X) q-2
(q-x' a N.q) (aq,q-N;q)x '" () q; q x
= Dm,n, hn
m,n=0,1, ... ,N,
where hn
=
(aq, q-N; q)n (_1)n( )N-n Nn-m () aq q . q;q n
(Delsarte [1976a,b], Dunkl [1977]) 7.12 The q-Meixner polynomials are defined by
=
Mn(x;a,c;q)
with 0 < aq relation
2¢1
(q-n,x;aq;q,_qn+1jc),
< 1 and c > O. Show that they satisfy the orthogonality
~M ( -x )M ( -x ) ~ m q ; a, c; q n q ; a, c; q (
x=o
where
hn
=
(aq;q)x x G) . ) c q q, -acq, q x
Dm,n =h ' n
(-acq; q)oo(aq; q)n qn (-c; q)oo(q, -qc- 1; q)n .
(When a = q-r-1, the q-Meixner polynomials reduce to the q-Krawtchouk polynomials considered in Koornwinder [1989].) 7.13 The q-Charlier polynomials are defined by cn(x;a;q) =
2¢1
(q-n,x;0;q,_qn+1ja).
Exercises
203
Show that
aX (X) L cm(q-X; a; q)cn(q-X; a; q)-(-.-) q x=o q, q 00
2
X
= (-a; q)oo(q, -qa-\ q)nq-nrSm,n' 7.14 Show that, for x
=
cos B,
Cn(x; qlq) = and
sin(n + I)B . B = Un(x), sm
n
. 1- qn hm ( iJ)Cn(x;iJlq)=cosnB=Tn(x), (3---71 2 1 -
~ 0
n~l,
where Tn(x) and Un(x) are the Tchebichef polynomials of the first and second kind, respectively. 7.15 Verify that formulas (7.6.2) and (7.6.14) follow from the q-analogue of the Fields-Wimp formula (3.7.9). 7.16 Let x
= cos B, It I < 1, and Iql <
1. Show that
fCn(x'iJlq) (>.;q)n tn = 2isinB (iJ,iJ;q)oo n=O ' (iJ2; q)n (1 - q) W(3 (xlq) (q, iJ2; q)oo x
l
e-
e iO
iO
(quei() , que- iO , >.tu; q) 00 ("0 iJue' ,iJue-'"0 ,tu; q) 00 dqu
and deduce that tn
00
~ Cn(x; iJlq) (iJ2; q)n
(i)
= (te- iO ; q)~}
2¢1
00
~ Cn(x; iJlq)q
(ii)
=
(iJ, iJe- 2iO ; iJ2; q, te iIJ ) .
(n) (iJt)n 2 (iJ2; q)n
(_te- iO ; q)oo
2¢1
(iJ, iJe 2iO ; iJ2; q, _te- iO ) .
7.17 Using (1.8.1), or otherwise, prove that if n is odd, if n is even. 7.18 If -1
< q,iJ < 1, show that
7.19 Derive the recurrence relations 1 - qn+1 1 _ iJ2 qn-1 2xCn(x; iJlq) = 1 _ iJqn Cn+1(x; iJlq) + 1 _ iJqn Cn- 1(x; iJlq),
204
Applications to orthogonal polynomials and 1 - j3 j3(1 - j3) Cn(x; j3lq) = 1 _ j3qn Cn(x; j3qlq) - 1 _ j3qn Cn- 2(X; j3qlq),
with C- 2(x; j3lq) = C-1(x; j3lq) = 0, Co(x; j3lq) = 1, and n ~ 0. (Ismail and Zhang [1994]) 7.20 Prove that
r Cn(cos 0; j3lq) cos(n + 2k)e Wf3(cosOlq) dO = 7r/j3;~; ~)= j3k q, ,q=
Jo
X
(j3-\ q)k(j32; q)n(q; q)nH 1 - qn+2k (q; qh(q; q)n(j3q; q)nH 1 - qnH '
n,k
~
0,
where Wf3(xlq) is defined in (7.4.6). 7.21 Using (7.4.15) and (7.6.14) prove that h(x;"() ( I) h( . r.i)Cn x;j3 q X,/-,
("t2,j3,j3q;q)= ~ (I ) r.i2.) ~dk,nCn+2k x;"( q ,
= (
"(,,,(q,/-, ,q
=
k=O
where h(x; a) is as defined in (6.1.2) and dk n ,
=
j3 k b / j3; q)k(q; q)n+2k(j32; q)nb; q)nH(1 - "(qn+2k) . (q; q)kb2; q)n+2k(q; q)n(j3q; q)nH(1- "()
(Askey and Ismail [1983]) 7.22 Prove that the continuous q-Hermite polynomials defined in Ex. 1.28 satisfy the orthogonality relation
r
(
I)
(
I) 1(2iIJ ) 12 e ;q = dO =
J o Hm cosO q Hn cosO q
27r(q; q)n (q;q)= Om,n.
7.23 Setting
in Ex. 7.19, show that 2x(1- j3qn)cn (x;j3lq) = (1- j32 qn)Cn+l(X;j3lq)
+ (1- qn)cn_l(x;j3lq),
for n ~ 0, with C-l (x; j3lq) = 0, co(x; j3lq) = 1. Now set j3 = s>.k and q = SWk, where Wk = exp(27ri/k) is a k-th root of unity, divide the above recurrence relation by 1- sW k and take the limit as S ---.. 1 to show that the limiting polynomials, c~(x; k), called the sieved ultraspherical polynomials of the first kind, satisfy the recurrence relation
2xc~(x; k) = C~+l (x; k) 2x(m
+ C~_l (x; k),
n -=I- mk,
+ >')C~k(X; k) = (m + 2>')C~k+l (x; k) + mC~k_l (x; k)
where k, m, n = 0,1, ... , cS(x; k) = 1 and cNx; k) = x. (Ai-Salam, Allaway and Askey [1984b])
Exercises
205
7.24 Rewrite the orthogonality relation (7.4.15) in terms of the sieved orthogonal polynomial cn(x; ,8lq) defined in Ex. 7.23 and set ,8 = sAk and q = SWk. By carefully taking the limits of the q-shifted factorials prove that
1 c~(x; k)c~(x; 1
8~,n,
k)w(x) dx =
-1
n
where
k-1 w(x)
=
II Ix
22A (k-1)(1_ x 2)-!
cos 2(7rjjk)IA
2 -
j=O
and
~(>'+1)1 (>.+lhn/kJ(2>')rn/kl r("2)r(>. +"2) (1) Ln/kJ (>')rn/kl
hn =
where the roof and floor functions are defined by
Ial = laJ =
smallest integer greater than or equal to a, largest integer less than or equal to a.
(AI-Salam, Allaway and Askey [1984b]) 7.25 The sieved ultraspherical polynomials of the second kind are defined by B~(x; k)
= lim Cn(x; sAk+1WklsWk), Wk = exp(27rijk). 8---+1
Show that B~(x; k) satisfies the recurrence relation 2xB~(x; k)
=
B~+l (x; k)
+ B~_l (x; k),
n
+ 1 -=I- mk,
2x(m + >')B~k-1 (x; k) = mB~k(x; k) + (m + 2>')B~k_2(X; k), where B6(x; k) = 1; Bf(x; k) = 2x if k 2:: 2; Bf(x; 1) = 2(>' + l)x. Show also that B~(x; k) satisfies the orthogonality relation
1 B~(x; k)B~(x; 1
k)w(x) dx =
-1
8~,n, n
where
k-1 W(X) = 22A (k-1)(1_ x 2)!
II Ix
2 -
cos 2(7rjjk)IA
j=O
and
_ hn
-
(l)Ln/kJ(>'+l)L~J
2r(>'+1) 1
1
(
nt J
r("2)r(>. +"2) >. + 1) Ln/kJ ( 2>' + 1) L
(AI-Salam, Allaway and Askey [1984b])
1
.
206
Applications to orthogonal polynomials
7.26 Using (2.5.1) show that
_ (ab,ae,be,q;q)n ( . b,e, dl) PnX,a, q - (bd-l.) aeq ,qn n (abeq-1 eiO;q)k (1- abee iO q2k-1) (aeiO,beiO,eeiO;q)k x {; (q; qh(l - abeq- 1eiO ) (be, ae, ab; q)k x (abedq-1; q)nH (de- iO ; q)n-k ei(n-2k)O
(abee 20 ; q)nH (q; q)n-k Deduce that the polynomials
Pn(x)
= q-+O lim Pn(x;a,b,e,dlq)
are given by
PO(X) = 1 = Uo(x), P1(X) = (1- 84)U1(X)
+ (83 - 81)UO(X), P2(X) = U2(x) - 81U1(X) + (82 - 84)UO(X), 4
Pn(X) = ~) -1)k8k Un _k(X), k=O
n ~ 3,
where 80
= 1,
81
= a + b + e + d, 83
82
= ab + ae + ad + be + bd + cd,
= abc + abd + aed + bcd,
84
= abed,
and Un (COS 0) = sin(n+1)ej sinO, U- 1(x) = O. When max(lal, Ibl, lei, Idl) < 1 show that these polynomials satisfy the orthogonality relation
211 7r
-1
Pm(X)Pn(x)(l - x 2)! dx (1- 2ax + a 2 )(1 - 2bx + b2 )(1 - 2ex + e2 )(1 - 2dx + d 2 )
0,
1- abed = { (1 - ab)(l - ae)(l - ad)(l - be)(l - bd)(l _ cd)' 1- abed, 1,
m#n, m m m
= n = 0, = n = 1, = n ~ 2.
(Askey and Wilson [1985]) 7.27 Prove that
· _ q, q! , _! _ ( n+2.,q) n sin(n. +0 1)e ' q Iq) - q S1n
(i)
Pn (cos 0, q,
(ii)
Pn(cos 0; 1, -1, q!, -q! Iq) = 2(qn; q)n cos nO,
(iii)
Pn(cosO,q,
(iv)
Pn(cosO,l,
n ~ 1,
· _ 1,q, ! _! _ ( n+ 1.,q) n sin( n + ~)e q Iq) - q sin(Oj2) , · _
! _!
_ ( n+1. ) cos(n + ~)e
q,q , q Iq) -
q
,q n
cos(Oj2) .
Exercises
207
7.28 Use the orthogonality relations (7.5.28) and (7.5.29) to prove the quadratic transformation formula (7.5.26). 7.29 Verify the orthogonality relations (7.5.28) and (7.5.29). 7.30 Verify formula (7.5.36). 7.31 Suppose that a,b,c,d are complex parameters with max (Ibl, Icl, Idl, Iql) < 1 < lal such that laqN+11 < 1 < laqNI, where N is a nonnegative integer. Use (6.6.12) to prove that 1
[1 Pm(X)Pn(X)W(X; a, b,
N
c, dlq) dx
+ 2n ~Pm(Xk)Pn(Xk)Wk
hn(a, b, c, dlq)' where Pn(x) by (7.5.22).
=
Pn(x;a,b,c,dlq), Xk
=
~
(aqk +a- 1 q-k) and Wk is given
7.32 Prove that
(i)
p~a,(3)(_x;q)
(ii)
p~a,(3)(_xlq)
= (-ltP~f3,a)(x;q), = (_ltq(a-f3)n/2p~f3,a)(xlq).
7.33 Using (7.4.1), (7.4.7) and (2.11.2) prove that
~ (~;;qJ)n On(x; ,8lq)On(Y; ,8lq)tn
(i)
(t 2,,8;q)= (,8t2, ,82; q)= x 8 W7
(~t2q-l. ~ jJ ,jJ,
teiiJ+i¢ , te -iO-i¢ , te iO - i¢ , tei¢-iO., q,jJ~)
,
~ (q; q)n 1 - ,8qn C ( ~I)O ( ~I) n f=' o (,82;q)n 1-,8 n X;{J q n Y;{J q t
(ii)
(t2,,8; q) = (,8teiiJ+ i¢, q,8te iO - i¢; q) = (q,8t2, ,82; q)= (teiiJ+i¢, te iO - i¢; q)= x
SW7
2
(,8t 2;,8, qteiiJ+ i¢, qte- iO - i¢, te iO - i¢, tei¢-iO; q,,8) ,
where -1 < q,,8, t < 1 and x = cosB, y = cos¢. (Gasper and Rahman [1983a], Rahman and Verma [1986a]) 7.34 Show that
Pn(cosB; a, b, c, dlq) x
l
qe - W / d
qeW
/d
D-l(B)(ab, ac, bc; q)n (dueiO,due- iO , abcdujq; q) =
(qju;q) (dU)n n du (daujq, dbujq, dcujq; q)= (abcdujq; q)n q q,
=
where
D( B) =
-iq(l - q) 2d (q, ab, ac, bc; q)=h( cos B; d)w( cos B; a, b, c, dlq).
208
Applications to orthogonal polynomials Hence show that 00 ( 11 ) n '""" a2c2) ; q nPn (() cos ; a, aq 1 2 , C, cq 2 q ~ 1 1 t n=O (q; q)n(a 2q2, ac, acq2; q)n
(at " ac2t -acteiIJ , -acte- iIJ ., q) 00 (-ct, -a 2ct, te iIJ , te- iIJ ; q)oo
x 8 W 7 ( -a 2ctq-\ -ac, -a/c, -ctq-~, ae iIJ , ae- iIJ ; q, -ctq~) . (Gasper and Rahman [1986]) 7.35 Show that Ex. 6.9 is equivalent to
1 1
w(y; a, b, p,eiIJ, J.Le-iIJlq)Pn(Y; a, b, c, dlq) dy
-1
27r(abJ.L 2; q)oo (ab; q)n n (q, ab, J.L2, aJ.Le iIJ , aJ.Le- iIJ , bJ.LeiIJ , bJ.Le- iIJ ; q)oo (abJ.L2; q)n J.L x Pn(cos (); aJ.L, bJ.L, CJ.L- 1 , dJ.L- 1 Iq),
where max(lal, Ibl, IJ.LI) 7.36 Show that if for Iql
<
l.
< 1 we define () a; q v
(a; q)oo
= (aq v ; q)
, 00
where 1/ is a complex number and the principal value of qV is taken, then (7.7.6) extends to
D (ae iIJ ae-iIJ'q) =_2a(1-qV) q , 'v 1- q
(aq~eiIJ,aq~e-iIJ;q)
v-I
.
7.37 Let n = 1,2, ... ,r, x = cos(), and
Un(x) = An,r (qv+le ilJ , qv+le- ilJ ; q)n X 6¢5
[
qn-r, qn+r+2v+2>-+I, qn+v+~ ,qn+2v, qn+v+leilJ, qn+v+le-ilJ . ] q2n+2v+l, qn+v+>-+I, qn+2v+l, _qn+v+>-+I, _qn+v+~ ,q, q
Show that Un(x) satisfies the q-differential equation
Dq[(qn+V+ 1 e iIJ , qn+v+l e -ilJ; q)-nUn(x)]
=X
1 - qn+2v 1 - qn+l
3
.
3.
(qn+v+2 e,IJ, qn+v+2 e-,IJ; q)
-2n-2v-l
Dq[(q-n-v e iIJ , q-n-v e- iIJ ; q)n+2v+l Un+1 (x)].
(Gasper [1989b])
Exercises
209
7.38 Show that the discrete q-Hermite polynomials
[n/2] (q; q)n (_l)k k(k-l) n-2k H ( . )= " n X, q ~ (2 2) ( ) q X k=O q; q k q; q n-2k satisfy the recurrence relation
and the orthogonality relation
where 1jJ(x) is a step function with jumps
Ixl (X 2q2, q; q2)= 2
(q2; q2)=
at the points x = ±qj, j = 0,1,2, .... (Ai-Salam and Garlitz [1965], Ai-Salam and Ismail [1988]) 7.39 Let a
< 0 and 0 < q < 1. Show that
I:
u$::)(x;q)u~a)(x;q) doJa) (x)
= (1- a)( _a)n(q; q) nq G)8m ,n, where
and oJa) (x) is a step function with jumps
qk
at the points x
=
qk, k
=
0,1, ... , and jumps
-aqk
at the points x = aqk, k = 0,1,.... Verify that when a = -1 this orthogonality relation reduces to the orthogonality relation for the discrete q-Hermite polynomials in Ex. 7.38. (See Ai-Salam and Garlitz [1957, 1965], Ghihara [1978, (10.7)], and Ismail [1985b, p.590]) 7.40 Show that if
(q; q)n k h ( . )- ~ n x, q - ~ ( ) ( ) x, k=O q; q k q; q n-k
210
Applications to orthogonal polynomials then h~(x;q) - hn+l(x;q)hn-1(x;q)
n
qn-kxk
k=O
q, q n-k
= (1 - q)(q; q)n-l L h~_k(X; q) ( .)
, n ;::: 1.
Deduce that the polynomials hn(x; q), which are called the Rogers-Szego polynomials, satisfy the Thnin-type inequality
for x ;::: 0 when 0 (Carlitz [1957b])
< q < 1 and n = 1,2, ....
7.41 Derive the addition formula
Pn(qZ; 1, 1; q) py(qZ; qX, 0; q) =
Pn(qx+y; 1, 1; q) Pn(qY; 1, 1; q) py(qZ; qX, 0; q)
+ Ln k=l X
(q. q) (q. q) qk(k+y-n) ' x+y+k , n+k (q; q)x+y(q; q)n-k(q; q)~
Pn_k(qx+y; l, qk; q) Pn-k(qY; l, l; q)
Z x x Py+k(q ; q ,0; q) X
n (q; q)y(q; q)n+kqk(x+y-n+l)
+ k=l L (.) (.) (. )2 q,qy-kq,qn-kq,qk
Pn_k(qx+y-k; l, l; q) Pn-k(qy-\ qk, qk; q) Py_k(qZ; qX, 0; q)
where x, y, z, n = 0,1, ... , and Pn(t; a, b; q) is the little q-Jacobi polynomial defined in Ex. 1.32. (Koornwinder [1991a]) 7.42 Derive the product formula 00
Pn(qX; 1, 1; q) Pn(qy; 1, 1; q)
=
(1 - q) LPn(qZ; 1, 1; q) K(qX, qy, qZ; q)qZ, z=O
where x, y, z, n = 0,1, ... , Pn(t; a, b; q) is the little q-Jacobi polynomial, and x+l y+l z+l. ) K( x y z. ) _ ( q ,q ,q ,q 00 xy+Xz+yz q ,q ,q ,q (1 -)( q q, q,.) q 00 q
x {3¢2(q-X,q-y,q-Z;0,0;q,q)}2. (Koornwinder [1991a]) 7.43 The q-Laguerre polynomials are defined by
Exercises
211
> -1 then these polynomials satisfy the orthogonality
Show that if a relation
(i)
and the discrete orthogonality relation (ii) where
A
=
(q, -c(1 - q)qe<+1, -1/ cq e«1 - q); q)oo . (q"'+1, -c(1 - q), -q/ c(1 - q); q)oo
(Moak [1981]) 7.44 Let v(y;a1,a2,a3,a4,a5Iq)
1, -1, q!, -q!, a1 a2a3a4a5) (1 _ y2)-! ,
= h(y;
y = cos cp.
h(y;a1,a2,a3,a4,a5)
and
Show that
1 1
-1
X
if)
(abC/-Leif),abC/-Le-if);q)n 2"¢ b 2 "¢ ) , a Cf-l e- t ; q n
Iq) (b
a Cf-l e t
Pn(y; a, b, c, dlq) dy -if) I ) (ab, ac, bc; q)n n q ( b 2 2 b 2 ) f-l a f-l , aCf-l , Cf-l ; q n Pn (x; af-l, bf-l, cf-l, df-l- 1Iq), x = cos (),
(
= g a, b, X
-if)
v(y; a, b, c, f-le , f-le
if)
c, f-le , f-le
where Pn(x; a, b, c, dlq) are the Askey-Wilson polynomials defined in (7.5.2) and max(lal, Ibl, Icl, If-ll, Iql) < l. (Rahman [1988a]) 7.45 Defining the q-ultrasphe'rical function of the second kind by Dn(x; (3lq) = 4
sin () h( cos 2(); (3) ~ h( ()) L..,. b(k, n; (3) cos(n + 2k cos 2 ; 1 k=O
+ l)e
212
Applications to orthogonal polynomials with b(k, n; (3) as in (7.4.9), prove that C~ (cos 0; (3lq)
=
+ D~ (cos 0; (3lq)
[4 sin 0 ((3, (3qn+1; q)ooh( cos 20; (3)] 2 (q, (32qn; q)ooh( cos 20; 1)
x ((qn+1,(32qn,qe2i(),qe-2i();q)OO ((3qn, (3qn+1, (3e 2i (), (3e- 2i (); q)oo rI..
[
(3,q/(3,-ylq,-q .] q1-n /(3, (3qn+1, qe2i(), eq-2i() , q, q
x
5'1'4
+
((3, q/ (3, (3e 2i () qn+1, (3e- 2i () qn+\ q)oo ((3qn+1, q-n /(3, (3e 2i(), (3e- 2i(); q)oo
X
5¢4
]) qn+1,(32 qn,(3qn+1/2,_(3qn+1/2,_(3qn+1. [ (3qn+1, (32q2n+1, (3e2i() qn+1, (3e-2i() qn+1 , q, q .
(Rahman [1992a]) 7.46 Let f(x) be continuous on [-1,1]. Show that
Dq
[/
1
-1
K(x, y)
f(y)
1= f(x),
~dy y 1- y2
where
K(
)
=
X, Y
(1 - q)(q, q, e 2i ¢, e- 2i ¢; q)ooh(x; _q1/4, _q3/4) 47rq1/4h(y; _q1/4, _q3/4, q1/2ei(), q1/2e-i()) (q, q, e 2i ¢, e- 2i ¢; q)oo
with x = cos 0, y = cos ¢. Note that this defines a formal inverse of the Askey-Wilson operator D q . (Ismail and Rahman [2002a,b]) 7.47 Bustoz and Suslov's q-trigonometric functions are defined by
where x = cos o. (i) Show that lim Cq(x; (1- q)w/2) = coswx and lim Sq(x; (1 - q)w/2) = sinwx. q--+1q--+1-
Notes
213
(ii) Prove the orthogonality relations
[11 W1/ 2(xlq)Cq(x; wm)Cq(x; wn )(1 - X2)-1/2dx
=
[11 W1/ 2(xlq)Sq(x; Wm)Sq(X; wn )(1 - X2)-1/2dx
=7f
where 7J
(q1/2; q)2 [8 ] ( . )2 00 Cq(7J; wn) -8 Sq(7J; w) 8m,n, q, q 00 W W=Wn
= (q1/4 + q-1/4)/2,
and Wm , Wn are two distinct roots of the equation
Sq(7J;w) = O. (Bustoz and Suslov [1998], Suslov [2003]) Notes §7.1 See also Atakishiyev and Suslov [1988a,b], Atakishiyev, Rahman and Suslov [1995], Nikiforov and Uvarov [1988], Nikiforov, Suslov and Uvarov [1991]' and Szego [1968, 1982]. For a classical polynomial system with complex weight function see Ismail, Masson and Rahman [1991]. The familiar connection between continued fractions and orthogonality was extended by Ismail and Masson [1995] to what they call R-fractions of type I and II, which lead to biorthogonal rational functions. See further work on continued fractions related to elliptic functions in Ismail and Masson [1999], Ismail, Valent and Yo on [2001], and Milne [2002]. In T.S. Chihara and Ismail [1993] extremal measures for a system of orthogonal polynomials in an indeterminate moment problem are examined. For orthogonal polynomials on the unit circle see Ismail and Ruedemann [1992]. Some classical orthogonal polynomials that can be represented by moments are discussed in Ismail and Stanton [1997, 1998]. §7.2 Andrews and Bressoud [1984] used the concept of a crossing number to provide a combinatorial interpretation of the q-Hahn polynomials. Koelink and Koornwinder [1989] showed that the q-Hahn and dual q-Hahn polynomials admit a quantum group theoretic interpretation, analogous to an interpretation of (dual) Hahn polynomials in terms of Clebsch-Gordan coefficients for SU(2). For how Clebsch-Gordan coefficients arise in quantum mechanics, see Biedenharn and Louck [1981a,b]. L. Chihara [1987] considered the locations of zeros of q-Racah polynomials and employed her results to prove non-existence of perfect codes and tight designs in the classical association schemes. The correspondence between q-Racah polynomials and Leonard pairs is outlined in Terwilliger [2003]. For the relationship between orthogonal polynomials and association schemes, see Bannai and Ito [1984], L. Chihara and Stanton [1986], Delsarte [1976b]' and Leonard [1982]. A multivariable extension of the q-Racah polynomials is considered in Gasper and Rahman [2003c], while a system of multivariable biorthogonal polynomials is given in Gasper and Rahman [2003a], which are q-analogues of those found in Tratnik [1991b] and [1989], respectively.
214
Applications to orthogonal polynomials
§7.3 Ai-Salam and Ismail [1977] constructed a family of reproducing kernels (bilinear formulas) for the little q-Jacobi polynomials. In Ai-Salam and Ismail [1983] they considered a related family of orthogonal polynomials associated with the Rogers-Ramanujan continued fraction. A biorthogonal extension of the little q-Jacobi polynomials is studied in Ai-Salam and Verma [1983a]. When b = 0 the little q-Jacobi polynomials reduce (after changing variables and renormalizing) to the Wall polynomials
Wn(x;b,q) = (-l)"(b;q)nq
(n+1) 2
L·[n]J
(j) . q 2 (_q-n x )1 (b. ).
n
j=O
q
,q
J
and to the generalized Stieltjes- Wigert polynomials
which are q-analogues of the Laguerre polynomials that are different from those considered in Ex. 7.43. Since the Hamburger and Stieltjes moment problems corresponding to these polynomials are both indeterminate, there are infinitely many nonequivalent measures on [0, (0) for which these polynomials are orthogonal. See Chihara [1978, Chapter VI], [1968b, 1971, 1979, 1982, 1985], Ai-Salam and Verma [1982b]' L. Chihara and T.S. Chihara [1987], and Shohat and Tamarkin [1950]. §7.4 An integral of the product of two continuous q-ultraspherical polynomials and a q-ultraspherical function of the second kind is evaluated in Askey, Koornwinder and Rahman [1986]. Ai-Salam, Allaway and Askey [1984a] gave a characterization of the continuous q-ultraspherical polynomials as orthogonal polynomial solutions of certain integral equations. Askey [1989b] showed that the polynomials Cn(ix; jJlq), 0 :::; n :::; N, are orthogonal on the real line with respect to a positive measure when 0 < q < 1 and jJ > q-N. Ismail and Rahman [1991] showed that the associated Askey-Wilson polynomials r;::'(x; a, b, c, dlq) defined in Ex. 8.26 have an orthogonality property on [-1,1]. A survey of classical associated orthogonal polynomials is in Rahman [2001], and an integral representation is given in Rahman [1996b]. A projection formula and a reproducing kernel for r;::'(x; a, b, c, dlq) is given in Rahman and Tariq Qazi [1997b]. Berg and Ismail [1996] showed how to generate one to four parameter orthogonal polynomials in the Askey-Wilson family by starting from the continuous q-Hermite polynomials. Also see Koelink [1995b] and Rahman and Verma [1987]. §7.5 Asymptotic formulas and generating functions for the Askey-Wilson polynomials and their special cases are derived in Ismail and Wilson [1982] and Ismail [1986c]. Kalnins and Miller [1989] employed symmetry techniques to give an elementary proof of the orthogonality relation for the Askey-Wilson polynomials. Following Hahn's approach to the classification of classical orthogonal polynomials N.M. Atakishiyev and Suslov [1992b] gave a generalized moment representation for the Askey-Wilson polynomials. Brown, Evans and Ismail [1996] showed that the Askey-Wilson polynomials are solutions of a
Notes
215
q-Sturm-Liouville problem and gave an operator theoretic description of the Askey-Wilson operator V q . L. Chihara [1993] extended her work on q-Racah polynomials [1987] to the Askey-Wilson polynomials. Floreanini, LeTourneux and Vinet [1999] also employed symmetry techniques to study systems of continuous q-orthogonal polynomials. A multivariable extension of Askey-Wilson polynomials is given in Gasper and Rahman [2003b] as a q-analogue of Tratnik [1991a], see Ex. 8.29. Also see Gasper, Ismail, Koornwinder, Nevai and Stanton [2000], Spiridonov and Zhedanov [1995-1997], Stokman [1997a-2003a], Stokman and Koornwinder [1998], Vinet and Zhedanov [2001], and Wilson [1991]. §7.6 For additional results on connection coefficients (and the corresponding projection formulas), see Andrews [1979a], Gasper [1974, 1975a]. Ex.7.5 A contiguous relation satisfied by a WP-balanced, generally nonterminating 8¢7 series is given in Ismail and Rahman [1991]. For more results on contiguous relations and orthogonal polynomials, see Gupta, Ismail and Masson [1992, 1996], Gupta and Masson [1998] and Ismail and Libis [1989]. Ex. 7.7 Ismail [1995] gave a simple proof of (1.7.2) by making use of iterations of (7.7.6) and evaluating them at Xj = ~(aqj +q-j ja). An operator calculus for Dq is developed in Ismail [2001a]. Ex.7.8 Stanton [1981b] showed that the q-Krawtchouk polynomials Kn(x; a, N; q) are spherical functions for three different Chevalley groups over finite fields and derived three addition theorems for these polynomials by decomposing the irreducible representations with respect to maximal parabolic subgroups. In Koornwinder [1989] it is shown that the orthogonality relation for the q-Krawtchouk polynomials Kn(x; a, Nlq) expresses the fact that the matrix representations of the quantum group Sp,U(2) are unitary. Ex.7.11 The affine q-Krawtchouk polynomials are the eigenvalues of the association schemes of bilinear, alternating, symmetric and hermitian forms over a finite field (see Carlitz and Hodges [1955], Delsarte [1978], Delsarte and Goethals [1975], and Stanton [1981a,b, 1984]). L. Chihara and Stanton [1987] showed that the zeros of the affine q- Krawtchouk polynomials are never zero at integral values of x, and they gave some interlacing theorems for the zeros of q- Krawtchouk polynomials. Ex. 7.12 and 7.43 N.M. Atakishiyev, M.N. Atakishiyev and Klimyk [2003] found the connection between big q-Laguerre and q-Meixner polynomials and representations of the group Uq(SUl,l). Ciccoli, Koelink and Koornwinder [1999] extended Moak's q-Laguerre polynomials to an orthogonal system for a doubly infinite Jacobi matrix originating from analysis on SUq (l, 1), and found the orthogonality and dual orthogonality relations for q-Bessel functions originating in Eq(2). Ex.7.22 Askey [1989b] proved that the polynomials Hn(ixlq) are orthogonal on the real line with respect to a positive measure when q > 1. Exercises 7.23-7.25 Other sieved orthogonal polynomials are considered in Ai-Salam and Chihara [1987], Askey [1984b], Charris and Ismail [1986, 1987, 1993], Charris, Ismail and Monslave [1994]' Ismail [1985a, 1986a,b], and Ismail and Li [1992]. Exercises 7.38-7.40 For additional material on q-analogues of Hermite
216
Applications to orthogonal polynomials
polynomials, see Allaway [1980], Ai-Salam and Chihara [1976], Ai-Salam and Ismail [1988], Carlitz [1963b, 1972], Chihara [1968a, 1982, 1985], Dehesa [1979], Desarmenien [1982]' Hou, Lascoux and Mu [2003], Ismail [1985b], Ismail, Stanton and Viennot [1987], Lubinsky and Saff [1987], and Szego [1926]. Ex. 7.41 Rahman [1989a] gave a simple proof for this addition formula. For derivations of the addition formula for Jacobi polynomials, see Koornwinder [1974a,b] and Laine [1982]. Ex.7.43 See also Cigler [1981] and Pastro [1985]. In view of the two different orthogonality relations for the q-Laguerre polynomials, it follows that there are infinitely many measures for which these polynomials are orthogonal. The Stieltjes- Wigert polynomials (see Chihara [1978, pp. 172-174], Szego [1975, p. 33] and the above Notes for §7.3)
sn(x) =
(_I)nq(2n+l)/4(q;q)~!
t [~] j=O
J
qj2(_q!X)j, q
which are orthogonal with respect to the log normal weight function
w(x)
=
k7r-! exp( -k 2 1og2 x),
0
<x<
00,
where q = exp[-(2k2)-1] and k > 0, are a limit case of the q-Laguerre polynomials. Askey [1986] gave the orthogonality relation for these polynomials (with a slightly different definition) that follows as a limit case of the first orthogonality relation in this exercise. Ai-Salam and Verma [1983b,c] studied a pair of biorthogonal sets of polynomials, called the q-Konhauser polynomials, which were suggested by the q-Laguerre polynomials. Ismail and Rahman [1998] studied two indeterminate Hamburger moment problems associated with L~(x; q), thus completing earlier work of Moak [1981]. For asymptotics of basic Bessel functions and q-Laguerre polynomials, see Chen, Ismail and Muttalib [1994]. Ex.7.46 A right inverse ofthe Askey-Wilson operator is derived in Brown and Ismail [1995]. Ex.7.47 The question of completeness of the q-trigonometric functions for use in a q-Fourier type analysis is dealt with in Ismail [200lb] and Suslov [200lc, 2003].
8 FURTHER APPLICATIONS
8.1 Introduction In this chapter we derive some formulas that are related to products of the q-orthogonal polynomials introduced in the previous chapter and use these formulas to obtain q-analogues of various product formulas, Poisson kernels and linearization formulas for ultraspherical and Jacobi polynomials. The method in Gasper and Rahman [1984] originates with the observation that since j-I
(q-X, aqX; q) j =
II (1 - qk(q-x + aqX) + aq2k)
k=O is a polynomial of degree j in powers of q-X +aqX, there must exist an expansion of the form Hk
(q-X,aqX;q)j (q-X,aqX;q)k =
L
Am(j,k,a;q) (q-X,aqX;q)m·
(8.1.1)
m=O
Since, for k
~
j,
3¢2
(q-j, l-x, al+ x ; aqk, q1+k- j ; q, q)
(q-X, aqX; q)J.
= --,-------,-----,-------,-"(q-k, aqk; q)j
by the q-Saalschiitz formula (1.7.2), it is easy to verify that
(q-X,aqX;q)j (q-X,aqX;q)k = (q;q)j(q;q)k(a;q)Hk j+k X
L
(8.1.2)
m=max(j,k)
This linearizes the product on the left side and forms the basis for the product formulas derived in the following section. Suppose {B j } ~o and {Cj } ~o are arbitrary complex sequences and b, c are complex numbers such that (b; q)k, (c; q)k do not vanish for k = 1,2, ... Then, setting
Fn
=
t
j=O
t
(q-n,a.qn;q)j B j (q-n,a.qn;qh Ck , (q, b, q)j k=O (q, C, q)k
we find by using (8.1.2) that
217
(8.1.3)
218
Further applications k (q-k, aqm; q) j
x
L ( b m-k.,q) j=O q, q
j
B m - k+j
(8.1.4)
for n = 0,1,2,.... This formula does not extend to noninteger values of n because, in general, the triple sum on the right side does not converge.
8.2 A product formula for balanced
4¢3
polynomials
Since we are mainly interested in q-orthogonal polynomials which are expressible as balanced 4¢3 series or their limit cases, we shall now specialize (8.1.4) to such cases. Set B. (b l , b2; q) j j C. _ (CI , C2; q) j j () J - (b 3 , qab I b2 jbb 3 ; q)j q, J - (C3, q ac IC2j cc3; q)j q , 8.2.1 where it is assumed that the parameters are such that no zero factors appear in the denominators. Then formula (8.1.4) gives
n m in=LL
(-n n) ( ) q ,aq;qm cI ,c2;qm-k
qk 2 -mk+m
(q, b; q)k(q, C, C3, qacIC2jcc3; q)m-k
m=Ok=O (b l ,b2 ;q)k X (b 3 ,qab I b2 jbb 3 ;q)k
¢ [ qk-m,aqm,bIqk,b2qk 4
3
] bqk,b3qk,abIb2qk+Ijbb3;q,q ,
(8.2.2) where in is the right side of (8.1.3) with B j and C j as defined in (8.2.1). The crucial step in the next round of calculations is to convert the 4¢3 series in (8.2.2) into a very-well-poised 8¢7 series by Watson's formula (2.5.1), i.e., [
4¢3
qk-m, aqm, bIqk, b2qk ] bq k , b3q k ,abI b2q k+Ijbb3 ;q,q
(bb3qk-m jab l , bb3qk-m jab2 ; q)m-k (bb3q2k-m ja, bb3q-m jab I b2; q)m-k X
8W 7 (
bb3q m-k) bb3q 2k-m-1 b3q k-m bq k-m k k k-m ; , - - , b I q ,b2 q ,q ;q, b b . a a a I 2
(8.2.3)
8.2 A product formula for balanced
41>3
219
polynomials
Then, replacing j by j - k in the sum on the right side of (8.2.4), we obtain 41>3 [
q-n, aqn, bl , b2 ] j ;q,q b, b3 , qab l b2 bb3
~
[ q-n, aqn,
CI, C2
j
41>3
]
C3, qacIC2 CC3
C,
;q,q
(q-n, aqn, CI, C2, qabdbb3, qab 2 jbb 3 ; q)m m (q, C, C3, qajbb 3 , qacIC2jcC3, qab l b2 jbb 3 ; q)m q
=~
~ (bb 3q-m-1 ja; q)j(l - bb3q 2j-m-1 ja)(b l , b2 , bq-m ja, b3q-m ja, q-m; q)j
X
~
(q; q)j(l - bb3q-m-1 ja)(bb 3q-m jab l , bb3q-m jab 2 , b3 , b, bbJ/a; q)j
bb3qm )j
x ( --
[q-j, ql-m jc, ql-m j C3, bb3qj-m-1 ja, CC3q-m j acIC2. 51>4
bl b2
q
I
-m JCI,
q
I
-m j C2,
'
bq-m ja, b3q-m ja
]
q, q . (8.2.5)
Note that the 51>4 series in (8.2.5) is balanced and, in the special case C = aqjb and C3 = aqjb3 , becomes a 31>2 which is summable by (1.7.2). Thus, we obtain the formula
bb3c2jaq, q
where). formula
=
I -m
bq-m ja,
j C2,
b3q-m ja,
b3 ,
q-m
b,
bb3 ja,
;
q,
a q2 bl b2CIC2
]
,(8.2.6)
bb3q-m-1 ja. This formula is a q-analogue of Bailey's [1933] product 2FI ( -n,
a
+ n; b; x)
2FI ( -n,
a
+ n; 1 + a -
b; y)
= F 4 ( -n, a + n; b, 1 + a - b; x(l - y), y(l - x)),
(8.2.7)
~ ~ (a)m+n(b)m+n m n F ( b d ) 4 a, ;C, ;x,y = ~~ m!n!(c)m(d)n x y .
(8.2.8)
where
However, even though (8.2.6) is valid only when the series on both sides terminate, (8.2.7) holds whether or not n is a nonnegative integer, subject to the absolute convergence of the two 2FI series on the left and the F4 series on the right. Application of Sears' transformation formula (2.10.4) enables us to transform one or both of the 41>3 series on the left side of (8.2.6) and derive a number of equivalent forms. Two particularly interesting ones are
220
Further applications
(8.2.9)
bb3 C2/ aq, ql-m /C2,
b3 ,
(8.2.10)
b,
where ,\ = bb3 q-m-l/a. Either of the formulas (8.2.9) and (8.2.10) may be regarded as a q-analogue of Watson's [1922] product formula for the Jacobi polynomials 2FI( -n, a + n; b; x) 2FI( -n, a + n; b; y)
(-1 )n(1+a-b)n (b)n F4 (-n,a+n;b,l+a-b;xy, ( I-x )( l-y )) ,
=
(8.2.11) where n = 0,1, .... The special case in which the lOcP9 series in (8.2.6), (8.2.9) or (8.2.10) become balanced is also of interest in some applications. Thus, if we set C2 = aq/bIb2CI, then by using Bailey's transformation formula (2.10.8) we may express (8.2.9) in the form 4cP3
[
q-n, aqn, bl , b2 ] b, b3, qa bb/bb ;q,q I 2 3
= (aq/b,aq/b 3 ;q)n (bb 3 )n (b, b3 ; q)n X lOcP9
[
fJ'
aq
4cP3
t
] [q-n, aqn, bb3 cd aq, bb3 /bl b2CI b, b3, bb/bb ;q,q 3 I 2
(q-n,aqn,bICI,b2CI,aq/bIb2CI;q)mqm m=O (q, aq/b, aq/b 3 , bIb2CI, bb3 /b Ib2; q)m bIb2CI/b,
bIb2CI/b 3 ,
b,
b3 ,
8.3 Product formulas for q-Racah and Askey-Wilson polynomials
bb3cI/aq,
aqm,
qab 1b2/bb 3,
b1b2c1q-m la,
221
(8.2.12)
where /-l = b1b2c1q-1. This provides a q-analogue of Bateman's [1932, p. 392] product formula 2F1 ( -n, a + n; b; x) 2F1 ( -n, a + n; b; y)
= (_I)n (1 + a (b)n
b)n:t (-n)k(a + n)k (1- x _ y)k k=O k!(1 + a - bh
x 2F1 (-k, a + k; b; -xy/(1 - x - y)).
(8.2.13)
8.3 Product formulas for q-Racah and Askey-Wilson polynomials Let us replace the parameters a,b,b 1,b2,b3,C1,C2 in (8.2.9) byabq,aq,q-X, cqx-N, bcq, c- 1q-Y, qy-N, respectively, to obtain the following product formula for the q-Racah polynomials introduced in §7.2: VVn(x;a,b,c,lV;q) VVn(y;a,b,c,lV;q) n (q-n abqn+1 qx-N qy-N c- 1q-X c- 1q-Y. q) (b q, qac -1.) ,q n n '"""' ' , , , , 'm m = (aq, bcq; q) n c m=O ~ (b q q, q, qac -1 ,c -1 ,q -N , q-N ; q) m ,/,. [cq- m , q(cq-m)~, _q(cq-m)~, ca- 1q-m, b- 1q-m, q-m, q-X, X 10'1-'9
1
1
1
(cq-m)2, _(cq-m)2, aq, bcq, cq, cqx+ -m,
q -y , cq x-N , cq y-N b 2N +3 ] y+1-m N -x+1-m N -y+1-m ; q, a q , cq ,q ,q
(8.3.1)
where VVn(x;a,b,c,lV;q) _ -
41>3
[q-n, abqn+1, q-X, cqx-N . ] -N b ,q,q aq,q ,cq
(8.3.2)
is the q-Racah polynomial defined in (7.2.17). This is a Watson-type formula. Two additional Watson-type formulas are given in Ex. 8.l. Letting c ----t 0 in (8.3.1) gives a product formula for the q-Hahn polynomials defined in (7.2.21): Qn(x; a, b, lV; q) Qn(Y; a, b, lV; q) b ) n (-n b n+1 x-N y-N ) = (-aqtqG) ( q;q n q ,a q ,q ,q ;q m (aqx+y)-m (aq;q)n m=O (q,bq,q-N,q-N;q)m
L
X 41>3
[
q-X, q-Y, b- 1q-m, q-m b 2N+3] N-x+1-m N-y+1-m; q, a q . aq,q ,q
(8.3.3)
222
Further applications
To obtain a Watson-type product formula for the Askey-Wilson polynomials defined in (7.5.2) we replace a, b, bl , b2, b3 , Cl, C2 in (8.2.9) by abcdq-l, ab, ae i(} , ae- i (}, ac, de i ¢, de- i ¢, respectively, where x = cos B, y = cos ¢. This gives Pn(x; a, b, c, dlq) Pn(Y; a, b, c, dlq) = (ab, ac, ad, ad, bd, cd; q)n(ad)-n n (q-n abcdqn-l de i() de- i(} de i ¢ de-i¢. q)
x
L
'(q, ad, ad, , bd,, cd, da-
'1
m=O A.
' m qm
')
;q
m
[aq- m /d, q(aq-m /d)~, -q(aq-m /d)~, ql-m /bd, ql-m lcd, q-m,
X 10'1"9
1
1
(aq-m /d)'l, -(aq-m /d)'l, ab, ac, aq/d,
ae i(} , ae- i (} , ae i ¢, ae- i¢ bcq ] ql-me-i() /d, ql-mei(} /d, ql-me-i¢ /d, ql-mei¢ /d; q, ad·
(8.3.4)
When b = aq~ and d = cq~, the 1O¢9 series in (8.3.4) becomes balanced and hence can be transformed to another balanced 1O¢9 via (2.9.1). This leads to a Bateman-type product formula Pn(x; a, aq~, c, cq~ Iq) Pn(Y; a, aq~, c, cq~ Iq)
1
X 1O¢9
[v'
qv'l, 1
v 'l ,
ae- i(} , acq~ ei¢+i(} ,
1
-qv'l, 1
-v'l,
ae i ¢,
aq~ ei ¢,
ce i¢,
ac,
a2q~ ,
1
acq'l,
a 2c2qm, .¢ q'l- m e' /c, 1
2
~:mm+!; q, q]
a ce q
2
ae i (} , acq~ ei¢-i(} , ,
(8.3.5)
where v = a2cei¢q-~. In fact, if we replace a and c by q(2o:+l)/4 and _q(2,6+l)/4, respectively, then this gives a Bateman-type product formula for the continuous q-Jacobi polynomials (7.5.24) which, on letting q ----t 1, gives Bateman's [1932] product formula for the Jacobi polynomials:
t
PAOI ,,6)(x)PAOI ,,6)(y) = (_1)nC8 + 1 )n (-nh(n+a+,8+1)k (x+2 y)k OI OI PA ,,6) (l)PA ,,6) (1) (a+1)n k=o k!(,8+1h
x p(OI,,6) k
(1 +
x Y ) /p(a,,6)(l), x+y k
(8.3.6)
which is equivalent to (8.2.13). For terminating series there is really no difference between the Watson formula (8.2.11) and the Bailey formula (8.2.7) since one can be transformed into the other in a trivial way. However, for the continuous q-ultraspherical polynomials given in (7.4.14), there is an interesting Bailey-type product formula that can be obtained from (8.2.6) by replacing a, b, bl , b2, b3 , Cl, C2 by
8.4 A product formula
223
a 4, a2q~ , ae i() , ae- i () " -a2q~ aeiq, and ae-iq, , respectively'.
_
n
'"'
(q-n a 4qn aeiq, ae-iq, -aei () -ae-i(). q) '
-~ -0 mX 1O¢9
(
[
,
,
,
1
,
'm m
1
2 -a 2. q q , a 2q2 , -a 2q2 " -1 -a "
_q-m,
q
)
m
q( _q-m)~,
_q(_q-m)~
-q -m)!2,
_(_q-m)~
(
,
q~-m la2,
_q~-mla2,
q-m,
,
_a2q~ ,
a2q~,
_q,
ae i () , _q1-m e -i() la,
(8.3.7)
For further information about product formulas see Rahman [1982] and Gasper and Rahman [1984].
8.4 A product formula in integral form for the continuous q- ultraspherical polynomials As an application of the Bateman-type product formula (8.3.5) for the AskeyWilson polynomials we shall now derive a product formula for the continuous q-ultraspherical polynomials in the integral form Cn(X; ,8lq)Cn (y; ,8lq)
(r; ~)n q, q
n
,8-n/2j1 K(x, y, Z; ,8lq)Cn (z; ,8lq) dz,
(8.4.1)
-1
where K(
x
. r-II ) = (q,,8,,8;q)=1 (,8e2i(},,8e2iq,;q)=12 x, y, Z, {J q 2n(,82; q)= W
(z;,8~eiO+iq",8~e-i(}-iq",8~ei(}-iq",8~eiq,-i(})
(8.4.2)
with w(z; a, b, c, d) defined as in (6.3.1) and x = cos e, y = cos ¢. First, we set c = -a in (8.3.5) and rewrite it in the form !
!
!
!
rn(x; a, aq2, -a, -aq2Iq)rn(Y; a, aq2, -a, -aq2Iq)
2q = 11++a a 2n(-q _!)n 2
~ (q-n, a 4qn, -a2q~ ei(}+iq" -a2q~ eiq,-i() , -aq~ e-iq,; q) x
~
m=O X
(q, a2q~, -a2q~, -a 2q, -a3q~ eiq,; q) m
mqm
lOW9 (-a3q-~eiq,;aeiq" -aeiq"aq~eiq"aei(},ae-i(},a4qm,q-m;q,q),
(8.4.3) where
224
Further applications
rn(X; a, b, c, dlq) _ -
4¢3
[q-n, abcdqn-1, aeil:l, ae-il:l . ] b d ,q, q . a ,ac,a
(8.4.4)
The key step now is to use the d = -(aq)! case of Bailey's transformation formula (2.8.3) to transform the balanced 1O¢9 series in (8.4.3) to a balanced 4¢3 series:
(a 2e- 2i ¢, -a3q! ei¢; q) m (a 4 , -aq! e- i¢; q) m
X4¢3
[
a2e2i¢, _q! eil:l+i¢, _q! ei¢-il:l, q-m ] 21""'"1:1 2 1 Z"1:1 "'" 1 2"'" 2;q,q· -a q'2 eZ
(8.4.5)
So (8.4.3) reduces to !
!
!
!
rn(X; a, aq2, -a, -aq2Iq) rn(Y; a, aq2, -a, -aq2Iq)
2 = 1 + a q2 n(_q-'2l)n
1+a
~ (q-n, a4qn, a2e-2i¢, _a 2q! eil:l+i¢, _a2q! ei¢-il:l; q)
x
mqm
~
(q, a 4 , a 2q!, -a 2q!, -a 2q; q) m
m=O
X 4¢3
[
a2e2i¢, _q! eil:l+i¢, _q! ei¢-il:l, q-m ] 2 1 ""'"1:1 2 1 "1:1+"'" 1 2"'" 2;q,q . -a q'2 eZ
(8.4.6)
Transforming this 4¢3 series by Sears' transformation formula (2.10.4), we obtain a further reduction
(8.4.7) Now observe that, by (6.1.1), w (z. aei¢-il:l aeil:l- i ¢ aeil:l+ i¢ ae-il:l- i¢) (aei¢-il:l+i'I/J aei¢-il:l-i'I/J. q) dz j -1 1 ' , " " j
= j1
W (z; aqjei¢-il:l, aeil:l- i ¢, aeil:l+ i ¢, ae-il:l- i ¢) dz
-1
(8.4.8)
8.4 A product formula where
lal < 1 and
z
225
cos'ljJ. Hence q-m, a 2 , a2e2i¢, a2e-2iO ] [ 4 2 1 i¢-iO I-m i¢-iO; q, q a,-aq2e ,-q2 e
41>3
=
(q, a 2 , a 2; q)oo I (a 2e 2i ¢, a 2e- 2iO ; q)oo 12 2n(a 4 ; q)oo X
11
W
(z', aei¢-iO , ae iO - i ¢ , ae iO +i¢ , ae- iO - i¢)
-1
(q, a 2 , a 2; q)oo I (a 2e 2i ¢, a 2e- 2iO ; q)oo 12
x
11
2n(a 4 ; q)oo W
(z', aei¢-iO , ae iO - i ¢ , ae iO +i¢ , ae- iO - i¢)
-1
( -aq~ ei'ljJ, -aq~ e-i'ljJ; q) X
m
( -a2q~ ei¢-iO, -q~ e iO - i ¢; q) m
dz.
(8.4.9)
Substituting (8.4.9) into (8.4.7) and using (2.10.4) we finally obtain 1
1
1
1
rn(X; a, aq'2, -a, -aq'2lq) rn(Y; a, aq'2, -a, -aq'2lq)
=
11
K(x, y, z; a 2 lq) rn(Z; a,
aq~, -a, -aq~ Iq) dz.
(8.4.10)
-1
This yields (8.4.1) if we replace a by f3~ and use (7.4.14). By setting f3 = qA in (8.4.1) and taking the limit q --+ 1, Rahman and Verma [1986b] showed that (8.4.1) tends to Gegenbauer's [1874] product formula
O~(x)O~(y) 0;(1)0;(1) =
11
-1
O~(z) K(x, y, z) 0;(1) dz,
(8.4.11)
where (8.4.12) according as 1 - x 2 - y2 - z2 + 2xyz is positive or negative. Rahman and Verma [1986b] were also able to derive an addition formula for the continuous q-ultraspherical polynomials corresponding to the product formula (8.4.1). This is left as an exercise (Ex. 8.11).
226
Further applications
8.5 Rogers' linearization formula for the continuous q- ultraspherical polynomials
Rogers [1895] used an induction argument to prove the linearization formula
Cm(X; ,6lq)Cn (x; ,6lq)
mi~,n) (q; q)m+n-2d,6; q)m-k(,6; q)n-k(,6; q)k(,62; q)m+n-k k=O
(,62; q)m+n-2k(q; q)m-k(q; q)n-k(q; q)d,6q; q)m+n-k
(1 - ,6qm+n-2k) (1 _ ,6)
X
(8.5.1)
Cm+n- 2k (X; ,6lq)·
Different proofs of (8.5.1) have been given by Bressoud [1981d], Rahman [1981] and Gasper [1985]. We shall give Gasper's proof since it appears to be the simplest. We use (7.4.2) for Cn(x; ,6lq) and, via Heine's transformation formula (1.4.3) ,
',61 ) = (,6e- 2iO ; q) 00 (,6; q)m imO X, q (q,6-1e-2iO; q)oo (q; q)m e
c ( m
x
where x
= cos e.
2cPl (q,6-1,,6-2ql-m;,6-1ql-m;q,,6e- 2iO ) ,
Then, temporarily assuming that Iql
(8.5.2)
< 1,61 < 1, we have
(8.5.3) where A
(,6e- 2iO ;q) 00 (,6; q)m(,6; q)n i(m+n)O m,n - (q,6-1e-2iO; q)oo (q; q)m(q; q)n e . _
(8.5.4)
The crucial point here is that the 4cP3 series in (8.5.3) is balanced and so, by (2.5.1),
8.6 The Poisson kernel for Cn(x; ,8lq)
227
Substituting this into (8.5.3) and interchanging the order of summation, we obtain
min(m,n)
X
L
k=O
X
x
(,8-2 ql-m-n; q)2k (q,8-1 e -2iO)k (,8-1 ql-m-n; q)2k A,
2'/'1
(
(./ -2iO) ql-'(./-1 ,I-'(./-2 q1+2k-m-n.,I-'(./-1 q1+2k-m-n.,q,l-'e,
(856) ..
which gives (8.5.1) by using (8.5.2) and observing that both sides of (8.5.1) are polynomials in x. Notice that the linearization coefficients in (8.5.1) are nonnegative when -1 < ,8 < 1 and -1 < q < 1. For an extension of the linearization formula to the continuous q-Jacobi polynomials, see Ex. 8.24.
8.6 The Poisson kernel for Cn(x; ,8lq) For a system of orthogonal polynomials {Pn(x)} which satisfies an orthogonality relation of the form (7.1.6), the bilinear generating function 00
(8.6.1) n=O is called a Poisson kernel for these polynomials provided that h n = CVn for some constant C > O. The Poisson kernel for the continuous q-ultraspherical polynomials is defined by
Kt(x, y; ,8lq)
=
~ ~~;;;)~(~ ~~i Cn(x; ,8lq)Cn (y; ,8lq)tn ,
(8.6.2)
where It I < 1. Gasper and Rahman [1983a] used (8.5.1) to show that
-qt,8! , qte iHi , qte- iO - i , -t,8! , ,8te- iO - i , ,8teiHi , te iO - i , q,8tei-iO,
tei-iO, q,8teiO - i ,
where x = cos(),y = cos¢ and max(lql, Itl, closely related kernel
(8.6.3)
1(81) < 1. They also computed a
228
Further applications ((3, t 2 ; q) 00 ((32, (3t 2; q)oo
teiO+iq, ,
te-iO-iq, ,
(3te- iO - i q, , (3te iHi q, , (8.6.4) Alternative derivations of (8.6.3) and (8.6.4) were given by Rahman and Verma [1986a]. In view of the product formula (8.4.1), however, one can now give a simpler proof. Let us assume, for the moment, that It(3-~ 1< 1 and 1(31 < l. Then, by (7.4.1) and (8.4.1), we find that, with z = cos'lj.;,
1
(t(3~eiW,t(3~e-iw;q)
1
Lt(x, y; (3lq)
=
-1
K(x, y, z; (3lq) (
1.
1.
00)
t(3-2 e2W , t(3-2 e- 2W ; q
dz
00
(q,(3,(3;q)ool ((3e2iO,(3e2iq,;q)00 12 27r((32; q)oo
By (6.3.9),
x 8 W7 ((3t 2q-1., teiO+iq, , te-iO-iq, , teiO-iq, , teiq,-iO , (3." q (3)
.
(8.6.6)
Formula (8.6.4) follows immediately from (8.6.5) and (8.6.6). It is slightly more complicated to compute (8.6.1). Consider the generating function 00
Gt(z)
=
L
1 _ (3qn 1 _ (3 Cn(z; (3lq)tn
n=O ((3te iW , (3te- iw ; q) 00 (l-(3)(te iw ,te- iW ;q)oo
_
~ ((3qte iW , (3qte- iw ; q) 00
1-(3 (qteiW,qte-iW;q)oo
= 1 _ (3t2 ((3qte 2W , (3qte- iw ; q) 00 • (
)
(te 2w , te- 2W ; q) 00
(8.6.7)
8.7 Poisson kernels for the q-Racah polynomials
229
Then Kt(x, Y; ,8lq) =
11
-1
K(x, y, Z; ,8lq) G 13_1 (z) dz t
2
(1- t 2)(q,,8,,8;q)= I (,8e2il1,,8e2i
1
7r h (cos"p; 1, -1, q~, -q~, qt,8~ ) x o h ( cos"p;,8~ eil1+i
(8.6.8) This gives (8.6.3) via (7.4.1) and an application of (2.10.1). By analytic continuation, formulas (8.6.3) and (8.6.4) hold when max(lql, Itl, 1,81) < l. Even though it is clear from (8.6.3) and (8.6.4) that these kernels are positive when -1 < q, t < 1 and -1 ::; x, y ::; 1 if 0 ::; ,8 < 1, it is not clear what happens when -1 < ,8 < 0 since both SrP7 series in (8.6.3) and (8.6.4) become alternating series. It is shown in Gasper and Rahman [1983a] that the Poisson kernel Kt(x, y; ,8lq) is also positive for -1 < t < 1 when -1 < q < ,8 < 0 and when 23 / 2 - 3 ::; ,8 < 0, -1 < q ::; O. For the nonnegativity of the Poisson kernel for the continuous q-Jacobi polynomials, see Gasper and Rahman [1986].
8.7 Poisson kernels for the q-Racah polynomials
For the q- Racah polynomials
Wn(x; q) == Wn(x; a, b, c, N; q) we shall give conditions under which the Poisson kernel N
L
hn(q)Wn(x; q)Wn(y; q)tn,
0::; t
< 1,
(8.7.1)
n=O and the so-called discrete Poisson kernel
Z (q-Z; q)n L ( hn(q)Wn(x; q)Wn(y; q), n=O q ,q n -N.
Z =
)
(8.7.2)
0,1, ... , N, are nonnegative for x, y = 0,1, ... , N. Let us first consider a more general bilinear sum
PAx, y) == PAx, y; a, b, c, a, ry, K, M, N; q) =
r
Z (q-Z; q) L ( hn(a,b,c,N;q)Wn(x;a,b,c,N;q) n=O q ,q n -K.
X
Wn(y; a, aba-I, ry, M; q),
(8.7.3)
230
Further applications
where z = 0,1, ... ,min(K,N) and N:::::: M. If a = a,ry = c and M = N, then (8.7.3) reduces to (8.7.2) when K = N, and it has the Poisson kernel (8.7.1) as a limit case. From the product formula (8.2.5) it follows that Wn(x; a, b, c, N; q)Wn(y; a, aba-I, ry, M; q) (bq, aqlc; q)n cn ~ ~ (q-n, abqn+\ q)r+s (q-X, cqx-N, q-Y, ryqy-N; q)r ~~ (-N) (aq, b c; q) n r=O s=O q ; q r+s (q, aq, q -M ,abryqa -1 ,cq l-s ; q )r ( qx-N , c -1 q I-x ; q) s 1 - cq r - s r+s-rs A (8.7.4) x (q, bq, ac-1q; q)s 1 _ cq-S q r,s,
where
[ q-S, qr- y , ryqr+y-M, aqr+l, bcqr+l.
=
A r,s
] r+l r-M b -1 r+l l+r-s ,q, q . aq ,q ,a rya q , cq
5¢4
(8.7.5)
Using (8.7.5) in (8.7.4) and changing the order of summation, we find that
x (_1)r+s q (r+s)(2N-r-s+l l /2-rs A T,S B T,S'
(8.7.6)
where B
with>'
r,s -
5¢4
>. q>,~ _q>,~ , qr+s-N , qr+s-z . N -r-s ] [ " >.~, _>.~, ab q N+r+s+2, qr+s-K' q, q ,
= abq2r+2s+1.
We shall now show that when K
(8.7.7)
= N,
( N-z ) q ; q z-r-s A, (qr+s-z ql+r+s-z. ql+N-z. q abqN+z+2) , , , , . ( abq N+r+s+2.,q) z-r-s 2'1"1 (8.7.8) To prove (8.7.8) it suffices to show that Br,s
=
4¢3
[
a,
1
1
-qa 2 , 1 -a2,
qa2, 1 a2,
b w] ; q,w aq
(wblaq, wlb; q)oo 2'1"1, A, (b b' q,w bl' a,q,w Ib) ( w I aq,w;q ) 00
(8.7.9)
whether or not b is a negative integer power of q, provided the series on both sides converge. Since 1 - aq2k = 1 - qk + qk(l - aqk), the left side of (8.7.9) equals 2¢I(aq,b;w;q,wla)+ (wbla,wlb;q)oo (I ) w a,w;q 00
w(l - b) ( ) 2¢I(aq,bq;wq;q,wlaq) aq 1-w
2¢1
(
I I ) b, bq; wb a; q, w b
8.7 Poisson kernels for the q-Racah polynomials wlb; q)oo + w(l(b)(wbla, I ) aq w aq,w;q
rI-.
(wblaq, wlb; q)oo (w I aq,w;q )00
rI-.
(b b· bl· Ib) q, W a, q, w
2'1-'1,
00
=
(b b· q, W bl· a, q, w Ib)
2'1-'1,
by (1.4.5). Also, for 0: = a the which, by (2.10.4), equals
5¢>4
231
series in (8.7.5) reduces to a
(8.7.10) 4¢>3
series
( M+1-y-s b-1 -1 -y-s. ) _' "( q ,q s (b"(qr+s+y-M) s q (qr M, b"(qr+1; q)s X 4¢>3
[
q-S, qr- y , b-1q-S, C"(-l q M+1- y -s ] l+r-s M+1-y-s b- 1 -1 -y-s; q, q . cq ,q ,,,( q
(8.7.11)
From (8.7.6), (8.7.8) and (8.7.11) it follows that JDz(x,y; a,b,c, a,,,(,lV, AI, lV; q)
20
(8.7.12)
for x = 0,1, ... , lV, y = 0,1, ... , AI, z = 0,1, ... , lV when 0 < q < 1, 0 < aq < 1, 0 ::; bq < 1, 0 < c < aqN and cq ::; "( < qM -1 ::; qN -1. Hence the discrete Poisson kernel (8.7.2) is nonnegative for x, y, z = 0,1, ... ,lV when 0 < q < 1, o < aq < 1, 0 ::; bq < 1 and 0 < c < aqN. If in (8.7.3) we write the sum with lV as the upper limit of summation, replace (q-Z; q)n by (tq-K; q)n and let K ----t 00, it follows from (8.7.6) that Lt(x, y; a, b, c, 0:, ,,(, AI, lV; q) N
=
L tnhn(a, b, c, lV; q)Wn(x; a, b, c, lV; q)Wn(y;
0:,
abo:- 1 , ,,(, AI; q)
n=O
(b q,aq I c;q ) N
x N-x
~ ~
(b a q 2.) ,q 2r+2s (-x q ,cqx-N ,q -y ,,,(q y-M.) ,q r
(abq2,c 1;q)N ~ ~ (abqN+2;q)r+s (q,o:q,q-M, ab,,(qo:-1, cql-s; q)r
-
X
( q x-N ,c -1 q -x ; q) s (1 - cq r-s) A C (_t)r+s (r+s)(2N -r-s+1)/2-rs (q,b q,ac- 1 ) s (1-cq-S ) rs q , q;q " rs
(8.7.13) for x
=
0,1, ... , lV, y
C
=
-
r,s -
4¢>3
0, 1, ... , AI with Ar,s defined in (8.7.5) and \! r+s-N \ \ ! [ A, qA2, -qA2, q \! \! b N+r+s+2
A2,-A2,a q
. t N-r-s ] ,q, q ,
(8.7.14)
where A = abq2r+2s+1. However, by Ex. 2.2, rI-.
4'1-'3
[A,
qA~, A~ ,
-qA~, b- 1 ; q, tb] -A~
,
Abq
(t, Aq; q)oo ( ) bAb.) 2¢>l b ,tb;tq;q,Aq (t, q, q 00 for max(ltbl, IAql)
(8.7.15)
< 1. Use of this in (8.7.14) yields
Gr,s = (t, abq 2r+2s+2; q)N-r-s
2¢>1
(qN-r-s, tqN-r-s; tq; q, abq2r+2s+2) ,
(8.7.16)
232
Further applications
from which it is obvious that er,s 2: 0 for 0 :::; t < 1, r + s :::; N when Combining this with our previous observation that Ar,s equals the expression in (8.7.11) when a = a, it follows from (8.7.13) that
o :::; abq2 < 1.
Lt(x,y;a,b,c,a,,,!,M,N;q) > 0
(8.7.17)
for x = 0,1, ... ,N, y = 0,1, ... ,M, 0 :::; t < 1 when 0 < q < 1, 0 < aq < 1, bq < 1, 0 < c < aqN and cq :::; "! < qM-l :::; qN-l. In particular, the Poisson kernel (8.7.1) is positive for x,y = O,l, ... ,N, 0:::; t < 1 when o < q < 1, 0 < aq < 1, 0 :::; bq < 1 and 0 < c < aqN. For further details on the nonnegative bilinear sums of discrete orthogonal polynomials, see Gasper and Rahman [1984] and Rahman [1982].
o :::;
8.8 q-analogues of Clausen's formula Clausen's [1828] formula
{2Fl [a +ai~ ! ;z]
r
F [ 3
2
2a, 2b, a + b 2a + 2b, a + b +
! ;z
]
,
(8.8.1)
where Izl < 1, provides a rare example ofthe square of a hypergeometric series that is expressible as a hypergeometric series. Ramanujan's [1927, pp. 2339] rapidly convergent series representations of l/n, which have been used to compute n to millions of decimal digits, are based on special cases of (8.8.1); see the Chudnovskys' [1988] survey paper. Clausen's formula was used in Askey and Gasper [1976] to prove that 3
F [-n,n+a+2,!(a+1).1-x] > 0 2 a+1,!(a+3) '2 -
() 8.8.2
when a > -2, -1 :::; x :::; 1, n = 0,1, ... , which was then used to prove the positivity of certain important kernels involving sums of Jacobi polynomials; see Askey [1975] and the extensions in Gasper [1975a, 1977]. The special cases a = 2,4,6, ... of (8.8.2) turned out to be the inequalities de Branges [1985] needed to complete the last step in his celebrated proof of the Bieberbach conjecture. In this section we consider q-analogues of (8.8.1). Jackson [1940, 1941] derived the product formula given in Ex. 3.11 and additional proofs of it have been given by Singh [1959], Nassrallah [1982]' and Jain and Srivastava [1986]. But, unfortunately, the left side ofit is not a square and so Jackson's formula cannot be used to write certain basic hypergeometric series as sums of squares as was done with Clausen's formula in Askey and Gasper [1976] to prove (8.8.2). In order to obtain a q-analogue of Clausen's formula which expressed the square of a basic hypergeometric series as a basic hypergeometric series, the authors derived the formula
a,b,abz,ab/z . { 4¢3 [ b! a q2, -ab! q2,-a b,q,q
]}2
5¢4
a2,b2,ab,abz,ab/z ] [ 2b2 b! b! b; q, q , a ,a q2,-a q2,-a (8.8.3)
8.8 q-analogues of Clausen's formula
233
which holds when the series terminate. See Gasper [1989b], where it was pointed out that there are several ways of proving (8.8.3), such as using the Rogers' linearization formula (8.5.1), the product formula in §8.2, or the Rahman and Verma integral (8.4.10). In this section we derive a nonterminating q-analogue of Clausen's formula which reduces to (8.8.3) when it terminates. The key to the discovery of this formula is the observation that the proof of Rogers' linearization formula given in §8.5 is independent of the fact that the parameter n in the 21>1 series in (7.4.2) is a nonnegative integer. In view of (7.4.2) let
J(z)
= 21>1(a,(3;aql(3;q,zql(3),
(8.8.4)
which reduces to the 21>1 series in (7.4.2) when a = q-n and z = e- 2iIJ . Temporarily assume that Iql < 1(31 < 1 and Izl ::::; 1. From Heine's transformation (1.4.3) , (8.8.5) Hence, if we multiply the two coefficients of zj, we get
21>1
series in (8.8.4) and (8.8.5) and collect the
(8.8.6) where
Ak =
41>3
[
q-k,(3,(3q-kla,a ] (3q-k, (32 q-k la, aql (3; q, q
(8.8.7)
is a terminating balanced series. As in (8.5.5) we now apply (2.5.1) to the series in (8.8.7) to obtain that
2ql(32;q)00 A k -- (aql(3,a (a 2 q1(3 , aq 1(32.) , q 00
8
w:7 (2/(3· (3 2 k+l/(32 -k. 1(3) a ,a,a"a q ,q ,q,q .
41>3
( ) 8.8.8
Using (8.8.8) in (8.8.6) and changing the order of summation we get the formula
f=
J2(z) - ((3z;q)oo (1- a 2q2k l(3)(a 2/(3,a,a,(3;q)k - (zql (3; q)oo k=O (1 - a 21(3)(q, aql (3, aql (3, a 2ql (32; q)k (a 2ql(32;q)zk (Zq)k x (a 2ql (3; q)zk
/3
21>1
[a2q2k+l/(32,ql(3 ] a 2 q2k+l 1(3 ; q, (3z . (8.8.9)
Observe that since the 21>1 series in (8.8.9) is well-poised we may transform it by applying the quadratic transformation formula (3.4.7) to express it as an 81>7 series and then apply (2.10.10) to get the transformation formula
1> [a2q2k+l/(32,ql(3 (3] 2 1 a2q2k+l 1(3 ; q, z [ X 41>3
=
(k+l) (azql(3;q)oo(-al(3z)k q 2 ((3zla; q)oo(aql (3z, azql (3; q)k
] aqk, aqk+! 1(3, -aqk+! 1(3, _ aqk+ 1 1(3 a2q2k+l 1(3, azqk+l 1(3, aqk+l 1(3z ; q, q
234
Further applications
1
X
rI-.
4'f'3
[
1
/3z, zq"2 , -zq"2, -zq ] ; q, q . qz2, o: zq k+l, /3 zq l-k /0:
(8.8.10)
We can now substitute (8.8.10) into (8.8.9) and change the orders of summation to find that
f
J2(Z) = (/3z,o:zq//3;q)oo (0:; q)m(0:2q//3 2; qhm (zq/ /3, /3z / 0:; q)oo m=O (q, o:zq/ /3, o:q/ /3z, 0:2 q/ /3, o:q/ /3; q)m X
qm 6W 5(0:2//3;0:,/3,q-m;q,o:qm+l//32)
+
(0:, 0:2q/ /3, zq, o:zq; q)oo (/3z, zq!, -zq!, -zq; q)m m (o:q/ /3, 0:2q/ /3, zq/ /3,0:/ /3z; q)oo m=O (q, qz2, o:zq, /3zq/o:; q)m q
f
X6W5(0:2 //3; 0:, /3, o:q-m / /3z; q, zqm+l /(3). Summing the above {
6W5
(8.8.11)
series by means of (2.7.1), we obtain the formula
2 (/3z, o:zq/ /3; q)oo 2¢>1 ( 0:, /3; o:q / /3; q, zq / /3 )} = (zq //3 , /3 Z/ 0:,. q) rI-. [0:, o:q/ /3 2, o:q! //3, -o:q! //3, -o:q/ /3 ] X 5'f'4 o:q/ /3, 0:2q/ /3 2, o:q/ /3z, o:zq/ /3 ; q, q (0:, o:q/ /3 2, zq, zq, o:zq/ /3; q)oo 00
+~~~~~=-~~~~~-
(o:q//3,o:q//3,zq//3,zq//3,o://3z;q)oo rI-.
x 5'f'4
[/3z,Zq//3,Zq!,-Zq!,-Zq ] 2 ; q, q , zq, z q, /3zq/o:, o:zq/ /3
(8.8.12)
which gives the square of a well-poised 2¢>1 series as the sum ofthe two balanced 5¢>4 series. By analytic continuation, (8.8.12) holds whenever Iql < 1 and
Izq/ /31 < l. To derive (8.8.3) from (8.8.12), observe that if 0: 0 and (8.8.12) gives
q-n, n
=
=
0,1, ... , then
(0:; q)oo =
J2(Z) (/3z, zql-n//3;q)oo ¢> [q-n,q!-n//3,ql-n//32,_q!-n//3,_ql-n//3 ] = (/3zqn,zq//3;q)oo 5 4 ql-n//3,ql-2n//32, zq l-n//3,ql-n//3z ;q,q _ (/3 2,/32;q)n (.:.)n rI-. [q-n,/32 qn,/3,/3z,/3/z. ] - (/3 , /3.) 5'f'4 /32 , /3 q2,1 /3 q2,1 /3' q, q (8.8.13) ,q n /3 by reversing the order of summation. Since
1
2;q)n (.:.)n rI-. [q-n,/32 qn, (/3z)!, (/3/z)! . J( z ) = (/3 (/3) /3 4'f'3 1 1 , q, q ;q n /3q"2, -/3q"2, -/3 by (7.4.14), it follows from (8.8.13) that rI-. {
4'f'3
l}2
[q-n,/32 qn,(/3z)!,(/3/Z)!. /3 q2,1 /3 q2,1 /3 ,q,q
(8.8.14)
8.8 q-analogues of Clausen's formula
235 (8.8.15)
for n = 0,1, ... , which is formula (8.8.3) written in an equivalent form. Now note that 2¢1(a,~;aq/~;q,zq/~)
(z(aq)!, -z(aq)!, zqa! I~, -zqa! I~; q)oo (zq!, -zq!, zq/~, -azq/~; q)oo X 8 W 7 ( -az/~;
a!, -a!, (aq)! I~, -(aq)! I~, -z; q, -zq),
(8.8.16)
by (3.4.7) and (2.10.1), and set a = a!,b = (aq)!/~ to obtain from (8.8.12) the following q-analogue of Clausen's formula:
_1
}2
(a2z2q, b2z 2q; q2)00 { (2 2b 2 . 2) 8 W7( -abzq 2; a, -a, b, -b, -z; q, -zq) z q,a 2z q,q 00 _ (azq! Ib, bzq! la; q)oo ,j., [ a2, b2, ab, -ab, -abq! . ] 1 1 5¥'4 1 1 1 , q, q (zq'i lab, abzq'i; q)oo a2b2, abq'i, abzq'i, abq'i I z
+
(zq, zq, a 2, b2; q)oo 1 1 1 (abq'i,abq'i,ablzq'i,abzq'i;q)oo 1
1
1 -zq'i, 1 -zq azq'i1 Ib, bzq'i1 la, zq'i,
X 5¢4
[
2
1
3
zq,z q,abzq'i,zq'ilab
;
q, q ,
(8.8.17)
where Iql < 1 and Izql < 1. To see that (8.8.17) is a nonterminating q-analogue of Clausen's formula, it suffices to replace a by qa, b by qb and let q ----+ 1-; then the left side and the first term on the right side of (8.8.17) tend to the left and right sides of (8.8.1) with z replaced by -4z(l- Z)-2 and so, by (8.8.1), the second term on the right side of (8.8.17) must tend to zero. It is shown in Gasper and Rahman [1989] that the nonterminating extension (3.4.1) of the Sears-Carlitz quadratic transformation can be used in place of (8.8.10) to derive the product formula 2¢1
(a,b;c;q,z ) ,j.,
x
6¥'5
2¢1
(a,aq I c;aq I b;q,z )
(az,abzlc;q)oo
= (z, bz I c; q )
00
[a, clb, (aclb)!, -(aclb)!, (acqlb)!, -(acqlb)! ] I I I ; q, q aq b, c, ac b, az, cq bz
+ (a,clb, az, bz,azqlc;q)oo (c,aqlb,z,z,clbz;q)oo ,j.,
x
6¥'5
[z, abzlc, z(ablc)!, -z(ablc)!, z(abqlc)!, -z(abqlc)! ] I I 21 ; q, q , az, bz, azq c, bzq c, abz c (8.8.18)
where Izl < 1 and Iql < 1. This formula reduces to (8.8.12) when a ~, c = aql ~ and z is replaced by zq/~.
= a, b =
236
Further applications
By applying various transformation formulas to the 2¢1 series in (8.8.12) and (8.8.18), these formulas can be written in many equivalent forms. For instance, by replacing b in (8.8.18) by c/b and applying (1.5.4) we obtain
2¢2(a, b; c, az; q, cz/b) 2¢2(a, b; abq/c, az; q, azq/c) ] _ (z,az/b;q)oo " [a,b,(ab)!,-(ab)!,(abq)!,-(ab q)!. - (az, z /b) ' q, q ; q 00 6'1'5 abq / c, c, ab, az, bq / z
+ (a, b, cz/b, azq/c; q)oo (c, abq/c, az, b/ z; q)oo " [z, abz, z(a/b)!, -z(a/b)!, z(aq/b)!, -z(aq/b)! ] ( ) ; q, q , 8.8.19 az, cz/b, aqz/c, zq/b, az 2/b
X 6'1'5
where max(lql, lazq/cl, Icz/bl) < 1. If we replace a, b, z in (8.8.19) by qa, bb, z/(z - 1), respectively, and let q ----+ 1-, we obtain the Ramanujan [1957, Vol. 2] and Bailey [1933, 1935a] product formula
2Fl(a, b; c; z) 2Fl(a, b; a + b - c + 1; z) = 4F3(a, b, (a + b)/2, (a + b + 1)/2; c, a + b, a + b - c + 1; 4z(1 - z)), (8.8.20) where Izl < 1 and 14z(l- z)1 < 1. This is an extension of Clausen's formula in the sense that by replacing a, b, c, 4z(1 - z) in (8.8.20) by 2a, 2b, a + b + ~,z, respectively, and using the quadratic transformation (Erdelyi [1953, 2.11 (2)]) 1
1
2Fl(2a, 2b; a + b + "2; z) = 2Fl(a, b; a + b + "2; 4z(1 - z)),
(8.8.21)
we get (8.8.1). See Askey [1989d].
8.9 Nonnegative basic hypergeometric series
Our main aim in this section is to show how the terminating q-Clausen formula (8.8.3) can be used to derive q-analogues of the Askey-Gasper inequalities (8.8.2) and of the nonnegative hypergeometric series in Gasper [1975a, Equations (8.19), (8.20), (8.22)]. As in Gasper [1989b], let us set 'Y Cl
= q2b,al = qb,a2 = qb eiO,a3 = qbe-iO,b 1 = q2b,b2 = _qb,
= q-n ,C2 = qn+a ,d1 = q2!(a+l) = -
d 2, el
= qH!2 = -e3, x = q, W = 1 ,
in the r = 3, s = t = u = k = 2 case of (3.7.9) to obtain the expansion
8.9 Nonnegative basic hypergeometric series
x 5rP4 [
237
q-j,qj+2b, qb,qbeiO, qbe-iO ] 2b HI HI b ;q,q, q ,q 2 , -q 2 , -q
(8.9.1)
where, as throughout this section, n = 0,1, .... By Ex. 2.8 the 4rP3 series in (8.9.1) equals zero when n - j is odd and equals ( q, q a-2b.,q 2) k 2k(n-2k+b+l/2) (q2n-4k+a+l, q2n-4k+2H2; q2)k q
when n - j = 2k and k = 0,1, .... Hence, using (8.8.3) to write the 5rP4 as the square of a 4rP3 series, we have 5rP4
[
q-n, qn+a, qb, qbeiO, qbe-iO ] 2b I (a+l) I (a+l) b ; q, q ,-q2 ,-q q ,q2
[n/2]
=
~
bIb
1
(_l)n(q-n,qn+a,q +"2,-q +"2;q)n-2k (q, q! (a+I), _q! (MI), qn-2k+2b; q)n-2k
(8.9.2)
(8.9.3) when a ~ 2b > -1 and 0 follows from (8.9.3) that 3
F
2
1. By setting x
-n, n + a, b . 1 - x] [ 2b,~(a+1)'-2-
= cos e and
>0 - ,
-1
letting q
:s; x :s; 1,
----+
1-, it
(8.9.4)
when a ~ 2b > -1 which shows that (8.9.3) is a q-analogue of (8.9.4). When a = a + 2 and b = ~(a + 1), (8.9.4) reduces to (8.8.2). Special cases of (8.9.4) were used by de Branges [1986] in his work on coefficient estimates for Riemann mapping functions. Another q-analogue of (8.9.4) can be derived by using (8.9.1), (8.8.3) and Ex. 2.8 to obtain 6rP5
[
q-n, qn+a, qb, _qb, q!aeiO, q!ae-iO ] 2b l(a+l) l(a+l) Ia Ia; q, q ,-q2 ,-q2,-q2 q ,q2
n
=
.?=
+
1
·0
1
1
"0
(q-n, qn a, q"2 a , q"2a e 2 ,q"2 ae- 2 ; q)j (-l)jqj+(~) (q q!(a+l) _q!(a+l) _q!a qJ+a-l. q)"
J=O'
,
"
'J
238
Further applications
X
( q, q a-2b.,q2) k q2k 2 +k+2kb (qa, q2b+1; q2h {
X
4¢3
[
]}2
q2k-n, qn+2k+a, qk+-iae!il:l, qk+-iae-!il:l q2k+!(a+l) , _q2k+!(a+1), _q2k+!a ; q, q
, (8.9.5)
which shows that (8.9.6) when a ~ 2b > -1 and 0 < q < 1. The expansions (8.12) and (8.17) in Gasper [1975a] are special cases of the q ----+ 1- limit cases of (8.9.2) and (8.9.5), respectively, when (7.4.14) and Gasper [1975a, (8.10)] are used. A q-analogue of the expansion (Gasper [1975a, 1989b])
[-n,n+a+2,~(a+1)'(1_x2)(1_
F 3
a+1,~(a+3)'
2
n _
-
n!(n+a+2).(0+2).
'"'
~ "( j=O J. n
x {
2
J
J
_ J. ')' (0+3) (. + a + 1).J 2 j J j!(n - j)!
(a + 1)j(2j + a
+ 2)n-j
2)] y
(1 _ 2)j Y
c!(n+1\X)CH !(n+2)(y)}2 J n-J
(8.9.7)
is easily derived by employing (3.7.9), (8.8.3) and (7.4.14) to obtain
¢ [q-n, qn+n+2, q! (n+1), q! (n+1) e2il:l, q! (n+1) e-2il:l, qn+1, q! (n+3), _q! (n+3) , _q! (n+2) ,
7 6
q! (n+2) e2ir, q! (n+2) e-2ir 1(n+2)
-q2
] ; q,
1(n+1)
,-q2
q
n (q-n qn+n+2 q! (n+2) q! (n+2) e2ir q! (n+2) e-2ir. q) .
= '"'
~
J=O
x {
'
,
,
,
(q q!(n+3) _q!(n+3) _q!(n+2) qj+n+l. q). '
,
,
,
'J
,
J
. . (j)
(-1)1
J+ 2
q
(q; q)j(q; q)n-j !(j+n+~) (qn+1; q)j(q2Hn+2; q)n-j q
x Cj (COS (); q! (n+1) Iq)Cn - j (COS T; qH! (n+2) Iq) } 2,
(8.9.8)
which is obviously nonnegative for real () and T when a > -2. If we proceed as in (8.9.5), but use the q-Saalschiitz summation formula (1.7.2) instead of Ex. 2.8, we find that
8.10 Applications in the theory of partitions of positive integers
239
}2
{ [qj-n " qj+n+a q!HiadiO , q!Hiae-!iO • ] I)J. 'J(b+l)+ (j) 2..!, x (q 4 '/-'3 J+1(a+l) J+1(a+l) J+1a ,q, q q 2 ,-q 2 ,-q 2 (8.9.9)
and hence (8.9.10) for real () when a ::::: c ::::: b, a (8.9.10) it follows that F [ 4
3
+ b > c > 0 and 0 < q <
-n, n + a, ~a, b + 1), c, a + b -
~ (a
1. By letting q ---+ 1 - in
.1 - x] > 0 c' -2- -
(8.9.11)
for -1 :::; x :::; 1 when a ::::: c ::::: band a+b > c > 0, which gives the nonnegativity of the 3F2 series in Gasper [1975a, (8.22)] when a = a + ~,b = ~a + and c = a + 1. Also see Gasper [1989d].
£
8.10 Applications in the theory of partitions of positive integers In the applications given so far we have dealt almost exclusively with orthogonal polynomials which are representable as basic hypergeometric series. These are important results and most of them have appeared in print during the last thirty years. They constitute the main bulk of applications as far as this book is concerned. However, our task would remain incomplete if we failed to mention some of the earliest examples where basic hypergeometric series played crucial roles. The simplest among these examples is Euler's [1748] enumeration p( n) of the partitions of a positive integer n, where a partition of a positive integer n is a finite monotone decreasing sequence of positive integers (called the parts ofthe partition) whose sum is n. To illustrate how p(n) arises in a q-series let us consider a formal series expansion of the infinite product (q; q)~} in powers of q: 00
k=O
00
(8.10.1) where
240
Further applications
n
=
kl . 1 + k2 . 2 + ... + k n . n,
(8.10.2)
p(O) = 1 and, for a positive integer n, p(n) is the number of partitions of n into parts::; n. In the partition (8.10.2) of n there are k m m's and hence 0::; k m ::; n/m, 1 ::; m ::; n. For small values of n, p(n) can be calculated quite easily, but the number increases very rapidly. For example, p(3) = 3, p( 4) = 5, p(5) = 7, but p(243) = 133978259344888. Hardy and Ramanujan [1918] found the following asymptotic formula for large n:
p(n)
~ 4n~ex+ C;) ']
(8.10.3)
Also of interest are the enumerations of partitions of a positive integer n into parts restricted in certain ways such as: (i) PN(n), the number of partitions of n into parts::; N, which is given by the generating function 00
(8.10.4) (q; q)"i/ = 2::PN(n)qn, n=O where n = kl . 1 + k2 ·2+ ... + kN . N has k m m's ; (ii) Pe(n), the number of partitions of an even integer n into even parts, generated by 00
II (1 _ q2k+2)-1
(q2; q2)~;;} =
k=O 00
(8.10.5)
= 2::Pe(n)qn; n=O
(iii) Pdist(n), the number of partitions of n into distinct positive integers, generated by 1
(-q; q)oo =
1
2:: 2:: ... qk ·Hk ·2+ .. 1
k1=O
2
k2 =O
00
=
2::
(8.10.6) Pdist (n )qn, n=O where n = kl . 1 + k2 ·2+ ... + k n . n, 0 ::; k i ::; 1, 1 ::; i ::; n, and (iv) po(n), the number of partitions of n into odd parts, generated by 00
(q;q2)~1= II(1-q2k+l)-1
k=O 00
(8.10.7) Euler's partition identity Pdist(n) = po(n)
(8.10.8)
8.10 Applications in the theory of partitions of positive integers
241
follows from (8.10.6), (8.10.7) and the fact that (-q; q)OO =
(q'l,1 -q'l;1)-1 q = (q; q2);;}. 00
(8.10.9)
Other combinatorial identities of this type can be discovered from q-series identities similar to, but perhaps somewhat more complicated than, (8.10.9). For example, let us consider Euler's [1748] identity involving the pentagonal numbers n(3n ± 1)/2: 00
n=-oo 00
00
n=l n=l which is given in Ex. 2.18. A formal power series expansion gives 00
k=O
=
1
1
k1=O
k2 =O
L L ... (_I)kl+k2+··qkl·Hk2·2+ .. ,
(8.10.11)
which differs from the multiple series in (8.10.6) only in the factor (-I)kl+k 2+··. This factor is ±1 according as the partition has an even or odd number of parts. Denoting these numbers by Peven (n) and Podd (n), respectively, we find that 00
(q; q)oo = 1 +
L
[Peven(n) - pOdd(n)] qn.
(8.10.12)
n=l
From (8.10.10) and (8.10.12) it follows that _{(-I)k Peven (n) -
Podd (n)
forn=k(3k±I)/2,
-
(8.10.13) otherwise. Thus Euler's identity (8.10.10) expresses the important property that a positive integer n which is not a pentagonal number of the form k(3k ± 1)/2 can be partitioned as often into an even number of parts as into an odd number of parts. However, if n = k(3k ± 1) /2, k = 1,2, ... , then Peven (n) exceeds Podd (n) by (-I)k. See Hardy and Wright [1979], Rademacher [1973] and Andrews [1976, 1983] for related results. Two of the most celebrated identities in combinatorial analysis are the socalled Rogers-Ramanujan identities (2.7.3) and (2.7.4) which, for the purposes of the present discussion, we rewrite in the following form:
o
1+
L 00
n2
-qn=l (q;q)n
1
= --------
fi (1- q5k+ 1) (1- q5k+4)
k=O
(8.10.14)
242
Further applications 00
1+
n=l
1
qn(n+l)
L ( )
=
TI (1 -
q; q n
q5k+2)
(1 -
(8.10.15) q5k+3)
k=O
It is clear that the infinite product on the right side of (8.10.14) enumerates the partitions of a positive integer n into parts of the form 5k + 1 and 5k + 4, while that on the right side of (8.10.15) enumerates partitions of n into parts of the form 5k + 2 and 5k + 3, k = 0,1, .... Following Hardy and Wright [1979] we shall now give combinatorial interpretations of the left sides of the above identities. Since
k2
=
1 + 3 + 5 + ... + (2k - 1),
we can exhibit this square in a graph of k rows of dots, each row having two more dots than the lower one. We then take any partition of n-k 2 into at most k parts with the parts in descending order, marked with x's and placed at the ends of the rows of dots to obtain a partition of n into parts with minimal difference 2. For example, when k = 5 and n = 32 = 52 + 7 we add 4 x's to the top row, 1 each to the 2nd, 3rd and 4th rows, counted from above. This gives the partition 32 = 13 + 8 + 6 + 4 + 1 displayed in the graph below.
•
•
•
•
•
•
•
•
•
•
•
•
•
•
x
•
•
•
•
x
•
•
x
•
x
x
x
x
• The identity (8.10.14) states that the number of partitions of n in which the differences between parts are at least 2 is equal to the number of partitions of n into parts congruent to 1 or 4 (mod 5). Observing that k(k
+ 1) = 2 + 4 + 6 + ... + 2k,
a similar interpretation can be given to the left side of (8.10.15). Since the first number in the above sum is 2, one deduces that the partitions of n into parts not less than 2 and with minimal difference 2 are equinumerous with the partitions of n into parts congruent to 2 or 3 (mod 5). For more applications of basic hypergeometric series to partition theory, see Andrews [1976-1988], Fine [1948, 1988], Andrews and Askey [1977], and Andrews, Dyson and Hickerson [1988]. Additional results on RogersRamanujan type identities are given in Slater [1951, 1952a], Jain and Verma [1980-1982] and Andrews [1975a, 1984a,b,c,d].
8.11 Representations of positive integers as sums of squares
One of the most interesting problems in number theory is the representations of positive integers as sums of squares of integers. Fermat proved that all
8.11 Representations of positive integers as sums of squares
243
primes of the form 4n + 1 can be uniquely expressed as the sum of two squares. Lagrange showed in 1770 that all positive integers can be represented by sums of four squares and that this number is minimal. Earlier in the same year Waring posed the general problem of representing a positive integer as a sum of a fixed number of nonnegative k-th powers of integers (positive, negative, or zero) with order taken into account and stated without proof that every integer is the sum of 4 squares, of 9 cubes, of 19 biquadrates, 'and so on'. More than 100 years later Hilbert [1909] proved that all positive integers are representable by s kth powers where s = s(k) depends only on k. For an historical account of the Waring problem, see Dickson [1920]' Grosswald [1985], and Hua [1982]. To illustrate the usefulness of basic hypergeometric series in the study of such representations we shall restrict ourselves to the simplest cases: sums of two and sums of four squares, where it is understood, for example, that n = xi + x§ = yi + y§ are two different representations of n as a sum of two squares if Xl -=1= Yl or X2 -=1= Y2· Let r2k (n) be the number of different representations of n as a sum of 2k squares, k = 1,2, .... We will show by basic hypergeometric series techniques that, for n ~ 1,
r2(n) = 4(d l (n) - d3 (n)),
(8.11.1)
L
(8.11.2)
r4(n) = 8
d,
dln,4id
where di(n), i = 1,3, is the number of (positive) divisors of n congruent to i (mod 4) and the summation in (8.11.2) indicates the sum over all divisors of n not divisible by 4. The numbers 4 and 8 in (8.11.1) and (8.11.2), respectively, reflect the fact that r2(1) = 4 since 1 = 02 + (±1)2 = (±1)2 +0 2, and r4(1) = 8 since 1 = 02 + 02 + 0 2 + (± 1)2 = 0 2 + 02 + (± 1) 2 + 0 2 = 02 + (± 1) 2 + 02 + 02 = (±1)2 + 02 + 0 2 + 02. Both of these results were proved by Jacobi by means of the theory of elliptic functions, but the proofs below are based, as in Andrews [1974a], on the formulas stated in Ex. 5.1, 5.2, and 5.3. Combining Ex. 5.1 and 5.2 we have (8.11.3) However, the bilateral sum on the left side is clearly a generating function of (-1 )nr2 (n) and so it suffices to prove that 00
00
"[dl(n) - d3 (n)](-qt ~
(-1 )nqn(n+l)/2
(8.11.4) l+qn n=l n=l By splitting into odd and even parts and then by formal series manipulations, we find that 00 (-1 )nqn(n+l)/2 00 qr(2r+l) 00 q(m+l)(2m+l)
L
n=l
1+
qn
=
L r=l
1+
00
00
r=l m=r
q2r -
= "
~
L
m=O
1+
q2m+l 00
00
m=Or=m+l
244
Further applications
(Xl
(Xl
m=Or=l (Xl
=L
(Xl
L(-lr (q(4m+l)r _ q(4m+3)r)
m=Or=l (Xl
=
L( -q)n(d (n) 1
d3 (n)),
(8.11.5)
n=l
which completes the proof of (8.11.4). To prove (8.11.2) we first replace q by -q in Ex. 5.1 and 5.3 and find that
nqn
(Xl
= 1 + 8 ~ 1 + (_q)n '
(8.11.6)
where the last line is obtained from the previous one by expanding (1 + (_q)n)-2 and interchanging the order of summation. Now,
-- L (Xl
n=l
nqn -+ 1 - qn
L
even
n~2
nq
nIl ----1 + qn 1 _ qn (
)
(8.11. 7) Thus, (8.11.8)
Exercises
245
Since T4(O) = 1, this immediately leads to (8.11.2). For a direct proof of (8.11.8) based only on Jacobi's triple product identity (1.6.1), see Hirschhorn [1987]. Ex. 5.1 and 5.4 can be employed in a similar manner to show that
T8(n) = 16( _l)n~) _1)dd 3,
n ~ 1.
(8.11.9)
din
See Andrews [1974a]. Some remarks on other applications are given in Notes §8.11.
Exercises
8.1 Prove for the little q-Jacobi polynomials Pn(x; a, b; q) defined in (7.3.1) that
(i)
(ii)
Pn(x;a,b;q)Pn(y;a,b;q) n (b ) n (q-n abqn+1. q) m = (-bq)-n q -(2) q;qn 'm(_b yq 2)m q (2) (aq; q)n m=O (q, bq; q)m
L
(q-m abqm+1. q) (' ) ' k ( xq )k k=O q, aq; q k m
X '""' ~
'
A. ( qk-m ,xq;; b 0 2'/-'1
q, b-1 y -1) .
8.2 Derive the following product formula for the big q-Jacobi polynomials defined in (7.3.10):
246
Further applications
8.3 Prove that (abq,aq;q)n(1-abq2n+l) (t)n L ( b ' ) (1 _ b) Pn(qX; a, b; q)Pn(qY; a, b; q) n=O q, q, q n a q aq 00
= (t, abq2; q)oo L 00
r=O X 2¢1
(-X
min(x,y)
L
-Y
)
-s
q ,q ; q sa q(x+y)(r+s) (q, bq; q)s (q, aq; q)r
s=O
(0,0; qt; q, abq2r+2s+2) q-2rs-s 2 t r +s ,
and that this gives the positivity of the Poisson kernel for the little q-Jacobi polynomials for x, y = 0,1, ... , 0:::; t < 1 when 0 < q < 1,0 < aq < 1 and O
(i) . .) (. . ) _ (bq, ab q N+2; q)n ( N+l)-n Qn ( x,a,b,N,q Qn y,a,b,N,q ( -N.) bq aq, q ,q n X
n
fo ~
X
6
(q-n, abqn+l, bqN -x+l, bqN -Y+\ q)m m q (q, bq, abqN+2, bqN+l; q)m (b-1q-N-m-\ qh(1 - b- 1q2k-N-m-l )(a-1b-1q-N-m-l, b-1q-m; q)k (q;q)k(1- b-1q-N-m-l)(aq,q-N;q)k
(q-m, q-X, q-Y; q)k ( b)k (x+y+2m-k)k (b-1q-N, b-1qx-N-m, b-1qy-N-m; q)k a q ,
X
(ii) Qn(X; a, b, N; q)Qn(Y; a, b, N; q) n
X
L m=O
(bq, ab q N+2; q) n (bqN+l)-n ( N) aq,q- ;q n
=
"
,
(q-n abqn+l bqN+l-x-y. q)
m
(q, bq, abqN+2; q)m
q (x+l)m
b -x x'"""' (-m q ,a q ,q ,q ;q k (bqN-X+2)k 6 (q, aq, q-N, bqN+l-x- y ; qh m
X 3¢2
m+l
[
-y
)
] qk-m,qk-x,bqN+1- X 0, bqN+l-x-y+k ; q, q .
8.5 Prove for the q-Racah polynomials that
(i)
247
Exercises
(ii)
(aqX+1, aqY+\ q) m X
X
m
q
(aq, aq; q)m -1 -N-m-1 10 W.9 (ca q ;ca -1 q -m ,a -lb-1 q -N-m-1 , q-m, q-X, cqx-N, q-Y, cqy-N; q, qba-1) .
8.6 Show that
(i) VVn (x;a,b,c,lV;q)VVn (y;a,b,c,A1;q) (bq,aqc-\q)ncn (aq, bcq; q)n
x
10
t
m=O
(q-n,abqn+l,qx-N,qy-M;q)m (q, bq, aqc-l, c- 1 ; q)m
W.9 (cq-m., ca- 1q-m , b- 1q-m 'q-m q-X cqx-N " q-Y cqy-M., q , ab q M+N+3) "
where n
= 0,1, ... , min(A1, lV),
x
= 0,1, ... ,lV,
and Y
= 0, 1, ... , A1;
(ii) VVn(x;a,a,c,lV;q)VVn(y;a,a,c,lV;q) (aqc-l,a 2qN+2;q)n (ac- 1qN+1)-n ~ (q-n,a 2qn+l,aqx+\q)m (acq, q-N; q)n ~o (q, a 2qN+2, aq; q)m X
( aq 1+ y-x, ac- 1q N-x-y+\ q)m m q (aqc- 1, aq1-x;q)m
x
lO VV9
(aq-X;a2qm+l,q-m,aqN-x+l,c-1q-X,q-X,q-Y,cqy-N;q,q).
8.7 For the q-Hahn polynomials prove that
(i)
,
248
Further applications
x Qn(x; a, b, N; q)Qn(Y; a, b, M; q)
= LZ
z-r (-Z) L q ; q r+s (b a q 2) ; q 2r+2s (-X q ,q
; q ) r (x-N q , q y-M ; q ) s (q-M, q-N, abqN+2; q)r+s (q, aq; q)r(q, bq; q)s
r=O s=O
for x
-Y
X
( N-z ) q ; q z-r-s (_1r+ sa- Sq(r+s)(2N-r-s+1)/2-(X+Y+1)S (abqN+r+s+2; q)z-r-s
x
21>1
(qr+s-z, q1+r+s-z; q1+N -z; q, abqN +Z+2)
= 0,1, ...
,N, Y
= 0,1, ...
o < aq < 1, 0 :::; bq < 1 and N
,M, z
= 0,1, ...
2 O.
,N, when 0
<
q
< 1,
:::; M;
(ii) N
" ~
(abq, aq, q-N; q)n(1- abq2n+1) ( t) n (n) -aq qNn- 2 (q, bq, abqN+2; q)n(1 - abq)
x Qn(X; a, b, N; q)Qn(Y; a, b, M; q)
=
t "'I: r=O s=O
(abq2; qhr+2s (q-X, q-Y; q)r (qx-N, qy-M;
qt
(q-M,abqN+2;q)r+s (q,aq;q)r(q,bq;q)s
x (t, abq2r+2s+2; q)N-r-s
21>1
(qN-r-s, tqN-r-s; qt; q, abq2r+2s+2)
x a- s q(r+s)(2N -r-s+l)/2-(x+y+1)s (-tr+s > 0, for x = 0, 1, ... ,N, Y = 0,1, ... ,M, 0 :::; t 1, 0 :::; bq < 1 and N :::; M. (Gasper and Rahman [1984])
< 1 when 0 < q < 1, 0 < aq <
8.8 Prove that
t
(aq, abq, ~cq; q)n( -c)-n An(Z)qG) n=O (q, bq, aqc 1; q)n(abq; qhn
x Wn(x; a, b, c, N; q)Wn(y; a, aba-I, ,,(, M; q) =
tI:
(q-Z;q)r+s (q-X,cqx-N, q-Y, "(qy-M; q)r r=Os=O (q-N;q)r+s (q,aq,q-M,ab,,(qa- 1, cq1-s; q)r
(qx-N, c- 1q-X; q) s (1 - cqr-s) -rs X (b Ar' s ILr+s, q, q,aqc- 1) ;q s (1- cq-S ) q
where z = 0,1, ... ,N,
N:::; M,
An(z)
= ~ (q-Z; q)n+k ILn+k
~ (q,abq2n+2;qh '
and {lLr} is an arbitrary complex sequence, Ar,s being the same as in (8.7.5).
Exercises
249
8.9 Deduce from Ex. 8.8 that
(abq, aq, d, e, bcq, q-Z; q)n (1 - abq2n+l)
Z
~ (q, bq, abq2ld, abq2 Ie, aqlc, abqz+2; q)n(1- abq)
(
ab qZ +2 cde
)n
x Wn(x; a, b, c, N; q)Wn(y; a, aba-I, ,,(, M; q)
LL
= ( abq2 ,abq21de; q) Z Z z-r
(-Z q , d ,e; q)r+s (-x q , cq x-N ; q) r (abq2 I d, abq2 I e; q) Z r=O s=o (q-N, deq-z-l lab; q)r+s (q, q-M; q)r
qt
(1 - cqr-s) r+s-rs A (q-Y, "(qy-M; q)r (qx-N, c-1q-X; x (aq, ab,,(qa-1, cql-s; q)r(q, bq, aqlc; q)s(1 _ cq-s) q r,s' 8.10 Use (7.4.14) to show that (8.8.3) is equivalent to the formula
{Cn(cos8; fJlq)}2 _ (fJ2,fJ2;q)n -n - (q,q,qn .) fJ
[q-n, fJ2qn,fJ,fJe2i6, fJe-2i6 . ] fJ2 , fJ! fJ! fJ ' q, q , q2,- q2,-
5¢4
where n = 0,1, .... 8.11 In view of the product formula (8.4.10) and Gegenbauer's [1874, 1893] addition formula for ultraspherical polynomials (see also Erdelyi [1953, 10.9 (34)] and Szego [1975, p. 98]) it is natural to look for an expansion of the form
Pn(z; a, aq~, -a, -aq~ Iq) n
=L
Ak,n( 8, ¢) Pk (z; ae i6+ i , ae- i6 - i, ae i6 - i , ae i-i6 Iq),
k=O where Pn(z; a, b, c, dlq) is the Askey-Wilson polynomial. By multiplying both sides by W(Z', ae i6 +i , ae- i6 - i , ae i6 - i , ae i -i6)
and then integrating over
Z
from -1 to 1, show that
(q; q)n (a 4qn , a4q-l, a2q~, _a2q~, _a 2; q)
A
an-Tn Tn - (q; q)Tn(q; q)n-Tn (a 4q-l; q)2Tn (a2q~, -a2q~, -a2; q
t
(8 ¢) Tn,n,
x Pn-Tn (x; aqTn/2, aq(Tn+l)/2, _ aq Tn/2, _a q(Tn+1)/2Iq) X
Pn-Tn
(Y; aqTn/2, aq(Tn+l)/2, _ aqTn/2, _aq(Tn+l)/2Iq) .
(Rahman and Verma [1986b]) 8.12 Prove the inverse of the linearization formula (8.5.1), namely, min(Tn,n)
CTn+n(x; fJlq)
=
L
k=O
b(k, m, n)CTn-k(x; fJlq)Cn-k(x; fJlq),
250
Further applications where
b(k, m, n) = (q; q)m(q; q)n(f3; q)m+n ((3; q)m((3; q)n(q; q)m+n ((3-2 q-m-n,(3-1;q)k (1- (3-2 q2k-m-n) ( 2 _l)k X (q,(3-1 q1-m-n;q)d 1 -(3-2 q -m-n) (3q . 8.13 Give alternate derivations of (8.6.3) and (8.6.4) by using the q-integral representation (7.4.7) of Cn(x; (3lq) and the q-integral formula (2.10.19) for an 8(P7 series. 8.14 By equating the coefficients of ei (m+n-2k)£! on both sides of the linearization formula (8.5.1) show that
(i) (q-m-n, (3-1 q 1-n; q) k (q-n, (3-1 q 1-m-n; q)k q-n, (3-1 q-m-n , q-k, q-m, (3-1 q 1-k, (3-1 q 1-m , (3-1 q 1-n, (3-2 q 1-m-n, (3-2 q 1-m-n, (3-2 q2-m-n, (3-1 q2-m-n qk-m-n: (3-1 q 1+k-m-n : q1-m-n, q -m-n ,
[(3,
X
.q q2. q] " '(3 ,
for k = 0,1, ... , n, and (ii)
x
q-k,
[(3,
(3-1 q 1-k,
q-m , (3 -1 q 1-m ,
(3-1 q -m-n, q-n , (3 -1 q 1-n , (3-2 q1-m-n,
qk-m-n: (3-1 q 1+k-m-n : for k = n, n + 1, ... ,n + m, where is the bibasic series defined in §3.9. (Gasper [1985]) 8.15 Show that, by analytic continuation, it follows from Ex. 8.14 that 4
rP3
[
a,
b, c, d q2] bq/a , cq/a , dq/a ; q, a 2
(a/d, bq/d, cq/d, abc/d; q)oo (q/d, ab/d, acid, bcq/d; q)oo 1
1
1
1
x 12WU(bc/d; (bcq/ad) 2 ,-(bcq/ad)2, q(bc/ad) 2 ,-q(bc/ad)2, ab/d, acid, a, b, c; q, q/a),
Exercises
251
where at least one of a, b, e is of the form q-n, n = 0, 1, .... (Gasper [1985]) 8.16 From Ex. 8.15 deduce that
lOW9 (a; q!, -q!, -q, a 2/b, a 2/e, b, e; q, q/a) _ (aq, a 2/b, a 2/ e, aq/be; q) 00 [a, 2 2 - (a , aq/b, aq/e, a /be; q)oo 4cP3 where one of a, b, c is of the form q-n, n
b, bq/a,
e,
bela
cq/a,
beq/a2
= 0,1, ....
8.17 Prove that
2 2]}2
b2, abz, ab/z . { 4cP3 [ a2, 2b2 a q, -a b,-ab' q q ,q
_ [a 2, b2, ab, abz, ab/z . ] - 5cP4 a 2b2 ,ab! b! b' q, q q2, -a q2,-a and
when the series terminate. (Gasper [1989b]) 8.18 With the notation of Ex. 7.37 prove that
Dq [(q1-veili,q1-ve-ili;q)2vUn(cosB)] when v > - ~ , ). ;::: 0, (Gasper [1989b])
°<
q
<
1, B is real and n
:s:
°
= 1, ... , r.
8.19 Prove the expansion formulas
(i) £ (x'iw) q,
=
(q; q)oow- V (qV;q)oo(_qw 2;q2)oo 00
x
L in(1- qv+n)qn2 /4 J~~n(2w; q)Cm(x; qVlq), n=O
2; q2)oo ~ n(n-2a-1)/4(' )n (ii) £ q (x,.'zw ) -_ ((-w . . 1/2) ~ q ZW -ZW, q 00 n=O (_q(a+!3+ 1)/2; q1/2)n(qa+I3+\ q)n X
X
~----~~~~7-----~-
(qa+!3+1; q)zn q(,6+n+1)/2 _q(a+n+1)/2 1/2.] (,6) 2cP1 [ q(a+,6+2n+2)/2 ; q ,ZW Pnn, (xlq).
(See Ismail and Zhang [1994]) for (i), and Ismail, Rahman and Zhang [1996] for (ii).)
252
Further applications
8.20 Prove the following linearization formula Pm(X; a, q/a, -b, -q/blq)Pn(x; a, q/a, -b, -q/blq) 2min(m,n)
L k=O
with a,b
CkPm+n-k(X; a, q/a, -b, -q/blq),
> O.
(Koelink and Van der Jeugt [1998], after transforming their formula to a symmetric form in m and n) 8.21 Prove the following convolution identity for the Askey-Wilson polynomials: m+n bm - k [m+n]
~
x
k
4¢3
q
(b2;q)n(a2b2e2qm+n-\q)n(e2;qh (e 2 , b2e 2qk-l; q)k(b 2e2q2k; q)m+n-k
q-k b2e2qk-l q-n a2b2qn-l ] [ b< a2b2e2q~+n-l, q-m-n ; q, q -1
-1
xPk ( W2 +2 W2 ; bWl,b/wl,es,e/slq ) Pm+n-k (WI +2WI -1
=Pn ( WI +2WI
;at,a/t,besq k ,beq k /slq )
-1
;at,a/t, bw2,b/W2Iq ) Pm (W2 +2W2 ;abtqn ,abqn /t,es,e/slq ) .
(Koelink and Van der Jeugt [1998]) 8.22 Show that 8 W7
("(iibc/ q; az, a/ z, ,,(a, "(b, "(c; q, qhd) ("(abc, be, bq/ d, eq/ d; q)oo
(abeq / d, q / "(d, qax"( / d, qa"( / xd; q) 00 00
xL n=O
x rn (
(1 - abeq2n /d) (abe/d, ab, ae; q) n(_ad)-n q (n+l) (1 - abc/d) (q, eq/d, bq/d; q)n 2
,,(+,,(-1
2
_ _ _
_)
;a,b,e,q/dlq rn
(z+z-l 2
where a = yiabed/q, ab = ab, ac = ae, ad = ad.
) ;a,b,e,q/dlq,
Exercises
253
(Stokman [2002]) 8.23 Prove the following product formula for the continuous q-Jacobi polynomials: rn(X; a, aq, -c, -cqlq2)rn(Y; a, aq, -c, _cqlq2)
=
/1
K(x, y, z)rn(z; a, aq, -c, -cqlq2)dz,
-1
where x = cos (), y = cos ¢, 0 ::::; (), ¢ ::::;
'if,
0
< q < 1, 0 ::::; a < c < 1, and
x (aei(}+i¢+i'I/J, aeilJ+i¢-i'I/J, aei'I/J-iIJ-i¢, ae-iIJ-i¢-i'I/J; q2)oo sin'ljJ
X/I
h(T; 1, -1, ql/2, _ql/2, -y'aC e i (IJ+¢)/2, _y'aCe- i (IJ+¢)/2)
-1
h(T; vcei'I/J/2, VCe-i'I/J/2, - vcei'I/J/ 2, -VCe-i'I/J/2,
X
dT
VI -
~ei(IJ-¢)/2, ~ei(¢-IJ)/2)
. T2
(Rahman [1986d]) 8.24 Let m, n be non-negative integers with n 2: m. Prove the following linearization formula: rn(x, ql/2, a, -b, _ql/2Iq)rn_m(x; ql/2, a, -b, _ql/2Iq) 2(n-m)
=
L
Ck r m+k(X;ql/2,a,-b,-ql/2Iq),
k=O
where (ab; qhm(ab; qhn-2m(q, _ aql/2, bql/2, -ab; q)n (abq; qhn(q, _ aql/2, bql/2, -ab; q)m( -q, aql/2, _bql/2, ab; q)n-m (1 - abq2m+2k)(abq2m _qm+l _b qm+l/2 ajb a 2b2q2n q2m-2n. q)k " " " qn-m+k/2 a -k (1- ab)(q, -abqm, _ aqm+l/2, b2q2m+l, q2m-2n+l jab, abq2n+l; qh
Ck=~--~~~=-~~~~--~~~~~~~--~=--------
X
X
lO W 9(b 2q2m; b2 , b2q2n+1, q2m-2n+l ja 2 , abq2m+k, abq 2m+k+l, ql-k, q-k; q2, q2).
(Rahman [1981]) 8.25 Let the little q-Legendre function be defined by b'q) P11 (q x'a '"
=
A,
2'/"1
[
q
-11
,
ab 11+1 ] q 'q qx+l aq " ,
v E C,
Further applications
254
for x a nonnegative integer. Derive the following addition formula:
Pv(qZ;I,I;q) VVy(qZ;qX,q) =
VVy(qZ; qX, q) Pv(qY, 1, 1; q) Pv(qx+y; 1, 1; q) (q-V qV+l qX+Y+1. q)k (k+l) '(_I)kqYk+ 2 k=1 (q,q;q)k
+L
'
(Xl
,
x Pv-k(qY; l, qk; q) Pv_k(qX+Y; qk, qk; q) VVY+k(qZ; qX, q) Y +L
(
) (-V v+l ) q; q Y q ,q ; q k (_I)k q(X+Y+l)k-m k=1 (q,q;q)k(q;q)y-k
x PV_k(qY-k;qk,qk;q) Pv_k(qX+Y-\qk,q\q) VVY+k(qZ;qX,q), where VVn(x; a, q) is the Wall polynomial, defined in Notes §7.3. (Rahman and Tariq M. Qazi [1999]) 8.26 The associated Askey- Wilson polynomials the 3-term recurrence relation
r~(x)
where n = 0,1,2, ... , r=:1 (x) = 0, rO'(x) = 1, ter, and
0:
= r~(x; a, b, c, dlq) satisfy
is the association parame-
AA = a- 1(1 - abqA)(1 - acqA)(1 - adqA)(1 - abcdqA-l) (1- abcdq2A-l)(I- abcdq2A) ,
CA =
a(1 - bcqA-l) (1 - bdqA-l) (1 - cdqA-l) (1 - qA) (1 - abcdq2A-2) (1 - abcdq2A-l)
~----~~~--~~~~--~~~~~~
Verify that, with x
r
= cos e,
"'()x - Ln (q-n, abcdq2"'+n-l, abcdq2"'-I, ae
n
x
- k=O
10 VVg
iO , ae- iO ; q)k
(q, abq'" , acq'" , adq"',abcdq",-I; qh
q
k
(abcd q2",+k-2; q"', bcq",-I, bdq",-I, cdq",-I, qk+l, abcd q2",+n+k-l, qk-n; q, a 2).
(Ismail and Rahman [1991]' Rahman [2001]) 8.27 By applying Ex. 2.20 to the
10 VVg
series in Ex. 8.26, show that
Exercises
255
Hence show that <>() (abcdq2<>-1, q<>+\ q)n -n<> r x = q n (q,abcdq<>-l;q)n
x
ill
K(x,y)r n (y;aq<>/2,bq<>/2,cq<>/2,dq<>/2Iq)dy,
where
K (x
,
(q, q, q, abq<>-l, acq<>-l, adq<>-l, bcq<>-l, bdq<>-l, cdq<>-l, q<>; q)oo 4n2(q<>+l, abcdq2<>-2; q)oo x (e 2i , e 2i ; q)oo(1- y2)-1/2 h(y; q<>/2e iO , q<>/2e- iO )
Y)-~~~~~~~-7~~~~~~~--~~-
-
x
r
Jo
(e2i'lj;,e-2i'lj;;q)ooh(cos'l/J;q~eiO,q~ciO) d h(cos'l/J;aq"';;l ,bq"';;l ,cq",;;l ,dq"';;l ,ql/2ei,ql/2e- i
x = cos e,
y = cos cp,
a-I
0:::;
e :::; n,
a-I
0::-1
0::-1
0:::;
max(laq-2 1,lbq-2 1,lcq-2 1,ldq-2 I)
cp :::; n,
0
< q < 1,
< 1, Re(a) > O.
(Rahman [1996b, 2001]) 8.28 The associated q-ultraspherical polynomials satisfy the 3-term recurrence relation 1 - aqn+l 1 - a(32 qn-l 2xC~(x;(3lq)= 1 - a (3 qn C~+l(x;(3lq)+ 1 - a (3 qn C~_1(x;(3lq),n20, with C:'\ (x; (3lq) formula
=
0, Co(x; (3lq)
Cm <> (x', (31 q)Cn<> (x', (3lq)
=
=
1. Prove the following linearization
min(m,n) "A(m,n)C<> (x' (3lq) , ~ k m+n-2k' k=O
where A(m,n) _ (q; q)m(q; q)n(aq; q)m+n-2k(a(3; q)m-k(a(3; q)n-k(a(3; q)k k - (aq; q)m(aq; q)n(a(32; q)m+n-2k(q; q)m-dq; q)n-k(q; q)k (a(32; q)m+n_k(qk-m-n ja; q)k k (1 - a(3qm+n-2k) x a (a(3q; q)m+n_k(qk-m-n; qh (1 - a(3) XlO W.9
(3 ( q k-m-n-l ;q k-m-nj a,
qk-m-n ja(3, a, a(32 jq, qk-m, qk-n, q-k; q, q). (Qazi and Rahman [2003]) 8.29 A multivariable extension of the Askey-Wilson polynomials in (7.5.2) is given by
Pn(xlq)
= Pn(x; a, b, c, d, a2, a3,···, aslq)
256
Further applications k
where Xk
= COS()k, X =
(Xl, ... ,X s ), n
(n1, ... ,ns ), Nk
=
=
L: nj,
j=l
k
Aj,k
II ai, A k+1,k
=
=
1, Ak
=
A 1,k, 1 ~ j ~ k ~ s.
i=j Prove that these polynomials satisfy the orthogonality relation
[11··· [11
Pn(xlq)Pm(xlq)p(xlq)dx1 ... dxs
=
An (q)6n,m s
with max(lql, lal, Ibl, lei, Idl, la21,···, lasl)
< 1, 6n ,m = TI 6nj ,mj' j=l
p(xlq) = p(x; a, b, e, d, a2, a3, ... ,as Iq) = (aei01 ae-i01 bei01 be-i01. q)-l ,
[
x X
S-l
IT
,
,
'cx)
(e2iOk,e-2iOk;q)00(I_x~)-1/2 (ak+1 eiOk+ 1+iO k , ak+1 eiOk + 1-iO k , ak+1 eiOk -iO k+ 1, ak+1 e- iOk + 1-iO k ; q)oo
1
(e 2iOs, e- 2iOa ; q)oo(1- X~)-1/2 (eeiOs, ee-iOs, deiOs, de-iOs; q)oo '
7-~~~~~~~~~~
= An(a, b, e, d, a2, a3,···, aslq)
An(q)
=
(2 )S [ 7r
q) (A2 q2Nk. q) 1 IIs (q ' A2k+1(q qNk+Nk-l-l. , nk k+1 A2 qNk+Nk- 1 a 2 qnk. q) ,00
k=l ' k 'k+1' 00 x (aeA 2,sqNs , adA 2,sqNs , beA 2,sqNa, bdA 2,sqNs ; q)~l,
where ai = ab and a~+l = cd. (Gasper and Rahman [2003b]) 8.30 Prove that
Pn (qk; q, 1; q)
=
r
. .
(qk+l. q)oo 2~ Jo U2n (cos ()) (Hk( cos ()lq))2 (e 2•O, e- 2•O; q)ood().
(Koelink [1996]) 8.31 Let max(lal, Ibl, lei, Idl) < 1, If I < min(lal, Ibl, lei, Idl) and let N be a nonnegative integer such that Ifq-NI < 1 < Ifq-N- 1 1. Prove the biorthogonality relation
1 1
-1
()
()
Rm X Sn X
( V
)
x; a, b, e, d, f dx
=
9o(a,b,e,d,l) , h ( b d I) um,n n a, ,e, ,
for m = 0,1, ... , n = 0,1, ... , N, where v(x; a, b, e, d, I) is the weight function defined in (6.4.3), 9o(a, b, e, d, I) is the normalization constant given by (6.4.1),
h ( b d f) = 27r(1 - abedfq2n-1 ) (abed/q, ab, ae, ad, l/af, bedf /q; q)n n n a, ,e, , ( / q) (q, cd, bd, be, a2 bedf, aq /f;) q , 1 - abed qn
Notes
257
and Rm(x), Sn(x) are the biorthogonal rational functions defined by
Rm(x)
=
Rm(x; a, b, c, d, 1)
(a 2 bcdj,
bc, dj, de i () , de- i (); q)m (ad, dja, abcdje i () , abcdje- i (); q)m d 1jd'; X 10 W.9 (aq -mjd ; q I-mjbd , q I-mj C, if ,a bc j ,ae i() ,ae -i() , q -m ; q, q) , Sn(x) = Sn(x; a, b, c, d, 1) (aqj j, qjaj, de i () , de- i (); q)n (ad, dja, qei () j j, qe- i () j j; q)n X 10 Wg(aq-n jd;
ql-n jbd, ql-n jcd, qjdj, abcj jq, ae i () , ae- i (), q-n; q, q)
with x = cos (). (Rahman [1991])
Notes
§8.5 For additional nonnegativity results for the coefficients in the linearization of the product of orthogonal polynomials and their applications to convolution structures, Banach algebras, multiplier theory, heat and diffusion equations, maximal principles, stochastic processes, etc., see Askey and Gasper [1977], Gasper [1970, 1971, 1972, 1975a,b, 1976, 1983], Gasper and Rahman [1983b], Gasper and Trebels [1977-2000], Ismail and Mulla [1987], Koornwinder and Schwartz [1997], and Rahman [1986d]. §8.6 and 8.7 The nonnegativity of other Poisson kernels and their applications to probability theory and other fields are considered in Beckmann [1973] and Gasper [1973, 1975a,b, 1976, 1977]. A complicated formula for the Poisson kernel for the Askey-Wilson polynomials Pn(x; a, b, c, dlq) in the most general case is given in Rahman and Verma [1991] (a typo is corrected in Rahman and Suslov [1996b]; also see Askey, Rahman and Suslov [1996] for a nonsymmetric extension of this kernel). Rahman and Tariq M. Qazi [1997a] contains a Poisson kernel for the associated q-ultraspherical polynomials. §8.8 and 8.9 A historical summary of related inequalities is given in the survey paper Askey and Gasper [1986]. For additional material related to de Branges' proof of the Bieberbach conjecture, see de Branges [1968, 1985, 1986], and de Branges and Trutt [1978], Duren [1983], Gasper [1986], Koornwinder [1984, 1986], and Milin [1977]. Sums of squares are also used in Gasper [1994] to prove that certain entire functions have only real zeros. §8.10 Additional applications of q-series are given in A.K. Agarwal, Kalnins and Miller [1987], Alder [1969], Andrews, Dyson and Hickerson [1988], Andrews and Forrester [1986], Andrews and Onofri [1984]' Askey [1984a,b, 1988a, 1989a,e, 1992-1996]' Askey, Koornwinder and Schempp [1984]' Askey, Rahman and Suslov [1996], Askey and Suslov [1993a,b], M. N. At akishiyev , N. M. Atakishiyev and Klimyk [2003], Berndt [1988, 1989], Bhatnagar [1998, 1999], Bhatnagar and Milne [1997], Bhatnagar and Schlosser [1998], Bressoud and Goulden [1985, 1987], W. Chu [1998b], Chung, Kalnins and Miller [1999],
258
Further applications
Cohen [1988], Comtet [1974], Frenkel and Thraev [1995], Geronimo [1994], Gessel and Krattenthaler [1997], Gustafson [1989-1994b]' Gustafson and Krattenthaler [1997], Gustafson and Milne [1986], Gustafson and Rakha [2000], Ismail [1990-2003b], Kadell [1987a-1998], Kirillov [1995], Kirillov and Noumi [1999], Kirillov and Reshetikhin [1989], Koelink [1994-2003]' Koelink and Rosengren [2001]' Koelink and Stokman [2001a-2003], Koelink and Swarttouw [1994], Koelink and Van der Jeugt [1998, 1999], Koornwinder [1989-1991a, 19922003], Koornwinder and Swarttouw [1992]' Koornwinder and Touhami [2003], Krattenthaler [1984-2001], Krattenthaler and Rosengren [2003], Krattenthaler and Schlosser [1999], Leininger and Milne [1999a,b], Lilly and Milne [1993], Milne [1980a-2002], Milne and Bhatnagar [1998], Milne and Lilly [1992, 1995], Milne and Newcomb [1996], Milne and Schlosser [2002]' Rahman and Suslov [1996b], Rota and Goldman [1969], Rota and Mullin [1970], and Stokman [2002-2003c]. §8.11 Mordell [1917] considered the representation of numbers as the sum of 2r squares. Milne [1996, 2002] derived many infinite families of explicit exact formulas for sums of squares. Another proof of (8.11.1) is given in Hirschhorn [1985]. Also see Cooper [2001], Cooper and Lam [2002]' and Liu [2001]. Ex. 8.3 For more on classical biorthogonal rational functions, see Rahman and Suslov [1993] and Ismail and Rahman [1996]. Ex.8.11 A second addition formula for continuous q-ultraspherical polynomials is given by Koornwinder [2003]. Addition formulas for q-Bessel functions are given in Rahman [1988c] and Swarttouw [1992]. For other addition formulas, see Floris [1999], Koelink [1994, 1995a, 1997], Koelink and Swarttouw [1995], and Van Assche and Koornwinder [1991]. Ex.8.19 A short elementary proof of the formula in (i) was found by Ismail and Stanton [2000], and is reproduced in Suslov [2003]. Other proofs of (ii) are given in Ismail, Rahman and Stanton [1999] and Suslov [2003, §4.8]. Ex. 8.20 The expression given for Ok is symmetric in m and n, but is equal to the one given in Koelink and Van der Jeugt [1998]. Ex. 8.29 This is a q-analogue of the orthogonality relation for the multivariable Wilson polynomials in Tratnik [1989]. Also see the multivariable orthogonal or biorthogonal systems in van Diejen [1996, 1997a, 1999], van Diejen and Stokman [1998, 1999], Gasper and Rahman [2003a,b,c], Rosengren [1999, 2001b], Stokman [1997a,b, 2000, 2001], and Tratnik [1991a,b].
9 LINEAR AND BILINEAR GENERATING FUNCTIONS FOR BASIC ORTHOGONAL POLYNOMIALS
9.1 Introduction
Suppose that a function F(x, t) has a (formal) power series expansion in t of the form (Xl
L fn(x)tn.
F(x, t) =
(9.1.1)
n=O
Then F(x, t) is called a generating function for the functions fn(x). A useful extension of (9.1.1) is the bilinear generating function (Xl
H(x, y, t)
=
L anfn(x)gn(y)tn .
(9.1.2)
n=O
We saw some examples of generating functions in Chapters 7 and 8. Some important linear generating functions for the classical orthogonal polynomials are listed in Koekoek and Swarttouw [1998]. In this chapter we restrict ourselves entirely to generating and bilinear generating functions for basic hypergeometric orthogonal polynomials. Of the many uses of generating functions the one that is most commonly applied is Darboux's method to find the asymptotic properties of the corresponding orthogonal polynomials which, in turn, are essential to determining their orthogonality measures. Darboux's method, as described in Ismail and Wilson [1982]' is as follows: If f(t) = 2::=0 fntn and g(t) = 2::=0 gntn are analytic in It I < rand f(t) - g(t) is continuous in It I :::; r, then fn = gn + O(r-n). In a slight generalization of this theorem Ismail and Wilson state, further, that if f(t) and g(t) depend on parameters al, ... , am and f(t) - g(t) is a continuous function of t, al, ... , am for It I :::; r and al, ... , am restricted to compact sets, then the conclusion holds uniformly with respect to the parameters. As far as the bilinear generating functions are concerned one of the most useful is the Poisson kernel K t (x, y) defined in (8.6.1). A related kernel, the so-called Christoffel-Darboux kernel, is defined for orthonormal polynomials Pn (x) by
~
()
() _ ~Pn+1(x)Pn(Y) - Pn(X)Pn+l(Y) ' x- y
~~x~y-k k=O n+l
(9.1.3)
where k n is the coefficient of xn in Pn(x). (9.1.3) is a fundamental identity in the theory of general orthogonal polynomials, see Szeg6 [1975], Chihara [1978], Dunkl and Xu [2001], Nevai [1979], and Temme [1996]. Some evaluations of Poisson kernels were given in Chapter 8. Here we shall derive some important 259
260
Linear and bilinear generating functions
bilinear generating functions for basic orthogonal polynomials and give some significant applications.
9.2 The little q-Jacohi polynomials For the little q-Jacobi polynomials, defined in (7.3.1), we shall first prove that
L 00
(
)
aq;q ntnqG)Pn(x;a,b;q) n=O (q;q)n =
(-t; q)oo (-xt; q)oo
[-xt, bxq ] -t, 0, 0 ; q, -aqt .
2¢3
(9.2.1)
Let G(x, t) denote the sum of the series in (9.2.1). Using (1.4.5) and (7.3.1) we have
Pn(x;a,b;q)
=
(~-\~)n(_xtq-G) aq;q n
2¢I(q-n, bxq; xqI-n; q, aqn+I),
(9.2.2)
which, when substituted into the left hand side of (9.2.1), gives
ft =f
G(x, t) =
n=O k=O
(bxq; qh(x-\ q)n-k (-xtt(aqn /x)k (q; q)k(q; q)n-k
f
(bxq; qh (_at)k qk2 (x-\ q)n (-xtqkt k=O (q; qh n=O (q; q)n
= ~ (bxq; qh (_tqk; q)oo (_at)k qk2
t='o
(q; qh (-xtqk; q)oo
'
(9.2.3)
from which (9.2.1) follows immediately. Note that while the q -+ 1- limit of
qa+ q) n (n) (1 - x ) L (. ' ) n t q Pn -2-;qa,qf3;q n=O q, q n 00
(
1.
2
is L~=o p;:,f3(x)t n , which has the value 2a+f3 R- I (l - t + R)-a(1 + t + R)-f3 with R = (1 - 2xt + t 2)!, see Szego [1975, (4.4.5)], the last expression on the right side of (9.2.3) does not give this limit directly. Nevertheless, it can be used to derive an asymptotic formula for Pn(x; a, b; q) by applying Darboux's method. Observe that, as a function of t, the pole of G(x, t) that is closest to the origin is at -x-I, the next one at -(qx)-I, the next at _(q2 x )-I, and so on. Clearly, (1
lim
+ xtqk)G(x, t)
t->-(xqk)-l
=
(bxq; qh (x-I; q)oo (a/x)k, (q; qh (q; q)oo
and so a suitable comparison function for asymptotic purposes is
f
(bxq; qh (x-\ q)oo (a/x)k k=O (q;q)k (q;q)oo l+xtqk
= (x-\ q)oo (q;q)oo
f
==0
(abq=+\ q)oo (-xt)=. (aq=/x;q)oo
(9.2.4)
9.2 The little q-Jacobi polynomials
261
Combining (9.2.4) and the left side of (9.2.1) we find, by Darboux's theorem, as extended by Ismail and Wilson [1982]' that
(x-I. q) (n) Pn(x;a,b;q) "-' ( ') 00 (_x)n q- 2 , aq; q 00
x
#
0,I,q,q2, ... ,
uniformly for x, a and b in compact sets. Now we shall reconsider the Poisson kernel for the little q-Jacobi polynomials given in Ex. 8.3, which has the right positivity properties but suffers from the flaw that it does not directly lead to the known q ----+ 1- limit:
f
n=O
=
(2n + a + (3 + 1) (a + (3 + 1 )nn! t n p(a,(3) (cos 2(})p(a,(3) (cos 2c/J) (a + (3 + 1)(a + l)n((3 + l)n n n
(1
1- t
+ t)a+(3+2 F4
[a +
(3 + 2 2 '
a+
(3 + 3 2 ;a
+ 1, (3 + 1;
a2
b2 ]
/1,2' /1,2
'
(9.2.5)
Hi!
where a = sin(}sinc/J, b = cos () cos c/J, /1, = +c~), 0:::; (},c/J:::; 7[, 0 < t < 1, see Bailey [1935, p. 102]. First, use the product formula Ex. 8.1(i) to obtain the following expression
1 - abq2n+l (abq, aq; q)n 1_ b (b .) (t/aq)n Pn(x; a, b; q) Pn(Y; a, b; q) n=O a q, q, q n
=L 00
Kt(x, y)
L L
min(x,y) (
00
=
b 2) (-X -y) a q ;q 2r+2s q ,q str+sa-Sq(x+y)(r+s)-s2-2rs (q,aq;q)r(q,bq;q)s
r=O
s=O
x
1 - abq2(n+r+s)+1 (abq2r+2s+\ q)s (~) -t n
00
~
1- abq2(r+s)+1
(q; q)n
q
(
).
(9.2.6)
What is needed to compute now is a series of the form
f
n=O
1 - aq2n (a: q)n qG) (_t)n == ht(a), say. I-a (q,q)n
(9.2.7)
Without the qG) factor the computation is trivial via the q-binomial theorem (1.3.2). But with this factor the best one can do seems to be to consider this as the b ----+ (Xl limit of the very-well-poised series 4 W3(a;
b; q, t/b)
which, by (3.4.8), transforms to
(t,aq 2t 2;q)00 SW7 ( aqt; ()1 2 ,-qa 1 2 ,bqt;q,t / b) aq 2,_ ()1 aq 2,qa 1 (qt2, atq 2) ; q 00 =
(1 _ t) (-qta~, -:q~ a~, t(aq)~ /~, tqa~ /b; q)oo (-tq'2, -tq, t/b, atq'2 /b; q)oo 1
X
1
1
1
1
1
1)
SW7 ( atq'2/b;tq'2,(aq)'2,qa'2,-a'2/b,-(aq)'2/b;q,tq'2 , (9.2.8)
262
Linear and bilinear generating functions
from (2.10.1). Since 0
< t < 1, it follows by letting b ----;
ht(a) = (1 _ t)
that
00
(-tqa~ 'l-tq~ d; q)oo (-tq'l, -tq; q)oo
,/-. X 3'1'2
[(aq)~,qa~,tq~.q tq~] 1 3 1" -qta'l,-tq'la'l
(9.2.9)
•
So, if we replace a and b in (9.2.6) by qa and q(3, respectively, and use (1.6) and (1.25), then Kt(x, y) of (9.2.6) can be written in a more suggestive form
Kt(x,y)
=
(1- t) 1
1
(-tq'l; q'l )a+(3+2 00
min(x,y) (q~(a+(3+2), q~(a+(3+3), _q~(a+(3+2), _q~(a+(3+3);
(_ tq~(a+(3+3) "_tq~(a+(3+4). q) r+s
xL L r=O s=O x
q)r+s
(q-X, q-Y; q)s (t qx+y)r+s q-s 2-2rs-2s ~~~~~~~~~~----
(q, qa+l; q)r(q, q(3+1; q)s qr+s+~ (a+(3+2) qr+s+~ (a+(3+3) tq~ [ _tqr+s+~(a+(3+3), _tqr+s+~(a+(3+4) ; q, tq2 .
1]
X 3¢2
(9.2.10)
It is obvious that the q ----; 1- limit of (9.2.10) is indeed (9.2.5) and that the expression on the right side of (9.2.10) is nonnegative when 0 < t < 1 and a,j3 > -1.
9.3 A generating function for Askey-Wilson polynomials There are many different generating functions for the Askey-Wilson polynomials Tn(X; a, b, e, dlq) defined in (8.4.4), of which G t (x; a, b, e, d I) q
(abed/q; =~ ~ (.) q)n t nTn (X; a, b, e, d I) q, n=O
q, q n
It I < 1,
(9.3.1)
is one of the simplest. By Ex. 7.34,
Gt(x;a,b,e,dlq)
= D-1(B) x
3¢2
r
e- iO / d (dueiO,due- iO , abedu/:; q)oo JqeiO /d (dau/q, dbu/q, deu/q, q)oo
q , be, q/u ] [ abed/ ad, abedu/q ; q, adut/q dqu,
(9.3.2)
where D(B) is defined in Ex. 7.34. However, by (3.4.1),
,/-. [abed/q,be,q/u. adut/] 3'1'2 ad, abedu/q ,q, q
= (abedt/q;q)oo (t; q)oo
¢ [(abCd/q)~, -(abed/q) ~, (abed) ~, -(abed) ~,adu/q .
x
5 4
ad, abedu/q, abedt/q, q/t
(abed/q, adt, abcdut/q, adu/q; q)oo (ad, abedu/q, adut/q, l/t; q)oo
+~~~~~--~~--~~~
] ,q, q
9.3 A generating function for Askey-Wilson polynomials
'"
[
X 5'1'4
263
] t(abcdjq) "21 , -t(abcdjq) "21 , t(abcd) "21 , -t(abcd) "21 ,adutjq . j 2j ,q, q . adt,abcdut q,abcdt q,qt (9.3.3)
Substituting this into (9.3.2), using (2.10.18) and simplifying the coefficients, we find that
Gt (x,. a, b,c, dl) q
=
(abcdtjq; q)oo () t; q 00
¢ [(abcdjq)~, -(abcdjq) ~, (abcd) ~, -(abcd) ~ ,aei (}, ae- i (} . ] 5 ab, ac, ad, abcdtjq, qjt ,q, q
x
6
+
(abcdj q, abt, act, adt, ae i (} , ae- i(); q)oo (ab, ac, ad, ate i (} , ate- iO , t- I ; q)oo
x
6'1'5
'" [t(abcdjq)~, -t(abcdjq)~, t(abcd)~, -t(abcd)~, ate iO , ate- iO . ] abt, act, adt, abcdt 2 jq, qt ,q, q . (9.3.4)
The generating function given in Ex. 7.34 is a special case of (9.3.4). A more difficult problem is to evaluate the sum of the series
Ht(x;a,b,c,dlq) =
f
n=O
((a~q))ntn Tn(x;a,b,c,dlq), q, q
(9.3.5)
n
where a is an arbitrary parameter. For a = ab this would, in particular, give a q-analogue of the generating function of Jacobi polynomials given in the previous section. Note that
Ht(x)
== Ht(x; a, b, c, dlq) ~ (abcdjaq; q)m
(a; q)oo
= (a bcdj q,.) q
~
00
m=O
(.) q, q m
a
m
( I ) ( ) Gtqrn x; a, b, c, d q. 9.3.6
So, use of (9.3.4) in (9.3.6) gives
Ht(x)
=
(a, abcdtjq; q)oo (t, abcdjq; q)oo X
'" 6'1'5
f m=O
(t, abcdjaq; q)m am (q, abcdtjq; q)m
[(abcdjq)~, -(abcdjq)~, (abcd)~, -(abcd)~, ae iO , ae- iO . a b,ac, a d ,abcdt qm-I , q I-mjt
+
(a, abt, act, adt, ae iO , ae- iO ; q)oo (ab, ac, ad, ate iO , ate- iO , t- I ; q)oo
x
~--~~~~~--~~~~
x
6'1'5
f
m=O
] ,q, q
(abcdjaq; q)m (q; q)m
. . (m+I) (ate,O,ate-,O;q)mq 2 (-at)m (abt, act, adt, qt; q)m
'" [tqm(abcdjq)~, -tqm(abcdjq)~, tqm(abcd)~, -tqm(abcd)~, abtqm, actqm, adtqm, tqm+ 1 , (9.3.7)
Linear and bilinear generating functions
264
Since, by (1.4.5) 2¢1(abed/aq, tq-n; abedtqn-\ q, aqn) =
(atqn, abedqn-l; q)oo (-n / n-l n) ( bd 1) 2¢1 q ,abed aq; abedq ; q, atq , (9.3.8) aqn, a e tqn- ; q 00
the first term on the right side of (9.3.7) becomes
f
(a, abed/ aq, ae iO , ae- iO ; q)nqG) (-at)n (t; q)oo n=O (q, ab, ae, ad, at, q/t; q)n
(at; q)oo
(9.3.9) Interchange of the order of summation followed by the use of (1.4.5) and simplification in the second term of (9.3.7) gives
(a, at 2 , abedt/q, abt, act, adt, ae iO , ae- iO ; q)oo (ab, ae, ad, at, abedt 2/q, t-I, ate iO , ate- iO ; q)oo x
f
(abed/aq, r\ q)n (at2t (q,abedt/q;q)n
n=O rI.
x 7'1'6
[at, t( abed/ q)! , -t( abed/ q)! , t( abed)! , -t( abed)! , ate iO , ate- iO . ] 2 1 1 ,q, q . abt, act, adt, at ,abedtqn- ,tq -n (9.3.10)
In the special case b = aq!, d = eq!, e --+ -e and a = ab = a 2q!, both 7¢6 series above become balanced 4¢3 's, which along with their coefficients, can be combined via (2.10.10). Denoting the sum of (9.3.9) and (9.3.10) in this combination by Gt(x; a, elq), we get Gt(x; a, elq) (a 2t a2tq! a 2ete iO a 2ete- iO . q) (aetq! , a 3 etq! ,ateiO , ate- iO ; q)oo ,
X
,
,
,(Xl
ae, aeq !2 ,e2 q - !2, ae iO ,ae -iO ; q ) n (a 3 etq!2 ; q) 2n (2 2 !)n 1 1 . 1. a t q2 n=O (q, a 2t, a 2tq2, a 2etq2 e'O, a 2etq2 e-'O; q)n(a 2e 2; qhn 00 "'"'
(
L...J
x 8 W7 ( a 3 etq2n-!; atq! /e, aeqn, aeqn+!, aqneiO, aqne-iO; q, act). (9.3.11) Unfortunately, neither the 8 W 7 series inside nor the outside series over n can be summed exactly except in the limit q --+ 1-. If we replace a and e by q!(o+!) and qH6+!), respectively, in (9.3.11), then lim G t (x;q!(o+!),q!(l3+!)lq)
q-->l -
9.4 A bilinear sum for the Askey-Wilson polynomials I
= (1 _ t)i3+ 1R-a-i3-1 ~ ~
n=O
265
((3)n [2t2 (1 - x)] n n! R2
x F [n+~(a+(3+1),n+~(a+(3+2).2t(1-X)] ( ) 2 1 a + (3 + 1 + 2n ' R 2 · 9.3.12 By the summation formula Erdelyi [1953 Vol. 1. 2.8(6)] we can evaluate the above 2F1 series, and then do the summation over n in (9.3.12) by the binomial theorem (1.3.1) to obtain the well-known generating function Szego [1975, (4.4.5)] for the Jacobi polynomials.
9.4 A bilinear sum for the Askey-Wilson polynomials I
We shall now compute the sum
F(x, ylq) :=
~ (abed)~: !~~~/g; q)n (abed/ fg)nknTn(X)Tn(Y),
(9.4.1)
where Tn(X) is the Askey-Wilson polynomial given in (8.4.4),
k n = K,(a, b, e, dlq)
(111 [Tn(xWw(x)dx) -1,
(9.4.2)
and f and 9 are arbitrary parameters such that the series in (9.4.1) has a convergent sum. By Ex. 7.34,
Tn( cos B)
= B-1 (B) lqe-i8/b qei8/b
Tn(COS¢)
= C- 1(¢) l
(buei!i, bue-i!i, abedu/q; q)oo(ed, q/u; q)n (bau/q, beu/q, bdu/q; q)oo(ab, abedu/q; q)n x (abu/q)ndqu, (9.4.3)
qe - i1> /e ( i¢ . ) 00 (bd ,q /v,q . )n eve ,eve -i¢ ,abedv / q,q qei1>/e (eav/q, ebv/q, edv/q; q)oo(ae, abedv/q; q)n x (aev/q)ndqv, (9.4.4)
where
B(B) = -
iq(l - q) 2b (q,ae,ad,ed;q)ooh(cosB;b)w(cosB;a,b,e,dlq),
(9.4.5)
iq(l - q) (q,ab,ad,bd;q)ooh(cos¢;e)w(cos¢;a,b,c,dlq). 2e
(9.4.6)
and
C(¢)
=-
Hence
F(x, ylq)
=
B-1(B)C-1 (¢) lqe-i8/b (buei!i, bue-i!i, abcdu/q; q)oo qeie /b (bau/q, beu/q, bdu/q; q)oo i qe - 1> /e (eve i¢, eve- i¢, abedv/q; q)oo l X qei1>/e (eav/q,ebv/q,cdv/q;q)oo x 8 W7(abcd/q; ad, f, g, q/u, q/v; q, ab2e2duv/ fgq2)d q ud q v. (9.4.7)
266
Linear and bilinear generating functions
However, by (2.10.10) the above
SW7
equals
(abed, be/ j, be/ g, abed/ j g; q)oo (be, abed/ j, abed/ g, be/ jg; q)oo
x
4¢3
ad,j,g,abeduv/q2 ] [ abedu/q, abedv/q, qjg/be; q, q
(abed, ad, j, g, abeduv/q2, ab 2e2du/ jgq, ab 2e2dv/ jgq; q)oo (be, abed/j,abed/g, abedu/q, abedv/q, ab 2e2duv/jgq2, jg/be; q)oo
+~~--~~~~--~~----~~~--~~~~=-~-
be/j,be/g,abed/jg,ab2e2duv/jgq2
]
x 4¢3 [ ab2e2du/ jgq, ab2e2dv/ jgq, beq/ jg ; q, q ,
(9.4.8)
which, when substituted into (9.4.7), leads to the sum of two terms, say, F Fl + F 2 , where F1(x, ylq)
=
=
(abed, be/ j, be/g, abed/ jg; q)oo B-l(B)C-1(¢) (be,abed/j,abed/g,be/jg;q)oo ~ (ad, j, g; q)n n q n=O (q, qjg/be; q)n
x~
l l
qe - W /b (bue i() , bue-i(} , abeduqn-l; q)oo
qeiO /b (bau/q, beu/q, bdu/q; q)oo i qe - ¢/e (eve i , eve- i , abedvqn-l, abeduvq-2; q)oo x qei¢ /e ( eav/ q, ebv / q, edv / q, abeduvq n-2) ; q 00 dqv dqu, X
(9.4.9) and F 2(x,ylq)
=
(abed,ad,j,g;q)oo B-l(B)C- 1(¢) (be, abed/ j, abed/ g, jg/be; q)oo ~ (be/ j, be/g, abed/ jg; q)n n q n=O (q, beq/ jg; q)n
x~
X
X
l l
qe - iO /b (bue i() , bue- i() , ab 2e2duqn-l / j g; q)oo
qeiO /b (bau/ q, beu/ q, bdu/ q; q)oo qe - i¢/e (eve i , eve- i , ab 2e2dvqn-l / jg, abeduv/q2; q)oo qei¢/e (eav/q, ebv/q, edv/q, ab2e2duvqn-2 / jg; q)oo
x dqv dqu
(9.4.10)
with x = cos B, y = cos ¢. The q-integral over v on the right side of (9.4.9) can be expressed as a terminating S¢7 series via (2.10.19), which can then be transformed to a balanced 4¢3 series that can be transformed back into a different S¢7 series. The final expression for this q-integral turns out to be C(¢) (dei, de- i , bdu/q; q)n (ad, bd, d/a; q)n X S W 7 (aq-n /d; ae i , ae- i , ql-n /bd, q-n, abu/q; q, q2 / adu). (9.4.11)
9.4 A bilinear sum for the Askey-Wilson polynomials I
267
The q-integral over u in (9.4.9) then has the form
l
qe -
iB
qe iB /b
/b
(bue iO , bue- iO , abeduqn-\ q)oo d (bauqk-l, beu/q, bduqn-k-l; q)oo qU,
which, by (2.10.18), gives
B(()) (ae iO , ae- iO ; q)k(de iO , de- iO ; q)n-k. (ad; q)n(ae; qh(ed; q)n-k
(9.4.12)
Substitution of (9.4.11) and (9.4.12) into (9.4.9) then gives F ( 1 X,
I) Yq
(abed, be/ f, be/ g, abed/ f g; q)oo
= (be, abed/ f, abed/g, be/ fg; q)oo x
L (f "9 de iO ,de- iO ,dei ,de-i.,q)n qn 00
n=O
x
10
(q, ad, bd, ed, d/a, qfg/be; q)n
W.9 (aq-n/d· ae iO , ae- iO , ae i , ae- i , ql-n/bd , q1-n/ ed " q-n. q'ad beq ) . \ '
(9.4.13) The q-integral over v in (9.4.10) equals
C ¢ (abdue- i /q, ab 2 edq n e- i / fg; q)oo ( ) (ab 2 eduqn-1 e -2 / fg, abde- 2; q)oo x 8 W 7 (abde- i /q; ae- i , be- i , de- i , q/u, fgq-n /be; q, ab 2 eduqn- 1ei / fg), (9.4.14) which is a bit more troublesome than the previous case because the 8 W 7 series is nonterminating unless fg/be is of the form q-k, k = 0,1, ... , which cannot be the case because of the factor (be/fg;q)oo in the denominator of F 1(x,ylq) and of the factor (fg/be; q)oo in the denominator of F 2 (x, ylq). So either we split up this 8 W 7 series into a pair of balanced 4¢3 series via (2.10.10) and get bogged down in a long and tedious computation, or seek an alternative shorter method. In fact, by (6.3.9) the expression in (9.4.14) can be written as 2 C(rI.) (q ae i , be i , de i ' ab cqn '1-', fg'
b2 cdqn abcdqn fg , fg'
abu/q , bdu/q , adu/q·, q) 00
assuming, for the time being, that (9.4.16)
268
Linear and bilinear generating functions
If we make the further assumption that ad = be, then the q-integral over u reduces to
j
qe- iO /b
qe iO /b
(bue iO , bue- iO , ab 2c2duqn-1 II g; q)oodqu 2 1 (ab eduqn- ei Ilg, q-1 (abd) !uei'I/J-!i, q-1(abd)!ue- i'I/J-!i; q)oo'
which sums to
(9.4.17)
x h(z; eiO-!i(adlb)!, e-iO-!i(adlb)!) . So the q-integral in u over the expression in (9.4.14) equals
B( 8)C( ¢)( q, ae i , bei , ee- i , de i ; q)oo 1(ae iO , ee iO , de iO ; q)oo 12 2n(ab,ae, ad, ad, bd, cd; q)oo (abedqn Ilg, ab 2eqn Ig, b2edqn Ilg; q)oo x (abedqnei+iO Ilg, abedqnei-iO Ilg; q)oo x
h(z;l,-l,q!,-q!)
1
/
-1
h (z; (abl d)! d i , (adlb)! e!i, (bdl a)! e!i
h(z; (abd)!e- i Ilg) h(z; (adlb)! eiO-!i, (adlb)! e- iO - !i, be( abd)! qne- !i I 1g) dz x v'f"=Z2' (9.4.18)
x
with ad = be. Since this integral is balanced we have, by (6.4.11), as its value the sum of multiples of two balanced very-well-poised 10 Wg series. Combining this result with (9.4.13), we then find the following bilinear summation formula
00
;
1 _ b2e2q2n-1
(b 2e2q-1, ab, ae, I, g; q)n ( b2e2 ) n 1 - b2e2q-1 (q, be 2a- 1, b2ea- 1, b2e21- 1, b2e2g- 1; q)n Iga 2
x T n (x;a,b,e,bca- 1Iq) T n (y;a,b,e,bea- 1Iq) _ (b 2e2, bel I, bel g, b2e2II g; q)oo (I, g; q)n 1(bca- 1eiO , bea- 1ei; q)n 12 n - (be, b2e2I I, b2e2I g, bell g; q)oo n=O (q, be, be2a- 1, b2ea- 1, bea- 2, 1gqlbe; q)n q
f
2 -n
X
+
10
W. (~. ae iO ae- iO ae i ae- i ab- 1e- 2q1-n ae- 1b- 2q1-n q-n. q q) 9
be'
,
,
"
(b 2e2,J, g, b2e2Ilg, b2e3 I al g; q)oo 2 1 2 (ae, b ea- , be a-1, be, cia, b2e2I I, b2e2lg, Iglbe, ab3 e2Ilg; q)oo
1(ee iO , bea- 1ei , b3 e2eiO I 1g, ab2e2ei I 1g; q) 001 2 X
,
~--------------------------~~~
1(b2e2eiHi Ilg, b2e2eiO-i Ilg; q) 00 12
"
,
9.5 A bilinear sum for the Askey-Wilson polynomials II
x
f:
(bel f, belg, ab3e 2I fg; q)
J
(b2e2eiHi¢ I fg, b2e2eilJ-i¢ I fg; q) ~12 qn
(q, b2c 3lafg, beql fg; q)
n=O
J
(b 3e2e ilJ I fg, ab2e 2e i¢ I fg; q)
10
b e qn ab eqn. . . . ) ., b3e3qn-1 w.9 ( ab3e2qn-1 fg fg '---yg' ---yg,ae,IJ ,ae -,IJ ,be'¢, be -'¢.,q,q (b 2e2 , f, g, b2e 2If g, ab 2el f g; q)oo
+
(ab, ae, be, b2ea- 1 , ale, b2e 2If, b2e2lg, fglbe, x
J
2
2 2
x
269
~3J;; q)
00
I(ae ilJ , bei ¢, b3e3e iIJ lafg, b2e3ei¢ I fg; q) 001 2
Xf: (bel f, belg, b3e4/afg; q) J (b2e2eiHi¢ lafg, b2e2eilJ-i¢ I fg; qf J2 qn (q, abe2 fg, beql fg; q)
n=O x
10
w.9 (
J
(b 3e3lafge ilJ , b2e3e i ¢ I fg; q)nl
b3e4qn-1 b2e2qn b3e 3qn-1 b2e3qn . .. . b '¢I a, bee -'¢/' ) f '-f-' f '-f-,ee ,IJ ,ee -,IJ ,ee a,q,q. ag 9 9 ag
.
(9.4.19) Note that by analytic continuation we may now remove the restrictions in (9.4.16) and require only that no zero factors appear in the denominators on the right side of (9.4.19). Note also that we have tacitly assumed that a, b, e and the product f 9 are real, although it is not necessary. An application of this formula is given in Rahman [1999]. For a more complicated formula without the restriction ad = be, see Rahman [2000b].
9.5 A bilinear sum for the Askey-Wilson polynomials II
Let {wdk=O be an arbitrary complex sequence such that Suppose
2::%"=OIWkl <
00.
00
G 1 (x, ylq) :=
L vnknTn(X)Tn(Y),
(9.5.1)
n=O
where Tn(X)
=
Tn(X; a, b, c, dlq),
_ (be,ad;q)n( d)n (n)~(adqn,adqn;q)j Vn - ( ) -a q 2 ~ ( 2 ) wn+j, abed; q 2n j=O q, abedq n; q j
(9.5.2)
and k n is the normalization constant given in (9.4.2). We shall prove that
The proof is actually quite simple. Use (8.3.4) to write
Linear and bilinear generating functions
270
Tn(X)Tn(Y) (bd, cd; q)n (~)n (ae,ab;q)n d
=
t
(q-n, abedqn-\ qh (q, ad; q)k
k=O i i de () de-i() de x ( ' , , de-i.,q)k qk (ad,bd,ed,dja;q)k x 10 Wg(aq-k jd; q1-k jbd, q1-k jed, q-k, ae i() , ae- i (), ae i , ae- i ; q, beqjad). (9.5.4) Observe that
f
1 - abedq2n-1 (abedq-l, ad; q)n (ad)-n . (be, ad; q)n (-ad)nqG) n=O 1 - abedq-1 (q, be; q)n (abed; qhn X
=
(q-n,abeqn-\q)k 00 (adqn,adqn;q)j ~..,...:.....,,----=:-----,--:....::..:..c.. ' " W +. (ad, ad; qh ~ (q, abedq2n; q)j n J
L 00
n=O
1
b d 2k+2n-1 ( b d -1 ) -a e q a e q ;q 2k+n (-lt q G)-k 1 - abedq-1 (q; q)n(ad, ad; qh
~ (ad, ad; q)k+n+j x j=O ~ ( q; q)j (abed; q) 2n+2k+j Wn+k+j
_ (abed; qhk -k ~ (ad, ad; q)mwm W ( b d 2k-1. k-m. m-k) - (d d ) q ~ ( ) (b d) 4 3 a e q ,q ,q, q a ,a ; q k m=k q; q m-k a e ; q m+k =
wkq-k,
by (2.3.4), which immediately gives (9.5.3).
9.6 A bilinear sum for the Askey-Wilson polynomials III
We shall now consider another bilinear sum, namely, 00
G 2(X, ylq)
=L
7r n k nTn(X)Tn(Y),
(9.6.1)
n=O where 7r
n
= (be, adj f; q)n (ad, bef; q)n
r~
rrk
~ (q, befqn, f q1-n jad; qh'
(9.6.2)
f i=- 0 being an arbitrary complex parameter and {rrdk=O an arbitrary complex sequence such that ~~=o Irrkl < 00. Changing the order of summation in (9.6.1), we can write it as 00
G 2(x, ylq)
=
L (q, be f ,qrr; j a,d. q). j=O
J
9.6 A bilinear sum for the Askey-Wilson polynomials III
x
~ (be, adq-j I j; q)n (j j)nk ~
(d b j j.) n=O a, e q ,q n
q
nrn
() X
() rn Y .
271
(9.6.3)
In the case ad = be, which is what we shall assume to be true, the sum over n in (9.6.3) is the special case of the left side of (9.4.19) with j and 9 replaced by be and beq-j I j, respectively. Note that in this special case the first term on the right side of (9.4.19) vanishes while the other two double series become single series. Thus,
(ae, be, be2a- l , b2ea- l , ea- I , bej, abj; q)oo (je iH , je iO - i ; q)0012 x (abj, bej; q)j 1(je iHi ,jeiO - i ; q)j 12 (j, ej la; q)j 1(aje i , bje iO ; q)j 12 1
. I . I . ajqj '0 '0 . . ) XIOW9 ( abjql- ;bcjql- ,jql,--,ae z ,ae- z ,bez ,be- z ;q,q e + (b 2e2, j, aj Ie; q)oo (ae iO , be i , beja-Iei()., ejei ; q)00.12 (ab, ae, be, b2ea- l , ae- I , bej, be2 ja-I; q)oo (je zHz , jez(}-z; q)0012 x (bej, be2ja-\ q)j 1(je iHi , je iO - i ; q)j 12 (j, aj I e; q)j 1(beja-Ie z(), eje z; q)j 12 1
1
I' I . I . ejqj 2 x lOW9 ( be ja- ql- ;bejql- ,jql,-a-'
eei () " ee- iO bea-Iei , bea-Ie- i " . q q) . (9.6.4) Hence,
(ae, be, be 2a- l , b2ea- l , cia, bej, abj; q)oo (je iHi , je iO - i ; q)0012 x 00 (abj; q)j 1(je iHi , je iO - i ; q)j 120j (q, j, ej la, qj lac; q)jl(bje iO , aje i; q)jl2 1
~
x
+
bejqj-\ jqj, ajqj Ie, ae iO , ae- i (} , be i , be- i ; q, q) (b 2e2, j, aj I c; q)oo 1(ae iO , be i , beja-IeiO , ejei ; q)oo 12 (ab, ae, be, b2ea-1, ale, bej, be2 ja-I; q)oo (je zHz , jez(}-z; q)0012
10 Wg(abjqj-\
1
x
x
00
~
(bc 2ja-\ q)j (je iHi , jei(}~i; q)jl20j (q, j, qj lac, aj Ie; q)jl(beja-Ie zO , eje z; q)jl2
10 W g (bc 2ja-Iqj-\
1
bejqj-\ jqj,
e~qj ,eeiO , ee- i , bca-Iei , bca-Ie- i; q, q). (9.6.5)
Linear and bilinear generating functions
272
Exercises 9.1 Show that
~ (cq; q)n p, ( b )n ~ ( b') n x; a, ,c; q t n=O q, q, q n
= 2(/J1(aq/x, 0; aq;q, qxt) l¢l(bx/c;bq;q,cqt), when Pn(x; a, b, c; q) is the big q-Jacobi polynomial defined in (7.3.10). Prove also that n ( b ) Tn x;a, ,c;q
rv
(aq/x, cq/x; q)oo n ( ) x aq, cq; q 00
as n ----t 00, for fixed x i=- 0, aqm+l, cqm+l, m a, band c in compact sets. (Ismail and Wilson [1982])
=
0,1, ... , uniformly for x,
9.2 Prove that
Pn(qm; a, b; q) as n ----t 00, m 9.3 Show that
=
rv
q(n-;m) (-aqt-m(bqm+\ q)oo/(aq; q)oo
0,1, ... , uniformly for a and b in compact sets.
(i)
(ii)
9.4 For the q-Hahn polynomials defined in (7.2.21) derive the following generating functions:
(i)
Exercises
273
(See Koekoek and Swarttouw [1998]) 9.5 For the dual q-Hahn polynomials defined in (7.2.23), prove that N
(-N
)
(i)
L n=O
(ii)
~ (q-N,bq;q)n R ( ( ) b N )tn ~ ( -1 -N.) n J.l X; ,e, ; q n=O q, e q ,q n
q( . ;)q n Rn(J.l(x);b,e,N;q)t n q, q n = 2 (PI (q-X, e- 1 q-X; bq; q, betqx+1 )(tq-N; q)N-x,
= 2¢1 (qx-N, bqx+\ e- 1 q-N; q, tq-X) (beqt; q)x. (See Koekoek and Swarttouw [1998])
9.6 Al-Salam-Chihara polynomials Qn(x; a, blq) are the e = d = 0 special case of the Askey-Wilson polynomials defined in (7.5.2). Prove that
~ Qn(x; a, blq)t n
=
(at, bt; q)oo , (te 2 (},te- 2 (};q)00
(i)
f::o
(ii)
~ (e; q)n Q ( bl) n ~ ( b.) n x; a, q t n=O q, a ,q n
(q;q)n
_ (at,et/a;q)oo - (i(} te ,te -i(}.) ,q 00
A,
3'1'2
[ab/e, ae iO , ae- iO . t/] ,q, e a , ab,at
where x = cosO, let/al < 1, It I < l. (See AI-Salam and Chihara [1976], Askey and Ismail [1984]' and Koekoek and Swarttouw [1998].) 9.7 Using the recurrence relation of 00
n=O
Deduce that
(Bustoz and Ismail [1982])
C~(x;fJlq)
in Ex. 8.28 show that
274
Linear and bilinear generating functions
9.8 Prove that
where
IAI < 1, It I < 1.
Using Ex. 7.35 show that
00
L p~a,t3)(xlq)tn n=O (q, q(a+t3+2)/2, q(a-t3 )/2, qt3; q) 00 (q(2a+3)/4 eiO, q2a+3)/4 e -iO; q1/2) 00
27r( qa+1, qa+t3+1; q)
1
h(Y; 1, -1, q1/2, _q1/2, tq(5a+t3+5)/4)
1
X
-1
00
h (Y; q(a+t3+1)/4, q(a+t3+3)/4, q(a-t3)/4eiO, q(a-t3)/4e-iO, tq(a+t3+1)/4)
x 3CP2 [
qa+1 tq(3a-t3+3)/4ei tq(3a-t3+3)/4e-i ] dy tq(5a+t3+5)/4 ei, tq(5a+t3+5)/4 e -i ; q, qt3 -Y--=l=-==y=2'
where y = cos cP, provided 0 < (3 < a. 9.9 Show that
f n=O =
Pn(x; a, b, c, dlq) tn (q,ab,cd;q)n
2CP1 (aj z, bj z; ab; q, zt) 2CP1 (cz, dz; cd; q, tj z),
where x = ~(z+z-1), Itzl < 1, formula (7.5.13). (Ismail and Wilson [1982]) 9.10 (i) Prove that, for a(3
=
Itz- 1 1<
1, and deduce the asymptotic
ab,
(a 2t, ateiHi j a, ate iO - i j a, atei-iO j a, ate- iO - i j a; q) 00
x sW7(a 2tjq; ae iO , ae- iO , ae i , ae- i , atj(3; q, btja),
Ibtjal
< 1.
(Askey, Rahman and Suslov [1996]; Koelink [1995a]) (ii) Show that the continuous q-Hermite polynomial defined in Ex. 1.28 is related to the Al-Salam-Chihara polynomial by
Exercises
275
(iii) Using (i), (ii) and the summation formula Ex. 2.17(ii) deduce the qMehler's formula tn
00
~ Hn(xlq)Hn(Ylq) (q; q)n (t 2 ; q)oo ( teiO+ i , teiO-i , tei-iO , te- iO - i ., q) 00 ,
It I < 1,
where x = cos (), Y = cos ¢. (See Carlitz [1957a]) 9.11 Use Ex. 7.33(ii) to prove that n
00
LL
1
m+ 1
1 -=- ~n+;
q~nCm(COS (); q~ Iq)Cm(cos ¢; q~ Iq)
n=Om=O (qei(O+
2
(q~ ei(O+
(Ismail and Rahman [2002b]) 9.12 Show that
f
n=O
(~;';~~)) '
Cn(cos();(3lq)Cn(cos¢;(3lq)t n
,q n
((3, ,,(, t 2 , (3te i (O+-O) , (3te- i (O+-O) , te-i(O+
x
00 ~
~
((32/"1I , tei(O+-O) , te-i(O+
n
(q, (3te i (O+-O) , (3te- i (O+
x 8 W 7 ((3t 2q2n-\ (3, tqnei(O+-O) , tqne-i(O+
7r.
276
Linear and bilinear generating functions
x (-
iq(1-v-n)/2 e i(()+¢) , _iq(1-v-n)/2 e -i(()+¢); q) n
X (iq(1+3V+n)/2 e i(O-¢) , iq(1+3v+n)/2 e i(¢-O); q) 00
(iq(1+v+ n )/2 e i(O-¢) , iq(1+v+ n )/2 e i(¢-O); q) 00 X 8W 7 ( -
qV; qV, iq(1+v+n)/2 e i(O+¢) , iq(l+v+n)/2e- i (O+¢) ,
iq(1-v-n)/2 e i(¢-O) , iq(1-v-n)/2 e i(O-¢); q, qV).
(Ismail, Masson and Suslov [1997]) 9.14 Show that
where Re v
> 0, 0 < b < a, and 0 ::::; 'Ij! ::::; 7f.
(Rahman [1988c]) 9.15 If k = 0,1,2, ... , and af3 = ab, prove that
where 0 ::::;
e : : ; Jr and 0 < ¢ < Jr.
(Ismail and Stanton [1997])
Exercises
9.16 For k
=
277
0,1,2, ... , show that 00
n=O
e
where 0 :::; ¢ :::; 7r and 0 < < (Ismail and Stanton [1997])
7r.
9.17 (i) Use (9.5.1)-(9.5.3) to prove that
t
n=O
(q-N, O!(3"(Oq~ -1; q)n (1 - abcdq2n-~)1 (abcdq~l, ab, ac, .-ab; q)n (0!(3, -,,(0, q)n (1 - abcdq )(q, bd, q)n(abcd, qhn
x (nt1) dan q ( I )
4¢3
[qn-N,0!(3"(oqn+N-1,ab qn ,-cdqn . ] abcdq2n, O!(3qn, _"(Oqn , q, q
x Tn(X; a, b, c, dlq) Tn(Y; a, b, c, dlq) =
(-ab,,,(o:q)N (_l)N (ab, -,,(0, q)N
t
m=O
(q-N,O!(3,,(oqN-l, -ab;q)ml(deil:/,~ei1>;q)mI2qm (q, ad, ad, bd, -0!(3, ,,(0, dla, q)m
x lOWg(aq-mld;q 1 -mlbd,q 1 -mlcd,q-m,ae'"1:/"1:/ ,ae-' ,ae'"1> ,ae-'"1> ;q, bcq ad)' (ii) From (i) deduce the projection formula m
p:::,b)(x;q) = Lg(n,m)p~o:,f3)(x;q), n=O where p:::,b) (x; q) is the continuous q-Jacobi polynomial defined in (7.5.25), and
(Rahman [1985])
278
Linear and bilinear generating functions
9.18 In (9.5.2) and (9.5.3) take ad = be and Wn =
(ex,(3,"(,o;q)n (be, be, beql f, ex(3"(of Ib 3e3; q)n'
--,----------=-------=--'--------"-'----,---0----=----,-----
Also, in (9.6.2) and (9.6.3) take ad = be and
an
=
(exJlbe,(3J1be,,,(Jlbe,ofl be;q)n n q (ex(3"(oj2 Ib 4 e4 ; q)n .
Choosing appropriate multiples of (9.5.1) and (9.6.1), show that
f
(1 - b2e2q2n-l) (b 2e2q-l, ab, ae, ex, (3, ,,(, 0; q)n (ex(3"( f Ibe; qhn -0 (1 - b2e2q-l) (q be2a- 1 b2ea- 1 f3'Yf a'Yf af3f af3'Y8f. q) (b 2e2. q)2 n'" be ' be ' be ' b3 e 3 , n ,n b2e2qn Of) (f)n ( ex(3"( fbeq2n-l ; ex(3"(f X SW7 b3e3 '-o-,ex qn ,(3qn,,,(qn;q, be a2 x rn(x;a,b,e,bea-1Iq) rn(y;a,b,e,bea-1Iq) (exf Ibe, (3 f Ibe, "( f Ibe, ex(3"( f Ibe; q)oo (ex(3f Ibe, ex"(f Ibe, (3"(f Ibe, f Ibe; q)oo 00
x
~
(ex, (3, ,,(, 0; q)nl(bea-1eil:l, bea-1e i¢; q)nl 2qn (q, be, be, b2ea-1, be2a-1, bea- 2, ex(3"(of Ib 3e3, beql f; q)n
X 10 W g ( a 2q-n Ibe;
aql-n Ib 2e, aql-n Ibe 2, q-n, aeil:l, ae-il:l, ae i¢, ae- i¢; q, q)
(ex, (3, ,,(, 0, ex(3"( f Ibe, ex(3"(oP Ib 4 e4 , f, af Ie; q)oo + (ex(3 f Ibe, ex"( f Ibe, (3"( f Ibe, of Ibe, ab, ae, be, b2el a; q)oo x (aeil:l, bei¢, befeil:l la, efe i¢; q)0012 . 2 (ale, be f I a, bel f, ex(3"(o f Ib 3e3; q)oo 1(Je'CO+¢), fe,CI:I-¢); q)oo 12 X (exJlbe, (3f Ibe, ,,(JIbe, of Ibe, be2f la; q)n 1(JeiCO+¢) ,jeiCO+¢); q)nl 2qn n=O (q, f, qf Ibe, af Ie, ex(3"(oj2 Ib 4 c4; q)n (efe i¢, befeil:l fa; q)nl 2 1
f
1
x
10 W g(be 2fqn-l
fa; fqn, befqn-l, efqn la, eeil:l, ee-il:l, beei¢la, bee- i¢ la; q, q)
(ex, (3, ,,(, 0, ex(3"( f Ibe, f, ef I a, ex(3"(oP Ib 4 e4 ; q)oo (ex(3f Ibe, ex"( f Ibe, (3"( f Ibe, of Ibe, ae, be, b2ea- 1, be2a- 1; q)oo 1(eeil:l, beei¢ I a, bfeil:l, afe i ¢; q)oo 12 x (cia, abf, bel f, ex(3"(o f Ib 3e3; q)oo 1(JeiCO+¢), feiCI:I-¢); q)oo 12
+--,-------,-----~-~--,-------,-----~~--,-----~---,-----~~~~~~--
~ (exf Ibe, (3f Ibe, "( f Ibe, of Ibe, abf; q)n (JeiCO+¢), feiCI:I-¢); q)nl 2qn 1
x
~ (q, f, qJlbe, eJla, ex(3"(oj2 Ib 4 e4 ; q)n
x
10 Wg(abfqn-\
I(afe i¢, bfeil:l; q)nl 2
fqn, befqn-l, afqn Ie, aeil:l, ae-il:l, bei¢, be- i¢; q, q).
(Rahman [1985]) 9.19 Show that the special case ex = be = -,,(, (3 = beq~ = -0, f = t of the formula in Ex. 9.18 gives the Poisson kernel for the special Askey-Wilson
Exercises
279
polynomials Tn(X; a, b, e, bca-1Iq):
(1 - b2 e2 q2n-1)(b2 e2 q -1 ,ab,ae,q .) n (tla 2)n 2 2 2 1 2 1 (1 - b e q-1 )(q, be a- , b ca- ; q)n
00
"
f='o
x Tn(X; a, b, e, bea-1Iq)(Tn(Y; a, b, e, bea-1Iq))
= (1 _ t 2 ) (beqt; q)oo
f=
(tlbe; q)oo n=O X 10 W g(a 2q-n Ibe;
+
(q, be 2a-1, b2ea-1, be, bea- 2 , bcqt, beqlt; q)n
aq1-n Ib 2e, aq1-n Ibe 2 , q-n, aei(i, ae-i(i, aeiq" a-iq,; q, q)
2 2 q) 12 (t 'dtle " b e . q) 00 I(aei(i , beiq, , cteiq, 'bea-1tei(i. . '.00
1 2 1 2 1 belt· q)> I(te'(O+q,) , te,(q,-(i)·,ex> q) 12 ( ab 'ae " be ae- , b ea- , be a- t " CX 00
x
(beq~, -be, -beq~; q)nl(bea-1ei(i, bea-1eiq,; q)nl 2qn
~
(-t, tq~, -tq~, be2a- 1t; q)nl (tei((iH) , tei((i-q,); q)nl 2 n (q, qtlbe, qt 2 , atle; q)nl(eteiq" tbea-1ei(i; q)nl 2 q
x lOWg(bc2tqn-1 la; tqn, betqn-l, etqn la, beeiq, la, bee-iq, la, eei(i, ee-i(i; q, q)
+
(t ctla b2e2.q) I(eei(i bca-1eiq, ateiq, btei(i.q) 12 '" 00, ' . ' . ' 00 (ab, be, b2 ca- 1, be 2a- 1, abt, belt; q)oo I(te'(O+q,) , te,((i-q,); q)oo 12
x
f='o
~ (-t, tq~ , -tq~ ,abt; q)n I(tei((iH) , tei(q,-(i); q)n 12 n (q, qtlbe, qt 2etla; q)nl(ateiq" btei(i; q)nl 2 q
X 10 W
g( abtqn-1; tqn, betqn-1, atqn Ie, aei(i, ae-i(i, beiq" be-iq,; q, q),
where 0 < t < l. (Rahman [1985]; Gasper and Rahman [1986]) 9.20 Prove that 00
L
q-n 2(Pl (stei(i, ste-i(i; S2; q, -q n r 2h¢1 (sei(i It, se-i(i It; S2; q, _qn)
n=-oo
(q, q, _q 2t 2I s2, -s2 Iqt 2; q) 00
(1 _ qt2) (1 _ 2'j,t cos 0 +
x
q;;2) (_qt2, _t-2, -1, -1, -q, -q, S2, S2, S-2, qs-2; q)oo
h( cos 0; st, -st, qst, sit, -qtl s) 1
1
h(cosO; q'2, -q'2, q, -q, qtls)
x 8 W 7 (qt 2; q, stei(i, ste-i(i, qtei(i Is, qte-i(i Is; q, q), where It I < 1, 0 ~ 0 ~ n, and appropriate analytic continuations of the 2 ¢1 series are used as necessary. (Koelink and Stokman [2003])
9.21 The continuous q-Hermite polynomial Hn(cosOlq) in base q-1 becomes
280
Linear and bilinear generating functions Show that
(i)
(ii)
(Ismail and Masson [1994])
10J,81 < 1, prove that
9.22 (i) For
C' (
'-q
0.) _ (q, a 2 1,82; q)=(q,82; q2)= cos ,a (2. 2) qa ,q = x ~ (e 2i ,p,e- 2i ,p;q)= Eq(cos7jJ;,8)d7jJ.
r
2n
io
h(cosnl0/,.. Q!.eilJ Q!.e-ilJ) (3 , (3
(ii) Prove that
Eq (cos 0; a )Eq(cos 0; ,8)
= ( ,8-1 (1-n)/2. ) L -a t) ,q n ,8nqn2 /4cn (cos 0; I'lq) q ,q = n=O 1', q n _,8a- 1q(n+1)/2, -1',8a- 1q(n+1)/2 . 2] 2CP1 [ n+1 ,q, a .
( 2. 2)
= ( ~;.q 2)= X
I'q
(See Bustoz and Suslov [1998] for (i), and Ismail and Stanton [2000, (5.8)] for an equivalent form of (ii).) 9.23 Prove that
L
= (abedq-1" ae ad" a' b' a' e' a'd'· q) (bb' edt·,q)2 n (_ t) n " n n=O (q, cd, be't, b'et, bd't, b'dt; q)n(abedq-1; qhn aa'
x 8 W7 (bb' edtq2n-1; beqn, bdqn, b' e' qn, b' d' qn, btl a'; q, a'tlb)
x rn(x; a, b, e, dlq) rn(Y; a', b', e', d'lq) (bb' edt, dt I e', btei<jJ, bte-i<jJ, ctei<jJ, cte-i<jJ, b'tei () , b' te- i(}, e'tei (} , e'te- i(}; q)= (be't , b' et , bd't , b'dt , bb't " edt te i (9+<jJ) , tei(IJ-<jJ) , te i (<jJ-IJ) , te- i (9+<jJ)., q) 00 X 8W 7
(bb'tlq; be ilJ , be- ilJ , b' ei<jJ, b' e-i<jJ, btl a'; q, a'tlb)
x 8 W 7 (ee'tlq; ee ilJ , ee- i (} , e' ei<jJ, e' e-i<jJ, e'tld; q, d'tle) , where ab = a'b', cd = e'd', x = cosO, and Y = coscp. (Koelink and Van der Jeugt [1999])
Notes
281
Notes
§9.3 The special case a = ab, b = aq!, d = cq!, c ----; -c of (9.3.7), in which case the expression on the right side of (9.3.7) can be combined into a sum of SW7 series, was given by Ismail, Masson and Suslov [1997]. However, formula (3.6) in their paper does not directly lead to the generating function
f: p~a,(3)(x)tn when one sets a
=
qa/2+1/4, c
= qf3/2+1/4
and takes the limit
n=O
q ----; 1-.
JS
2 ) (x; q), Ex.9.14 A product formula for Jackson's q-Bessel function which is obtainable from this formula, is used in Rahman [2000a] to evaluate a Weber-Schafheitlin type integral for these functions. Ex.9.18 This is a q-analogue of Feldheim's [1941] bilinear generating function for the Jacobi polynomials. Ex.9.20 Rosengren [2003e] computed a more general bilinear generating function in an elementary manner. Bustoz and Suslov [1998] gave a bilateral bilinear sum that generalizes the classical Poisson kernel for the Fourier series:
f
n=-oo
tlnlein(x-y) -
1 - t2 1- 2tcos(x - y)
+ t2 '
0::; t
<
l.
Ex. 9.21 The continuous q-Hermite polynomials in base q-l were introduced by Askey [1989b] who also gave an orthogonality measure for those polynomials on (-00,00) and computed the orthogonality relation. Ismail and Masson [1994] gave a detailed account of the family of extremal measures for these polynomials. See also Carlitz [1963b], Ismail and Masson [1993], and Ismail [1993].
10 q-SERIES IN TWO OR MORE VARIABLES
10.1 Introduction
The main objective of this chapter is to consider q-analogues of Appell's four well-known functions F l , F2, F3 and F4 . We start out with Jackson's [1942] (l), (2) , (3) and (4) functions, defined in terms of double hypergeometric series, which are q-analogues of the Appell functions. It turns out that not all of Jackson's q-Appell functions have the properties that enable them to have transformation and reduction formulas analogous to those for the Appell functions. Also, starting with a q-analogue of the function on one side of a hypergeometric transformation of a reduction formula may lead to a different q-analogue of the formula than starting with a q-analogue of the function on the other side of the formula. We find, further, that the alternative approach of using the q-integral representations of these q-Appell functions can be very fruitful. For example, it immediately leads to the fact that a general (1) series is indeed equal to a multiple of a 3¢>2 series (see (10.3.4) below). The q-integral approach can be used to derive q-analogues of the Appell functions that are quite different from the ones given by Jackson. In the last section we give a completely different q-analogue of F l , based on the so-called q-quadratic lattice, which has a representation in terms of an Askey-Wilson type integral. We do not attempt to consider Askey-Wilson type q-analogues of F2 and F3 because these are probably the least interesting of the four Appell functions and nothing seems to be known about these analogues. Instances of AskeyWilson type q-analogues of F4 have already occurred in Chapter 8 (product of two q-Jacobi polynomials) and then in Chapter 9 (§9.5, §9.6, Ex. 9.18, and Ex. 9.19). Since in this chapter we will be mainly concerned with deriving and applying q-integral representations of q-Appell functions, in many of the formulas it will be necessary to denote the parameters by powers of q. Once a formula has been derived via the q-integral techniques, if it does not contain any q-integrals or q-gamma functions, then the reader may simplify the formula by replacing the qa, qb, etc. powers of q by a, b, etc., as we did in the formulas in the exercises for this chapter.
10.2 q-Appell and other basic double hypergeometric series In order to obtain q-analogues of the four Appell double hypergeometric series (see Appell and Kampe de Feriet [1926])
282
10.2 q-Appell and other basic double hypergeometric series
283
00
F1(a; b, b'; c; x, y)
L
=
(10.2.1)
m,n=O D ( b b' , ) E2a;, ;C,c;x,y
D
E3
(
=
' . b b'.. )_ a, a, , ,C, x, y -
' ) F4 (b a, ; c, C ; x, y
=
~ (a)m+n(b)m(b')n
~ m,n= O
m n "() (') x y, m.n. C m C n
~ (a)m(a')n(b)m(b')n "( ) m. n. C m+n
~
m,n=O
~ (a)m+n(b)m+n
"() (') x
~ m.n. C m C n m,n= O
m
m n x y ,
n y ,
(10.2.2)
(10.2.3)
(10.2.4)
Jackson [1942] replaced each shifted factorial by a corresponding q-shifted factorial giving the functions
;r,.(ll( 'b b'· . . '±'
)-
a " , c, q, x, y -
cI>(2 l (a;b,b';c,c';q;x,y) ;r,.(3l( '±'
;r,.(4l( '±'
~ (a;q)m+n(b;q)m(b';q)n m n ~ ( ) ( ) () x y , m,n=O q; q m q; q n c; q m+n =
'. b b'· . . )a, a, , ,C, q, x, y -
a"
b'
'..
)_
c, C ,q, x, y -
f
((a;)q)(+nt~q))(b~;q)n)
m,n=O q; q
q; q n c; q
m
m
c'; q n
(1025) ..
xmyn, (10.2.6)
~ (a; q)m(a'; q)n(b; q)m(b'; q)n ~
m,n=O
(
)
q; q
(
m
) ()
q; q n c; q m+n
m n (1027) x y, ..
~ (a; q)m+n(b; q)m+n m n ~ (. ) (. ) ( . ) (" ) x y . m,n=O q,qmq,qnc,qmc,qn
(10.2.8)
The series in (10.2.5)-(10.2.8) are absolutely convergent when Iql, lxi, Iyl < 1, by the comparison test with the series n-O xmyn, and then their sums are called the q-Appell functions. Jackson also-considered q-analogues of the
z=:
Appell series with yn replaced by ynqG) in each term of the series. Similarly, one could obtain a q-analogue of the general double hypergeometric series
(10.2.9) where we use the contracted notations defined in §3.7 by replacing each shifted factorial by a q-shifted factorial, such as in Exton [1977, p. 36] and Srivastava [1982, p. 278]. However, such a basic double series might not be of the same form as its confluent limit cases or as the double series obtained by inverting the base as in (1.2.24) or in Ex. 1.4(i). Therefore, with the same motivation as given on p. 5 for our definition of the reps series hypergeometric series, we
284
q-series in two or more variables
define the q-analogue of (10.2.9) by
;r,.A:B·C [aA : bB; Cc ] '±'D'E"F dD: eE; I F ;q;x,y ., =
f
(aA; q)m+n(bB; q)m(CC; q)n m,n=O (dD; q)m+n(q, eE; q)m(q, IF; q)n X
D-A [(_I)mq (m)] I+E-B [(-ltq (n)] I+F-C xmyn, [(_I)m+nq (m+n)] 2 2 2 (10.2.10)
where q -=I=- 0 when min(D - A, 1 + E - B, 1 + F - C) < O. This double series converges absolutely for lxi, Iyl < 1 when min(D-A, I+E-B, I+F-C) ::::: 0 and Iql < 1. Also, confluent limit cases of (10.2.10) have the same form as the series in (10.2.10), and if one of the parameters b1 , ... ,bB , Cl, ... Cc equals 1, then the double series in (10.2.10) reduces to a single series of the form in (1.2.22). Since the series in (10.2.9) is called a Kampe de Feriet series when B = C and E = F, the series in (10.2.10) can be called a q-Kampe de Feriet series when B = C and E = F. 10.3 An integral representation for (1) (qa; qb, qb'; qC; q; x, y)
Since, by (1.11.7) and (1.10.13)
(q::q)m+n (q ,q)m+n for 0
r t a+m+n1
rq(c)
=
10
r q(a)r q(c - a)
1
(~~a~)oo
(tq, q)oo
dqt,
(10.3.1)
< Re a < Re c, we find that (I)(qa. qb qb'.qc. q. x y) , , , '"
rq(c) rq(a)rq(c-a)
=
1
b
1
X
b'
a-I (qt, xtq ,ytq ; q)oo d ( ) ( c-a) qt, 10.3.2 o xt, yt, tq ; q 00 t
which was given by Jackson [1942, p. 81]. This is a q-analogue of the wellknown integral representation of F 1 :
Fl(a; b, b' ; C; x, y) =
r(a)~~; _ a)
11
t a- 1 (1 - t)c-a-l(1 - xt)-b(1 - yt)-b' dt,
(10.3.3) Re a < Re c. It follows from (10.3.2) and the definition of Jackson's q-integral (1.11.3) that
o<
(I)(qa. qb qb'.qc. q. x y) = (qa ' xqb , yq~'q) ,00 q ,X,y,q 00 ,
,
,
"
,
(
C
.)
A.
3 '/'2
[ qc-a "y'q X qa ] xqb , yqb' " . (10.3.4)
Andrews [1972] gave an independent derivation of (10.3.4). Some reduction formulas, analogous to the ones for Fl listed, for example, in Erdelyi [1953, Vol. I], can be derived from (10.3.4).
10.3 An integral representation for q,(1)(qa;qb,qb';qc;q;x,y)
285
Case I. Let y = xqb. Then (10.3.4) and (1.4.1) give q,(1)(qa; qb, qb'; qC; q; x, xl) _ (qa,xqb+b';q)oo (c .) q,x,qoo
-
_
-
2cPl
[qc-a,~. 2cPl
a] xqb+b, q, q
[ qa, qb+b' . ] qC , q, X ,
(10.3.5)
which is a q-analogue of the reduction formula
F1(a; b, b'; C; x, x) Case II. Let y
=
=
2Fl(a, b + b'; c; x).
(10.3.6)
qc-a-b'. Then (10.3.4) and (1.4.1) give
(10.3.7)
(10.3.8)
Case III. Let c = b + b'. The 3cP2 series of (10.3.4) in this case becomes a series of type II, and hence by (3.2.7) qb+b'-a X y ] 3cP2 [ xqb, y~b,' ; q, qa _ (qb+b',yqa-b;q)oo [qb+b'-a,qb,x qb / y . a-b] (a xqb , qb+b' ,q,yq . q ,yqb'.) ,q 00 3cP2
(10.3.9) provided Iyqa-bl
< 1, which is a q-analogue of the reduction formula
F1(a; b, b'; b + b'; x, y)
=
(1 - x)-b(l - y)b- a2F1 (b, b + b' - a; b + b';
i =:). (10.3.10)
286
q-series in two or more variables
Integral representation. (2) (qa;
qb,
Since it follows from (10.2.6) that
l' ;qC, qC' ; q; x, y)
="
OO(ab) q, q ; q m xm A, (qa+m qb'. qC' . q y) ~ (q,q;q c) m 20/1 "'" m=O
and, by (1.11.9), a+m 2¢1(q
when 0
b'. c'. _ ,q ,q ,q,y)-
r q(c') r q(b')r q('-b') c
(10.4.1)
11 ( 0
qu, uyq a+m., q ) 00 b'-l (uy, uq c'-b'.) dqu, ,q 00 u (10.4.2)
< Re b' < Re c', we find that
(2) (qa;
qb,
l'; qC, qC'; q; x, y)
rq(C') r 1 (qu,uyqa;q)oo [qa,qb,o ] b'-l =rq(b')rq(c'-b')io (uy,uqC'-b';q)oo3¢2 qC,uyqa;q,x u dqu,
(10.4.3) provided 0
< Re b' < Re c'. By (1.11.7)
(qb;q)n= rq(c) r1vb+n-1 (vq;q)oo dv (qC;q)n rq(b)rq(c-b)io (vqc-b;q)oo q,
(10.4.4)
when 0 < Re b < Re c. Thus, by combining (10.4.3) and (10.4.4) we obtain the following integral representation: (2) (qa;
l, qb'; qC, qC'; q; x, y)
= B-1(b,c- b)B-1(b',c' _ b') r 1 r 1 ub'-lvb-1 (qU,~v,~yqa;q)oo q q io io (uy,uqc-b,vqc-b;q)oo
x 2¢1(qa,0;uyqa;q,xv)dqudqv, when 0
(10.4.5)
< Re b < Re c and 0 < Re b' < Re c', which is a q-analogue of
F2(a; b, b'; c, c'; x, y) = B- 1(b, c - b)B-1(b', c' - b') X
fa1 fa1 Ub'-l Vb-1(1_ u)c'-b'-l(l_ v)c-b-l(l_ uy - vx)-adudv,
(10.4.6) SInce
Transformation formulas. tion formulas for F2 note that
To derive q-analogues of the transforma-
+ ' b' , + b' " (yqa+m; q)oo 2¢1(qa +m,q b ;qC ;q,y) = ( ) 2¢2(qa m,qc - ;qC ,yqa m;q,yq ) y;q 00 (10.4.7)
287 by (1.5.4), and hence (10.4.1) and (10.4.7) give (2) (qa;
l, l'; qC, qC'; q; x, y)
ff
(qa;q)m+n(qb;q)m(qc'-b';q)n Xm(_ )n nb'+(~) (y; q)oo m=On=O (yqa; q)m+n(q, qC; q)m(q, qC'; q)n y q ,
= (yqa;q)oo
(10.4.8) which is a q-analogue of Erdelyi [1953, Vol. I, 5.11 (7)]. Similarly, we get a q-analogue of Erdelyi [1953, Vol. 1., 5.11 (6)] by interchanging x +-+ y, b +-+ b', c+-+c'.
by (II1.2) and (II1.7) , and hence, with a bit of simplification, we find that (2) (qa;
qb, qb'; qC, qa; q; x, y)
= ( yq b') ;q
LL
(b) ) (b' q;q m+n (qa-b' ,y;q,,:, q ;q )nxm+nqmb'. (y; q)oo m=On=O (q; q)m(q; q)n(qC, yqb ; q)m+n 00
00
00
(10.4.10) This can be written in the notation of (10.2.10) in the form (2) (qa;
=
qb, l'; qC, qa; q; x, y)
(yqb'; q)oo 2:2;1 [qb, 0 : qa-b', y; qb' b'] , ·q·xq x (y; q)oo 2:1;0 qC, yqb : 0; " , ,
(10.4.11)
which is a q-analogue of Bailey [1935,9.5(6)]. If we specialize further by setting c = c' = a, then we can use the transformation
(10.4.12) to get (2) (qa;
l, qb'; qa, qa; q; x, y)
= (yqb';q)oo
f ff
f
(qb;q)m m (q-m,qb';q)n (_ )n (a+m)n+G) (y; q)oo m=O (q; q)m X n=O (q, qa, yqb'; q)n y q
= (yqb'; q)oo (y;q)oo
m=On=O
(qb; q)m+n(q~'; q)n(xy)nxm qn(n-1)+an (q,qa,yqb ;q)n(q;q)m
288
q-series in two or more variables 00 (b+n _ ( yq b' ; q) 0 000" (b q, qb' ; q) n ( )n n(n-l)+an" q ; q )m m - (y;q ) 00 n=O ~ (q,qa,yq b') ~ (q;q )m x ;q n xy q m=O
b yq b' ; q) 0 000" (b _ ( xq, q, qb' ; q)n (xy ) n n(n-l)+an (x,y;q )00 n=O ~(a b)n q q,q ,yqb' ,xq;q
-
=
(xqb,yqb';q)oo (x,y;q)oo
2¢3
[qb,qb' a] qa,xqb,yqb' ;q,q xy ,
(10.4.13)
which is a q-analogue of Bailey [1935,9.5(7)]. Finally, let c = b. Then (10.2.6) gives (2) (qa;
=
l, l'; l, qC'; q; x, y)
L (aq ,q,b' ; q)n L
(a+n ) q ; q m xm n=O (q,qC ;q)n m=O (q;q)m 00
00
(xqa;q)oo
= (x; q)oo
3¢2
[qa,qb',o ] qC', xqa ; q, y ,
(10.4.14)
which is a q-analogue of Bailey [1935,9.5(3)].
From (10.2.7) it follows that (3) (qa,
qa'; l, l'; qC; q; x, y)
(a b) L 1'~ ;~ m Xm2¢1 (qa', l'; qc+m; q, y). m=O q, q ; q m 00
=
(10.5.1)
Using
r ub+m- 1 (uqc-b+n;q)oo (qu; q)oo d u= q 1
f q(b)f q(c - b) (qb; q)m(qc-b; q)n
Jo for 0
fq(c)
(qC;q)m+n'
(10.5.2)
< Re b < Re c, and
r
1
b'+n-l
Jo v provided 0
(qv;q)oo d _ fq(b')fq(c-b-b') (qb';q)n (vqc-b-b';q)oo qVfq(c-b) (qc-b;q)n'
(10.5.3)
< Re b' < Re (c-b), we obtain the following integral representation
(3) (qa,
qa'; qb, l'; qC; q; x, y)
rr
1 1 b-l b'-l fq(c) (qu,qv,xuqa;q)oo =fq(b)fq(b')fq(c-b-b')Jo Jo u v (uqc-b,vqc-b-b',xu;q)oo
x 2¢1(qa', uqc-b;0;q,vy)dq udq v.
(10.5.4)
Since (10.5.5) (10.5.4) is a q-analogue of Bailey [1935, 9.3(3)].
289 The function <[>(3) (qa, qa'; qb, qb'; qC; q; x, y) does not seem to have any useful transformation formulas, just as F3(a, a'; b, b'; c; x, y) does not, as was noted by Bailey [1935, §9.3]. Reduction formulas. Suppose a + a' = c. Then <[>(3) (qa,
qc-a; l, l'; qC; q; x, y)
_ ~ (qa,qb;q)m m ~ ( c ) X m=O q,q;q m
-
b'
'" (c-a 2'1'1 q
L L
b'. c+m. ) ,q, q ,y b
b'
(qa; q)m+n(q ; q)m(q ; q)n m( _ t n(c-a)+G) (y; q)oo m=O n=O (qC; q)m+n(q; q)m(q, yqb'; q)n X y q ,
= (yq ; q)oo
00
00
(10.5.6) by (1.5.4), which is a q-analogue of Bailey [1935, 9.5(4)]
F1 (a; b, b', c; x, y)
=
(1 - y)-b' F3 (a, c - a; b, b'; c; x, -y-). y-1
(10.5.7)
When a+a' = b+b' = c, the right hand side of (10.5.6) can be reduced to two 4¢3 series in the following way. Using
f
(t; q)m xm = (xt; q)oo , m=O (q; q)m (x; q)oo
it follows that <[>(3) (qa,
qc-a; qb, qC-b; qC; q; x, y) q-b( qa, qb, qc-a, qC-b; q)oo
The limit of the right side of (10.5.8) as q ---+ 1- is
(I-x)
-a
(l-y)
b-C{ r(c)r(b - a) ( 1 ) r(b)r(c_a)2 F 1 a,c-b;l+a-b;(l_x)(l_y)
290
q-series in two or more variables
r(c)r(a - b) a-b ( 1 )} +r(a)r(c-a) ((l-x)(l-y)) 2F1 b;c-a;l+b-a;(l_x)(l_y) y)a+b- c2F 1(a, b; c; x
= (1 -
+y-
xy),
(10.5.9)
by Bailey [1935, 1.4(1)]. So we can regard (10.5.8) as a q-analogue of the formula
F3(a, c - a; b, c - b; c; x, y)
= (1 -
y)a+b- c2F 1(a, b; c; x + y - xy),
(10.5.10)
Bailey [1935, 9.5(5)]. If instead of starting with (3) (qa, qc-a; qb, qC-b; qC; q; x, y) we started with (1) (qa; qb, qC-b; qC; q; x, y) we would obtain a different q-analogue of (10.5.10). For, (1) (qa;
qb, qC-b; qC; q; x, y)
~ ~ (qa; q)m+n(qb; q)m(qc-b; q)n
=~~ m=O n=O
L 00
=
m=O
=
1
(q
(qC; q)m+n(q; q)m(q; q)n b
m X
n Y
)
,qc.; ~ m xm21/JI(qa+m, qC-b; qc+m; q, y) q, q ,q m
ff
(yqC-b; q)oo (qa, qb; q)m(qc-a, qC-b~ q)n xm( _y)nqG)+(a+m)n (y;q)oo m=On=O (qC;q)m+n(q;q)m(q,yqC b;q)n (10.5.11)
by (1.5.4), and (1) (qa;
l, qC-b; qC; q; x, y)
a] _ (qa,xqb,yqc-b;q)oo A, [qc-a,x,y. (c .) 3'1'2 xqb yqc-b, q, q q ,X,y,q 00 ,
-
_ (xqa+b-c,yqc-b;q)oo [qc-a,qc-b,yqC-b/ x . a+b-C] (. ) 3rP2 qC yqc-b ,q, xq , (10.5.12) x,y,q 00 , by (10.3.4) and (3.2.7), and hence
ff
m=O n=O
(qa,qb;q)m(qc-a,qC-b~q)n
xm(_ytqG)+(a+m)n
(qC; q)m+n(q; q)m(q, yqC b; q)n
_ (xqa+b-c;q)oo [qc-a,qc-b,yqC-b/ x . a+b-c] (x; q)oo 3rP2 qC, yqc-b ,q, xq .
(10.5.13)
If we take the limit q ----+ 1- of (10.5.13) and replace y by y/(y - 1) we obtain (10.5.10). This is an example of a situation where one gets different q-analogues by starting with opposite sides of the same formula.
10.6 Formulas for a q-analogue of F4
Appell's F4 function is probably the most important of the four Appell functions because of its applications in the theory of classical orthogonal polyno-
10.6 Formulas for a q-analogue of F4
291
mials. It has the integral representation
F4(a, b; e, e'; x(l - y), y(l - x)) =
1111
f(e)f(e') ua-lv b- l (1 - u )c-a-l (1 - v)c'-b-l f(a)r(b)f(e - a)r(c' - b) 0 0 x (1 - ux)Ha-c-c' (1 - vy)Hb-c-c' (1 - ux - vy)c+c'-a-b-ldudv, (10.6.1)
which was derived by Burchnall and Chaundy [1940] from their expansion
F4(a, b; e, e'; x(l - y), y(l - x)) ~ (a)r(b)r(1+a+b-e-e')r
= ~
r=O
X 2Fl (a
I ( ) ( ')
T. ere r
( ) xr yr 2Fl a + T, b + T; e + T; X
+ T, b + T; e' + T; y).
(10.6.2)
Jackson's <1>(4), given by (10.2.8), does not seem to be useful in any applications that we have come across, and it does not have any formulas with its arguments x and y replaced by x(l - y) and y(l - x), respectively, so our approach will be to transform (10.6.1) and (10.6.2) into forms that we can find q-analogues of. First observe that, since
2Fl(a + T,b+ T; e+ T; x) = (1- x)-a- r 2Fl (a + T,e - b; e+ T;
_X_) , x-I
2Fl(a + T, b + T; e' + T; y) = (1 - y)-b-r 2Fl (b + T, e' - a; e' + T; -y-) y-1
by (1.5.5), the expansion formula (10.6.2) transforms to
F4(a, b; e, e'; x(l - y), y(l - x)) =
(1 _ x)-a(1- y)-b ~ (a)r(b)r(1 + a + b - e - e')r (_x_)r (_y_)r ~ T!(e)r(c')r x-I y- 1 X 2Fl
(a + T, e - b; e + T;
X
~ 1) 2Fl (b + T, e' - a; e' + T; Y ~ 1)·
(10.6.3)
Replacing x and y by x/(x - 1) and y/(y - 1), respectively, (10.6.3) becomes
F4 (a, b; e, e'; - (1 _ x ~1 - y) , - (1 - x ~(1 _ y))
=(1- )a(l_ )b~(a)r(b)r(1+a+b-e-e')r( )r X y r=O ~ ) ( ') r xy T.I( ere x2Fl(a+T,e-b;e+T;x) 2Fl(b+T,e'-a;e'+T';y). Analogously, the integral representation (10.6.1) transforms to
(1- x)-a(1- y)-b F4( a, b; e, e'; - (1- x~(1- y)' - (1- x~(1- y))
f(e)f(e') f(a)f(b)f(e - a)f(e' - a)
(10.6.4)
292
q-series in two or more variables
10 1 10 1 u a- 1v b- 1(1_ u)c-a-1(1_ v)C'-b-1(1 _ ux)b-c(1- vy)a-c'
X
x (1 - uvxy)c+c'-a-b- 1dudv, where 0
(10.6.5)
< Re a < Re c and 0 < Re b < Re c'. Let
. b ,.) = ~ (a, b, abq/cc'; q)r (CC'xy)r M( X,y ) = M( x,y,a, ,c,c,q ~ ( ') b r=O
q, C, C ; q r
a q
x 2¢h(aqr,c/b; cqr;q,x) 2(/JI(bqr,c'/a;c'qr;q,y),
(10.6.6)
which, via (10.6.4), is a q-analogue of the left side of (10.6.5) on replacing a, b, c, c' by qa , qb, c q ,qc' ,respect·lve1y. S·mce
(10.6.7) if 0
< Re
a
< Re c, and
'" (b+r c'-a. c'+r. )_ rq(c') (qC';q)r 2'1-'1 q ,q ,q ,q,y - rq(b)rq(c' - b) (qb;q)r X
if 0
1
Hr-1 ( qv, yvq c'-a) ;q d (10.6.8) o v (yv, vqc' -b; q)(Xl qV, 1
(Xl
< Re b < Re c', we have the integral representation M(
r r
l l a-I b-1 rq(c)rq(c') X,y - rq(a)rq(b)rq(c _ a)rc(c' _ b) io io u v
)_
( qu, qv, xuq c-b ,yvqc'-a ,xyuv; q )
(Xl
d
d
x (xu,yv,uqc-a,vqc'-b,xyuvqc+c'-a-b-1;q)(Xl qU qV, (10.6.9) which is an exact analogue of (10.6.5). However, (10.6.9) is, by the definition of q-integrals, the same as the double sum c-b c' -a ) a b M( ) = (q, q ,xq , yq , xy; q x, y (qC, qc' ,x, y, xyqc+c'-a-b-1; q)(Xl (Xl
X
LL (Xl
(Xl
m=O n=O
(
xyq
c+c'-a-b-1) ( c-a) ( c'-b ) ;q m+n x,q ;q m,y,q ;q nqam+bn (xy; q)m+n(q, xqc-b; q)m(q, yqC -a; q)n
(qa, qb, xqc-b, yqc'-a, xy; q)(Xl (qC, qC', x, y, xyqc+c'-a-b-1; q)(Xl xyqc+c' -a-b-1 : x, qc-a X c'p1:2;2 [ 1:1;1 xy : xqc-b
; y, qc'-b . . a b] ; yqc'-a ,q, q ,q ,
(10.6.10) by (10.2.10). This double series looks more like the series in (10.2.5) for c'p(1) than the series (10.2.8) for c'p(4). Note that the term-by-term q ----+ 1- limit can
10.6 Formulas for a q-analogue of F4
293
be taken on the expression on the right side of (10.6.9) but not on its double sum in (10.6.10). However, it is in the form given by (10.6.6) that the function M(x, y) seems to be most useful. First of all, when cc' = abq the series in (10.6.6) becomes a product
2¢1(a, cjb; c; q, x) 2¢1(b, c' ja; c'; q, y) which corresponds to a q-analogue of the right hand side of the product formula
F4 (a, b; c, a + b + 1 - C; x(l - y), y(l - x)) = 2F1(a, b; C; x) 2F1(a, b; a + b + 1 - C; y)
(10.6.11)
in Bailey [1935, 9.6(1)]. We shall now manipulate the series in (10.6.5) in order to obtain an expression that closely resembles the F4 function. First, it follows by combining the terms in the obvious manner, that
M(
) ~ ~ (a, cjb; q)m(b, c' ja; q)n m n x, y = ~ ~ ( ) ( ') x y m=O n=O
q, C; q
m
q, C ; q n
q-m q-n abqjcc'
]
x 3¢2 [ bq1-:n jc, ~q1-n jc'; q, q .
(10.6.12)
Since the 3¢2 series above is balanced it can be summed by (1.7.2). A bit of simplification then leads to
M(
) _ ~ ~ (a, cq-n jb; q)m(b, c'q-m ja; q)n m n mn ~ ~ ( ) ( ') x y q , m=O n=O q, C; q m q, C ; q n
x, y -
(10.6.13)
which has a straightforward q ----t 1- limit, but is not very useful because of the dependence of the terms (cq-n jb; q)m and (c'q-m ja; q)n on m and n. However, SInce
and
(c' q-m j a; q)n (c';q)n
A, (-n 2'/'1 q
m' ;q,cq , n-mj a,) ,aq;c
by (1.5.2), we have
M(x,y) =
(b) m n a;q m+s ;q n+r X Y m=O n=O r=O s=o (q; q)m-r(q; q)n-s(q, C; q)r(q, c'; q)s
L LLL 00
X
=
00
m
n
()
(-lr+S(cjbr(c' ja)sqmn-ms-nr+(;)+G)
ff
(a, b; q)r+s (_ cx)r (_ C'Y)S r=O s=o (q, C; q)r(q, c'; q)s b a
X
q(;)+G)-rs",,r+s,
(10.6.14)
q-series in two or more variables
294
where
'Yr =
=
~ ~ (aqr; q)m(bqr; q)n ~ ~
m=On=O
(q; q)m(q; q)n
(byqr;q)oo ( ) y;q 00
,/, (
2'/"1
r
m n mn x y q
b r
aq ,y; yq ; q, x
)
(axqr,byqr;q)oo ,/, ( r b r r b r ) (10.6.15) ( ) 2'/"2 aq , q ; axq , yq ; q, xy . x,y;q 00 Since (xqa+r+s, yqb+ r+s ; q)oo/(x, y; q)oo is simply a q-analogue of the product (l_x)-a-r-s(l_y)-b-r-s and the above 2¢2 series approaches 1 when q ---+ 1-, in view of (10.6.6), we may introduce the following function
=
ff
(qa, qb; q)r+s( --:x qC - bt ( _yqcl-a)s q(;)+G)-rs r=O s=O (q, qC; q)r(q, qC ; q)s(qax , qby ; q)r+s x 2¢2 (qa+r+s , qb+r+s; xqa+r+s, yqb+r+s; q, xy)
as an appropriate q-analogue of (1 - x )-a(1 - y) -b F4 (a, b; c, c'; - (1 _ x ~(1 _ y)' - (1 _ x ~(1 _ y)).
10.7 An Askey- Wilson-type integral representation for a q-analogue of F1
The q-integral representation for
r
r( )2 1 -/,
= r(a);("( _ a) Jo (1- cos7j1)':>-~(l + cos7j1)'Y-O<-~ X (
1-
XX
"2 + "2 cos 7j1
)-{3(1 -
y
y
"2 + "2 cos 7j1
)_{31
d7j1, (10.7.1)
provided 0
< Re a < Re 'Y. This suggests the following integral representation
=A X
l
1r
o
y)
(e i ,p , e-i,p·q~)a , - - -1 (_ei,p , 2
(>..ei,p, >"e-i,p; q)
_e-i,p.q~)
-{3 (f-lei,p , f-le-i,p; q)
-a 2
,~-!
4
_(31
4
d7j1,
(10.7.2)
where A is a normalizing constant, and >.., f-l are related to x and y by 4>.. x = - (1- >..)2'
4f-l y = - (1- f-l)2·
(10.7.3)
10.7 Askey-Wilson-type integral representation
295
A more general form of the integral in (10.7.2), namely, S(a,b,c,d,j,g;A,~) =
i7r h(cos1jJ; 1, -1, q~, -q~, A,~) dnl , ( ) 'f/ o
()
10.7.4
h cos 1jJ; a, b, c, d, j, 9
was evaluated in Nassrallah and Rahman [1986]. It was found that S(a,b,c,d,j,g;A,~) =
( K,
X
+
b d) (q/abcd, aq/b, aq/c, aq/d, Aa, Va, ~a, ~/a; q)oo a, ,c, (qa2, q/bc, q/bd, q/cd, aj,J la, ag, g/a; q)oo lOWg(a2; ab, ac, ad, aj, ag, aq/A, aq/~; q, AM/abcdjg) idem (a;j,g),
(10.7.5)
with IAM/abcdjgl < 1. The proof in Nassrallah and Rahman [1986] is long and tedious. We will give a shorter proof here. By (2.11.7) and (2.10.1) h(Z;A,~)
h(z;j,aq)
= cos 1jJ.
where z
(Aa,A/a,~a,~/a;q)oo
(qa 2, ja,j/a,q; q)oo x 8W7(a2; aj, aq/A, aq/~, aei'lj;, ae-i'lj;; q, A~/aJ) + h(z;a,jq) (Aj,A/j,~j,~/j;q)oo h(z;j,aq) (qj2,aj,a/j,q;q)oo x 8 W7U 2; aj, jq/A, jq/~, jei'lj;, je-i'lj;; q, A~/aJ), (10.7.6) Substituting (10.7.6) into (10.7.4) gives
S(a,b,c,d,j,g;A,~) =
f
(Aa,A/a,~a,~/a;q)oo 1- a 2q2n (a2,aj,aq/A,aq/~;q)n (A~)n 2 (qa ,ja,f/a,q;q)oo n=O 1-a2 (q,aq/j,Aa,~a;q)n aj
i 7r
X
+
h( cOS'f/, nl'·l , - 1 ! ! n+l) d1jJ ,q2,-q2,aq
o h( cos 1jJ; b, c, d, g, aqn) idem (a; J).
(10.7.7)
By (6.3.7) the integral displayed in (10.7.7) equals
2n(a 2q2n+l, abcdqn, acdgqn, abdgqn, abcgqn; q)oo (bc,bd,bg, cd,cg, dg,abqn, acqn, adqn, agqn, a 2bcdgq2n; q)oo x 8 W7(a 2bcdgq2n-l; abqn, acqn, adqn, agqn, bcdgq-\ q, q) 2n(abcd,q/abcd,aq/b,aq/c,aq/d;q)oo (ab, ac, ad, be, bd, cd, ag,g/a, q/bc, q/bd, q/cd; q)oo (ab, ac, ad, ag; q)n (q)n X (aq/b,aq/c,aq/d,aq/g;q)n bcdg 2nq 2n(bcdg / q, q2 /bcdg, gq/b, gq/ c, gq/ d; q)oo - bcdg (q, be, bd, cd, bg, cg, dg, qg2; q)oo(1- agqn) X
8W7(g2;bg,cg,dg,agqn,gq-n/a;q,q/bcdg),
(1 - ar) (10.7.8)
296
q-series in two or more variables
by (2.11.7) and (2.10.1). So we get
S(a,b,e,d,j,g;A,M) = ( b d) (q/ abed, aq/b, aq/ e, aq/ d, Aa, AI a, Ma, M/ a; q)oo ~ a, ,e, (q/be,q/bd,q/ed,aj,j/a,ga,g/a,qa 2;q)00 x lOWg(a2;ab,ae,ad,aj,ag,aq/A,aq/M;q,
a~~g)
+ idem (a; f) 2nq (bedg/q,q2/bedg,gq/b,gq/e,gq/d;q)00 bedg (q,be, bd,ed, bg,eg, dg,qg2; q)oo
X
f
1 - g2q2m
m=O
1-g2
(g2, bg, eg, dg, ag,
t;:q; q)m
(AM) m
(q,9q/b,gq/e,gq/d,9q/:,¥;q)m abedjg
x {(Aa'A/a'Ma'M/a,¥,~;q)oo (ja, j/a, ag, a/g, ~j,aAr;q)oo x
8W7(A~a; AI j, M/ j, AM/q, agqm, aq-m /g; q, q) +
idem (a; f)}. (10.7.9)
Using (2.11.7) to evaluate the expression in { } above and simplifying, we obtain (10.7.5), which can be regarded as an extension of (10.3.4) of AskeyWilson type. In the special case AM = abedjg the integral in (10.7.4) was evaluated 1 a: + 1 1 +1 m (6.4.11). If we replace a, b, e, d, j, 9 by q"2-"4, q"2 "4, -q-2--"4, -q-2- "4, Aq-f3 and Mq-f3', respectively, then this condition amounts to 'Y = {3 + {3', and therefore this case corresponds to (10.3.12) and (10.3.13). Note also that if A = 9 or M = j, which would imply A = M- f3 ' or M = Aq-f3, then the integral in (10.7.4) becomes an 8¢7 series (see §6.3) and hence an analogue of the Gaussian series 2F1. This, then, corresponds to (10.3.5) and (10.3.6). It can be shown that <[>1(0:; {3, (3'; 'Y; x, y) has a q-Appell type double series representation that corresponds to (10.2.5), see Ex. 10.16 for more on the general function S(a, b, e, d, j, g; A, M)' •
,,(-Ol
0:
Exercises
10.1 Show that
(i)
<[>(1)(b'/ . b b'. bb'.. ) x, , , , q, x, Y
(ii)
<[>(1)( -q/y; b, qx/y2; -qbx/y; q; x, y) (_q, bx; q)00(xq2 /y2, x 2q2 /y2; q2)00 (x, y, qx/y, -qbx/y; q)oo(x; q2)00 .
(Andrews [1972])
=
(b', bx, b'y/x; q)oo (bb' ,x, y,.) q 00 ,
,,{-ex
Exercises
297
10.2 Derive the transformation formulas (2) (a', b, b'·, c, c'·, q', x , y)
(i)
ff
= (b, ax; q)oo (c, x; q)oo
m=O
(c/b, x; q)m(a, b'; q)n bmyn, n=O (ax; q)m+n(q; q)m(q, c'; q)n
(3) (a , a'·" b b'·, c, q', x , y)
(ii)
=
(a, bx; q)oo ~ ~ (c/a; q)m+n(x; q)m(a', b'; q)n m n ~~ ay, (c,x;q)oo m=On=O (q,bx;q)m(q,c/a;q)n
and their special cases (iii)
(2) (a', b, b'·, c, a', q', x , y)
(iv)
_ (b,ax;q)oo (3)( / . '. .. ) ( ) c b, 0, x, b , ax, q, b, y , c,x;q 00 (3)(a a'· b b'· aa'· q' x y) -
, '"
'"
_ (a,bx;q)oo (2)('. '. .. ) - ( ) a ,x, b , bx, 0, q, a, y . aa', x; q 00 (Andrews [1972]) 10.3 Prove that (l)(a'b b'cq'x -x) , , , , , ,
=,j,
3 '1'2
2 [ a , C,aqcq, b ".q2 x 2] .
Deduce the quadratic transformation formula ,j,
3'1'2
[c/a,x,-x. ] _ (c;q)00(x 2;q2)00 bx , -bx ,q, a - (a,. q) 00 (b2 x, 2. 2) q 00
,j,
3'1'2
2 2] .
[a,a q ,b2 . C cq ,q, x ,
10.4 Show that (1) (bq!', b" b' b2". q' X , y) 1
(xq'i;q)oo 1 1 1 ( ) 2¢1((by/x)2,-(by/x)2;-b;q2,x). y;q 00
=
Hence, prove the quadratic transformation formula 3¢2 [
x,
:;,b~;! ; q, bq! ]
(x,y,b 2;q)00 (bx, by, bq'i; q)oo
-----'-----;-1-'------
2¢1 (( by /)1 x 2,
-
(by /)1 x 2; -b; q21,x) .
10.5 Derive the quadratic transformation formula (l)(a'b , , b'cx , , , y)
) = L a, b2,xq1/ 21 y; q n yn 00
(
n=O X 4¢3
(q, [
C,
bq2; q)n
! !] .
q-n/2, _q-n/2, (by/x)!, -(by/x)! . -b, ql/4-n/2(y/x)!, _ql/4-n/2(y/x)! ,q ,q
298
q-series in two or more variables
10.6 Prove that
, b, b'·, c, c'·, q', x , y)
(2) (a'
=
ff
(a; q)m+n(c/b; q)m(c' /b'; q)n (-bx)m( -b'y)nqG)+G) m=O n=O (ax; q)m+n(q, c; q)m(q, c'; q)n X
(~x;~)oo 2(/JI( aqm+n, O;axqm+n; q,y). x; q
00
Note that this gives a q-analogue of Bailey [1935, 9.4(8)]. 10.7 Show that a general 3(P2 series with an arbitrary argument is a multiple of Jackson's (1) series, in particular,
a,b,c. ] _ (ax, b,c; q)oo 3
(1)(.
/
/ ...
)
x,d b,e c,ax,q,b,c.
ff
(a, b; q)m+n( -xc/b)m( -yb/a)n q(r;')+(~)-mn m=O n=O (q, c; q)m(q, b; q)n(ax, by; q)m+n x 2 a,aq / c,c /b,c,q,cxy aq,x ) , ax, y; q 00 and that this is a q-analogue of Bailey's [1935, Ex. 20(ii), p. 102] reduction formula
F4
= 10.9 Show that
(a, b; c, b; -
(1 _
x~1 _ y) , - (1 _ x~(1 _ y))
(1 - x) a (1 - y) aF1 (a; 1 + a - c, c - b; c; xy, x).
ff
(a, b; q)m+n( -xa/b)m( -yb/a)n q(r;')+G)-mn m=O n=O (q, a; q)m(q, b; q)n(ax, by; q)m+n X
2
(by/a, ax/b; q)oo (by, ax; q)oo(1- xy/q) ,
Ixyl < q,
and that this is a q-analogue of Bailey's [1935, Ex. 20(iii), p. 102] formula
F4 (a, b; a, b; - (1 _ =
x~(1- y)' ---'---(l---X---':-~--'---(l---y---'--))
(1 - xy)-l(l- x)b(l _ y)a.
10.10 Prove that
ff
(a, b; q)m+n( -aqx/b2 )m( -yb/a)n qG)+(~)-mn m=O n=O (q, aq/b; q)m(q, b; q)n(ax, by; q)m+n X
2
_ (by/a; q)oo .t, [a, b, aq/by . /b] - (b y;q ) 00 3'f'2 aX,aq /b ,q,XY
Exercises
299
and that this is a q-analogue of Bailey's [1935, Ex. 20(v), p. 102] reduction formula
F4 (a, b; 1 + a - b, b; - (1 _ =
(1 - y) a
2Fl
(
X~l _
x~(1- y))
y)' - (1 -
y a, b; 1 + a - b; - X(l- )) . I-x
10.11 Using (10.6.15) prove that
ff
(a, b; q)m+n( -cx/b)m( _qy/c)n qC;')+G)-mn m=O n=O (q, c; q)m(q, aq/c; q)n(ax, by; q)m+n x 2
ff
(a; q)m+n(b; q)m(b; q)n (_ c~ )m( _ m=On=O (q, c, ax; q)m(q, aq/c, by; q)n bq'i
yq~)n q(m-n)2j2, C
where Icx/bl < 1 and Iy/cl < 1. This is a q-analogue of Bailey's [1935, Ex. 20(i), p. 102] reduction formula
F4 (a, b; c, 1 + a - c; - (1 _ x ~1 _ y) , - (1 - x ~(1 _ y)
= F2 (a;b,b;c, 1 + a 10.12 Show that if f(x, y)
(i) (ii)
= (f>(l) (a;
c; _x_, -y-). x - I y-1
b, b'; c; q; x, y), then
(abx - c/q)f(q2 x , qy) + (1 - bx)f(qx, y) + (c/q - a)f(qx, qy) + (x - l)f(x, y) = 0, (aby' - c/q)f(qx, q2y) + (1 - b'y)f(x, qy)
+ (c/q - a)f(qx, qy) + (y - l)f(x, y) = O. 10.13 Show that if f(x, y)
=
(f>(2)
(a; b, b'; c, c'; q; x, y) then
ab'xf(q2 x , qy) - ~ f(q 2x, y) - axf(qx, qy) q
+ (1 + ~ 10.14 Show that if f(x " y)
=
(f>(3)
b' x ) f (qx, y)
+ (x -
1) f (x, y) = O.
(a a'·" b b'·, c·, q., x , y) then
abxf(q2 x , y) - ~ f(q 2x, qy) + ~ f(qx, qy) q
+ (1 10.15 Show that if f(x,y)
ax - bx)f(qx, y)
q
+ (x -
= (f>(4)(a,b;c,c';q;x,y)
l)f(x, y) = O.
then
abxf(q2 x , q2y) - cf(q 2x, y) - (a + b)xf(qx, qy)
+ (1 + c)f(qx, y) + (x -
l)f(x, y)
=
O.
q-series in two or more variables
300
10.16 Using (6.3.3) twice show that S(a,b,e,d,j,g;A,~)
2n(A/a,A/j,~/b,~/g;q)=
ab(l - q)2(q, q, q, cd, fla, aq/ j, g/b, qb/g, aj, bg; q)=
r (qu/a,qu/j,Au;q)= r (qv/b,qv/g,~v,eduv;q)=d b
x
if
(eu,du,Au/aj;q)=
ig
(ev,dv,uv,~v/bg;q)=
d qU
2n(abed,A/a,Aa,~/b,~b;q)=
(q, ab, ae, ad, be, bd, cd, j la, ja, g/b, gb; q)= x
+
ff
(ab;q)m+n(ae,ad,Alj;q)m(be,bd,~/g;q)n qm+n m=O n=O (abed; q)m+n(q, aq/ j, Aa; q)m(q, qb/g, ~b; q)n idem (a; j)
2n(aedg,A/a,Aa,~/g,~g;q)= +-;-----'--.,--=-'----'=-....,....:-'c-:-.::..c....:....::-'-:--=:'--:-----:--
(q, ae,ag,ad,eg,dg, cd, j/a,aj,b/g, bg; q)=
x
+
ff
(ag; q)m+n(ae, ad, AI j; q)m(eg, dg, ~/b; q)n qm+n m=O n=O (aedg; q)m+n(q, aq/ j, Aa; q)m(q, gq/b, ~g; q)n idem (a; j).
10.17 Extend (10.3.4) to
where D is a q-Lauricella function. (Andrews [1972]) 10.18 Prove that
[q-n : ale, e/d, e/e ; e/a, bid, b/e.. ] 1:2;2 be/de : a, q1-n /b ; b, q1-n /a ,q, q, q
1:3;3
(e, able; q)n (a,b;q)n ' where n = 0,1, .... Deduce the summation formula
ff
j=O k=O
(a/e, e/d: e/e; q)j(e~a, bid, b~e; qh b1akqjk (be/de, q)j+k(q, a, q)j(q, b, q)k
(e, able; q)= (a, b; q)= . (Gasper [2000])
qV,
Notes
301
10.19 Show that
j,k,m?O (,x,qz/"~y,,y,axz/,,~yz;q)oo
(x, a~zh, ~,y, y, xz, yZ; q)oo X 8 W7(~,yq-\~,
I, ,y/a, 1/z, ~Y/X; q, axzh)·
(Krattenthaler and Rosengren [2003]) 10.20 Extend (1.9.6) to
ff
n=Om=O X
(blqm 1 ,
... ,
brqmr; q)n+m(a, b; q)n(c, d; q)m
(bl , ... , br ; q)m+n(q, bq; q)n(q, dq; q)m
(a-lql-Mt(c-lql-M)m
(q, q, bq/ a, dq/ C; q)oo (bd db; q)ml ... (b r /bd; q)mr (bd)M (bq, dq, q/a, q/c; q)oo(b l ; q)ml ... (b r ; q)mr
where ml, ... , mr are nonnegative integers, M Iql-MI < min(lal, Icl)·
=
ml
+ ... +
m r , and
(Denis [1988])
Notes §1O.2 B. Srivastava [1995] introduced some elementary bibasic extensions of (10.2.5)-(10.2.8). For a quantum algebra approach to multivariable series, see Floreanini, Lapointe and Vinet [1994]. Sahai [1999] considered the q-Appell functions from the point of view of universal enveloping algebra of 8£(2). See also Jain [1980a,b] and, for connections of Appell functions with Ben root systems, Beerends [1992]. §§10.3-1O.6 Agarwal [1974] evaluated some q-integrals of the double series in (10.2.5)-(10.2.8). Also see Nassrallah [1990, 1991]. Exercises 10.1-10.2 Upadhyay [1973] gave some transformation and summation formulas for q-J ackson type double series. Ex. 10.3 This is a q-analogue of a quadratic transformation formula for Fl(a; b, b; C; x, -x) obtained by Ismail and Pitman [2000]. Exercises 10.12-10.15 Agarwal [1954] gave some 3-term contiguous relations satisfied by <[>(1), ... ,<[>(4), including formulas connecting them with the q-derivatives. Ex.1O.17 For an application of the q-Lauricella series, see Bressoud [1978]. Ex.1O.18 See Van der Jeugt, Pitre and Srinivasa Rao [1994] for more summation formulas of this type. For similar results when q ---+ 1, see Pitre and Van der Jeugt [1996].
11 ELLIPTIC, MODULAR, AND THETA HYPERGEOMETRIC SERIES
11.1 Introduction
Since the series L Un is called a hypergeometric series if 9 (n) = U n +IiUn is a rational function of n, and a q-(basic) hypergeometric series if g( n) is a rational function of qn, it is natural to call this series an elliptic hypergeometric series if g( n) is an elliptic (doubly periodic meromorphic) function of n with n considered as a complex variable. One motivation for considering these three classes of series is Weierstrass's theorem that a meromorphic function J(z) which satisfies an algebraic addition theorem of the form P(f(U), J(v), J(u
+ v)) = 0
identically in u and v, where P(x, y, z) is a nonzero polynomial whose coefficients are independent of u and v, is either a rational function of z, a rational function of e AZ for some A, or an elliptic function (see, e.g., Erdelyi [1953, §13.11]' Rosengren [200la, 2003c], and, for a proof, Phragmen [1885]). Elliptic analogues of very-well-poised basic hypergeometric series were introduced rather recently by Frenkel and Turaev [1997] in their work on elliptic 6j-symbols, which are elliptic solutions of the Yang-Baxter equation found by Baxter [1973], [1982] and Date et al. [1986-1988]. Frenkel and Turaev showed that the elliptic 6j-symbols are multiples of the very-well-poised 12Vl1 elliptic (modular) hypergeometric series defined in §11.3 (recall from §7.2 that the ordinary q-analogues of Racah's 6j-symbols are multiples of balanced 4¢3 series, which are transformable to 8¢7 series) and then used the tetrahedral symmetry of the elliptic 6j-symbols and the finite dimensionality of cusp forms to derive elliptic analogues of Bailey's transformation formula (2.9.1) for terminating 1O¢9 series and of Jackson's 8¢7 summation formula (2.6.2). This quickly led to a flurry of activity on elliptic hypergeometric series, resulting in several related papers (listed in alphabetical order) by van Diejen and Spiridonov [20002003], Gasper and Schlosser [2003], Kajihara and Noumi [2003], Kajiwara, Masuda, Noumi, Ohta and Yamada [2003], Koelink, van Norden and Rosengren [2003], Rosengren [200la-2003f], Rosengren and Schlosser [2003a,b], Spiridonov [2000-2003c], Spiridonov and Zhedanov [2000a-2003], Warnaar [2002b2003e], and others. In particular, general elliptic hypergeometric series and their extensions to theta hypergeometric series were introduced and studied by Spiridonov [2002a-2003a], who also considered theta hypergeometric integrals in Spiridonov [2003bj. Transformations of elliptic hypergeometric integrals are considered in Rains [2003bj. We start in §11.2 with the elliptic shifted factorials, Spiridonov's [2002aj "multiplicative" r+lEr theta hypergeometric series notation and its very-well302
11.2 Elliptic and theta hypergeometric series
303
poised r+1 Vr special case, and then point out some of their main properties. The "additive" forms of these series r+1er and r+1Vr are presented in §11.3, along with their modular properties, and the Frenkel and Turaev summation and transformation formulas. In §11.4 we present a simpler derivation of the elliptic analogue of Jackson's 8¢7 summation formula via mathematical induction, which is a modification of proofs that were kindly communicated to the authors by Rosengren (in a June 13, 2002, e-mail message), Spiridonov (in a Nov. 22, 2002, e-mail message) and Warnaar (see §6 in his [2002b] paper). Central to this method of proof and, in fact, to all of the formulas in this chapter is the theta function identity given in Ex. 2.16(i) which is simply the elliptic analogue of the trivial identity
(1- xA)(I- xIA)(I- fLV)(I- fLlv) - (1- xv)(I- xlv)(I- AfL)(I- fLIA)
= ~ (1 - XfL)(1 - xl fL)(1 - Av)(1 - Ajv).
(11.1.1)
The elliptic analogue of Bailey's 1O¢9 transformation formula is derived in §11.5, along with some other transformation formulas. Theta hypergeometric extensions of some of the summation and transformation formulas in sections 3.6-3.8 are derived in §11.6. Some multidimensional elliptic hypergeometric series are considered in §11. 7, where we present Rosengren's [2003c] elliptic extension of Milne's [1985a] fundamental theorem and related formulas. Many additional formulas involving theta hypergeometric series and elliptic integrals are presented in the exercises at the end of the chapter.
11.2 Elliptic and theta hypergeometric series
Define a modified Jacobi theta function by
O(X;p)
=
O(X1, ... , xm;p) =
(x,plx;p)oo,
II O(Xk;P),
(11.2.1)
k=l
where X,X1, ... ,Xm -=I- 0 and Ipi < 1. Since O(x;p) = (x;p)oo(plx;p)oo is the product of two infinite p-shifted factorials, analogous to the name of the triple product identity (1.6.1), we will call O(x;p) the double product theta function (with argument x and nome p). From (1.6.9) and (1.6.14) it follows that (11.2.2) and
[a· (J "
T]
=
O(qa. p) 'q(1-a)/2
O(q;p)
(11.2.3)
with (11.2.4) where a, (J, T are complex numbers such that 1m (T) > 0 and (J -=I- m + nT for integer values of m and n. We set qa = e2niaa and pa = e 2nim . Following
304
Elliptic, modular and theta hypergeometric series
Warnaar [2002b], we define an elliptic (or theta) shifted factorial analogue of the q-shifted factorial by
(a; q,P)n
=
{ 1,rr~:~ B(aqk;p), 1/ rr- B( aq n+k.,p ) , n
k=O-
l
and let
n = 1,2, ... , n=O,
(11.2.5)
n=-I,-2, ... , m
(aI, a2,···, am; q,P)n
=
II (ak; q,P)n,
(11.2.6)
k=l
where a, al, ... , am -I=- O. Analogous to the name q-shifted factorial for (a; q)n, we also call (a; q, P)n the q, P -shifted factorial in order to distinguish it from the a, T-shifted factorial defined in the next section. Notice that B(x; 0) = 1- x and thus (a; q, O)n = (a; q)n. Since q is called the base in (a; q)n and P is called the nome in B( a; p), we call q and P in (a; q, P)n the base and nome, respectively. Similarly, in order to distinguish the modular parameters a and T, we call a the base modular parameter and T the nome modular parameter. For the sake of simplicity, we decided not to use Spiridonov's [2002a,b] notation B(a;p; q)n for the elliptic shifted factorial. Corresponding to Spiridonov [2002a], we formally define an r+IEr theta hypergeometric series with base q and nome p by
(11.2.7) where, as elsewhere, it is assumed that the parameters are such that each term in the series is well-defined; in particular, the a's and b's are never zero. Unless stated otherwise, we assume that q and p are independent of each other, but we do not assume that the above series converges or that the numerator parameters aI, a2, ... , ar+l, denominator parameters b1, ... , bTl and the argument z in it are independent of each other or of q and p. Note that if ajpk is a nonpositive integer power of q for some integer k and a j E {I, 2, ... , r + I}, then the series in (11.2.7) terminates. Clearly, if z and the a's and b's are independent of p, then lim r+1Er(al, ... , ar+l; b1, ... , br ; q,p; z)
p--70
= r+1Er(al, ... , ar+l; b1, ... , br ; q, 0; z)
= r+lcPr(al, ... , ar+l; b1,···, br ; q, z),
(11.2.8)
where the limit of the series is a termwise limit. As is customary, the notation r+1Er(al, a2, ... , ar+l; bl , ... , br ; q,p; z) is also used to denote the sum of the series in (11.2.7) inside the circle of convergence and its analytic continuation (called a theta hypergeometric function) outside the circle of convergence. Unlike nonterminating r+1Fr series and nonterminating r+lcPr series with 0 < Iql < 1, which have radius of convergence R = 1 (as can be easily seen by applying the ratio test to (1.2.25) and (1.2.26)),
11.2 Elliptic and theta hypergeometric series
305
for 0 < Iql, Ipl < 1 and any R E [0,00] there is a nonterminating r+lEr series with radius of convergence R. In particular, there are nonterminating r+lEr series with 0 < Iql,lpl < 1 that converge to entire functions of z, which is not the case for nonterminating r+lFr and r+l¢r series with r = 0,1, ... and 0 < Iql < 1. For example, consider a nonterminating r+lEr series with 0< Iql, Ipi < 1 and (11.2.9) where ml, ... , mr+l are integers and br + l function identity ()(apm;p) (apm; q,P)n
for m, n
=
q. Since the double product theta
= (-a)-mp-(';)()(a;p)
=
implies that
(a; q,P)n( _a)-mnp-nG)q-m(~)
= 0, ±1, ... , we find that 00
. Z) -- " ( ZPr )n qMrG) , (11210) 1'+1 E l' (al,···, ar+l,. alPml , ... , arp mr . ,q,p, ~ .. n=O
with ar+l = qp-m r + 1 , Pr = rr~~i(-ak)mkp(~k), and Mr = ml + ... + mr+l. From (11.2.10) it is clear that this series converges to an entire function of Z when Mr > 0, converges only for Z = 0 when Mr < 0, and converges to 1/(1 - zPr) when Mr = 0 and IZPrl < 1. As in Spiridonov [2002a], we call a (unilateral or bilateral) series L Cn an elliptic hypergeometric series if g(n) = cn+1/cn is an elliptic function of n with n considered as a complex variable, i.e., g(x) is a doubly periodic meromorphic function of the complex variable x. For the r+lEr series in (11.2.7)
()x = II akq ;p ()(bkqX;p) 1'+1 ()(
g
x
)
Z
(11.2.11)
k=l
with br+ l = q. Clearly g(x) is a meromorphic function of x. From (11.2.4) it is obvious that qx+a- 1 = qX and hence g(x + a-I) = g(x). Since, by (11.2.1), ()(aqx+ra- 1 ;p)
=
(_aqx)-l()(aqx;p),
(11.2.12)
it follows that (11.2.13) Thus g(x
+ Ta- l ) =
g(x) when ala2··· ar+l = (bl b2 ··· br)q,
(11.2.14)
in which case g(x) is an elliptic (doubly periodic meromorphic) function of x with periods a-I and Ta- l . Therefore, we call (11.2.14) the elliptic balancing condition, and when (11.2.14) holds we say that r+lEr is elliptically balanced (E-balanced). In Spiridonov [2002a] an r+lEr series is called, simply, "balanced" when (11.2.14) holds, but here we need to distinguish between the different balancing conditions that arise. Notice that, unlike the requirement
Elliptic, modular and theta hypergeometric series
306
that z = q in the definition of a balanced r+l ¢r series, no restrictions are placed on the argument z in the above definition of an E-balanced r+IEr series. If z = q, then, by the definition of a k-balanced r+l¢r series in §1.2, the r+l¢r series in (11.2.8) is (-1 )-balanced if and only if the elliptic balancing condition (11.2.14) holds. Analogous to the basic hypergeometric special case, we call the r+IEr series in (11.2.7) well-poised if qal
=
=
a2 bl
a3 b2 =
... =
(11.2.15)
ar+lbr,
in which case the elliptic balancing condition (11.2.14) reduces to 2 2 2 ala2···ar+1
Via (11.2.5) and the n p, we find that 2
=
B(aq n;p)
alq
(11.2.16)
.
limit case of Ex. 1.1(iv) with q replaced by
---+ 00
1
()r+l
=
111
11
(qa"2,-qa"2,qa"2/p "2,-qa"2 p "2;q,P)n(_ )-n
B() a;p
1
1
1
1
1
(11.2.17)
q
1
(a"2,-a"2,a"2p"2,-a"2/p"2;q,P)n
is an elliptic analogue of the quotient 1 - aq2n
(qd, -qd; q)n
1-a
(d,-d;q)n '
which is the very-well-poised part of the r+l Wr series in (2.1.11) with al = a; see Ex. 11.3. Therefore the r+IEr series in (11.2.7) is called ve'T"y-well-poised if it is well-poised, 'T" ~ 4, and a2
=
1
qai, a3
=
1
-qai, a4
=
!
1
qai /p"2, a5
=
!
1
(11.2.18)
-qai p"2.
Corresponding to Spiridonov [2002b, (2.15)]' we define the well-poised theta hypergeometric series by
r+l Vr
very-
r+I Vr(al;a6,a7, ... ,ar+l;q,p;Z) =
~ B(alq2n; p)
~ n=O
B(al;p )
(aI, a6, a7, ... , ar+l; q, P)n (qz)n. (q,alq / a6,alq / a7,···,alq / ar+l;q,P ) n
(
)
11.2.19
It follows that if (11.2.15) and (11.2.18) hold, then r+I Vr(al;a6,a7, ... ,ar+l;q,p;Z)
= r+IEr(al, a2,···, ar+l; bl , ... , br ; q,p; -z), and that
r+l Vr
(11.2.20)
is elliptically balanced if and only if 2 ( a62 a7···
2) 2 ar+l q =
(
alq )r-5 .
(11.2.21 )
As in Warnaar [2003c], when the argument z in the r+l Vr series equals 1 we suppress it and denote the series in (11.2.19) by the simpler notation r+l Vr(al; a6, a7, ... , ar+l; q, p). If aI, a6, a7, ... , ar+l are independent of p, then lim
p-+O
r+l Vr(al;
= r-l W r-
a6, a7,·· ., ar+l; q,p)
2(al; a6,···, ar+l; q, q),
(11.2.22)
11.2 Elliptic and theta hypergeometric series
307
which shows that there is a shift r ----t r-2 when taking the P ----t 0 limit, and that the P ----t 0 limit of an r+l Vr(al; a6, a7,"" ar+l; q,p) series with aI, a6, a7,"" ar+l independent of p is an r-l W r - 2 series. As mentioned in the Introduction, Frenkel and Thraev showed that the elliptic 6j-symbols, which are elliptic solutions of the Yang-Baxter equation (formula (1.2. b) in their paper) found by Baxter [1973], [1982] and Date et al. [1986-1988] can be expressed as 12 Vu series (in the additive notation discussed in the next section). Then they employed the tetrahedral symmetry of the elliptic 6j-symbols, which is analogous to the symmetry of the classical, quantum and trigonometric 6j-symbols (see Frenkel and Turaev [1995, 1997]), and the finite dimensionality of cusp forms (see Eichler and Zagier [1985]) to derive (in their additive form) the following elliptic analogue of Bailey's 1O¢9 transformation formula (2.9.1) 12 Vu(a;
b, c, d, e, I, Aaqn+l /el, q-n; q,p)
(aq,aq/el,Aq/e, Aq/l;q,P)n (aq/e,aq/I,Aq/el,Aq;q,P)n
x
12 VU(A;
Ab/a, Ac/a, Ad/a, e, I, Aaqn+l/el, q-n; q,p)
(11.2.23)
for n = 0,1, ... , provided that the balancing condition
(11.2.24)
bcdel(Aaqn+l/ef)q-nq = (aq)3,
which is equivalent to A = qa 2/bcd, holds. Note that both of the series in (11.2.23) are E-balanced when (11.2.24) holds. Setting A = aid in (11.2.23) yields a summation formula for 10 Vg series that is an elliptic analogue of Jackson's 8¢7 summation formula (2.6.2) and of Dougall's 7F6 summation formula (2.1.6), which, after a change in parameters, can be written in the form: TT (
lOv9
• b d -no ) _ (aq, aq/bc, aq/bd, aq/cd; q,P)n a, ,c, ,e,q ,q,p - (aq /b ,aq / c,aq /d ,aq /b c d ;q,p ) n
for n = 0,1, ... , provided that the balancing condition bcde can be written in the form
=
(
11.2.25
)
a 2qn+l, which
(11.2.26) holds. Clearly, if a, b, c, d, e are independent of p, then (11.2.25) tends to Jackson's 8¢7 summation formula (2.6.2) as p ----t O. Unlike in the basic hypergeometric limit cases of (2.6.2) discussed in Chapters 1 and 2, one cannot take termwise limits of (11.2.25) to obtain a 3E2 analogue of the q-Saalschiitz formula (1.7.2), 2El analogues of the q-Vandermonde formulas (1.5.2) and (1.5.3), or even a lEo analogue of the terminating case of the q-binomial theorem (1.3.2) in Ex. 1.3(i). Therefore one cannot derive (11.2.25) by working up from sums at the lEo, 2El, and 3E2 levels as was done in Chapters 1 and 2, and so one is forced to employ a different approach, such as in the abovementioned Frenkel and Thraev derivation, or by some other method. Rather than repeating the Frenkel and Thraev [1997] derivation of (11.2.25), we will present in §11.4 a simpler derivation of (11.2.25) via mathematical induction, which is a modification of those discovered independently by Rosengren, Spiridonov, and Warnaar. We will then show in §11.5 that (11.2.23) follows from
Elliptic, modular and theta hypergeometric series
308
(11.2.25) in the same way as in our derivation of Bailey's 1O¢9 transformation formula (2.9.1) from Jackson's 8¢7 summation formula (2.6.2). Observe that if we set e = ±(aq)~, then (11.2.25) reduces to V; ( . b
9 8
d -no ) _ (aq, aqjbc, aqjbd, aqjcdi q,P)n a, ,c, ,q ,q,p - (aq jb ,aq j c,aq jd ,aq jb cd iq,P ) n
(
11.2.27
)
for n = 0, 1, ... , provided that the balancing condition
(11.2.28) holds, which is equivalent to the elliptic balancing condition b2c2d 2 = a 3q2n+1. In view of (11.2.24), (11.2.26), (11.2.28), and of the required balancing conditions for other significant special cases of the Frenkel and Turaev transformation formula (11.2.23) to hold, analogous to the definition of a VWP-balanced series given in §2.1 we call the series r+l Vr(ali a6, a7, ... , ar+li q,Pi z) a verywell-poised-balanced (VWP-balanced) series when the very-well-poised balancing condition (11.2.29) holds. It follows that r+lVr(alia6,a7, ... ,ar+liq,P) is VWP-balanced if and only if (11.2.30) and that the summation formulas (11.2.25), (11.2.27) and the transformation formula (11.2.23) hold for n = 0,1, ... , when the series are VWP-balanced. Note that (11.2.30) reduces to (a6a7··· a2j+6)q = (alq)j when r = 2j + 5 is odd. It should also be noted that if either (11.2.30) or (a6··· ar+dq = -(±(alq)~t-5 holds, then r+l Vr(ali a6, a7, ... , ar+li q,p) is E-balanced, and hence an elliptic hypergeometric series. If r+l Vr(ali a6, a7, ... , ar+li q,p) is E-balanced, then it is VWP-balanced when r is even, but not necessarily when r is odd. In particular, the elliptic balancing condition (11.2.21) is not a sufficient condition for the Frenkel and Turaev transformation and summation formulas (11.2.23) and (11.2.25) to holdi the series in these formulas need to be VWP-balanced in order for these formulas to hold for n = 0,1, .... Since, if r+lEr is a well-poised series satisfying the relations in (11.2.15), r+lEr(al, a2,···, ar+li bl , ... , bri q,Pi z)
=
1.
1.
1.
1
1.
1
r+9 Vr+8 (ali at , -at, at p2, -at jp2 ,a2, a3, ... ,ar+li q,Pi -z), (11.2.31)
we find that the very-well-poised balancing condition for the above r+9 Vr+8 series is equivalent to the well-poised balancing condition (ala2··· ar+dz = - ( ± (alq) !)r+l 2
(11.2.32)
for the r+lEr series in (11.2.31). Hence, a well-poised r+lEr series is called well-poised-balanced (WP-balanced) when (11.2.32) holds. In particular, the well-poised 4E3 series in the transformation formula in Ex. 11.6 is WP-balanced. Clearly, every VWP-balanced theta hypergeometric series is WP-balanced and, by the above observations, every WP-balanced theta hypergeometric series can be rewritten to be a VWP-balanced series of the form in (11.2.31).
11.2 Elliptic and theta hypergeometric series
309
Analogous to the bilateral r'¢r series, we follow Spiridonov [2002a] in defining a rG r bilateral theta hypergeometric series by
(11.2.33) Note that rGr(al, ... ,ar;q,bl,b2, ... ,br-l;q,p;Z)
= rEr-l (al' ... ' ar; bl , ... , br- l ; q,p; z)
(11.2.34)
and, more generally, as in the bilateral basic hypergeometric case in (5.1.5), if the index of summation in an r+lEr series is shifted by an integer amount, then the resulting series is an r+l G r + l series multiplied by a quotient of products of q,p-shifted factorials. Also note that corresponding to (11.2.9) we can consider nonterminating rG r series with 0 < Iql, Ipi < 1 and
where ml, ... , mr are integers. As in (11.2.10) we find that 00
rGr(al, ... ,ar;alpml, ... ,arpmT;q,p;z)
=
L
(zur)nqNrG)
(11.2.35)
n=-oo
with U r = I1~=l (_ak)mkp("';'k) and N r = ml + .. ·+mr, which clearly converges for any z =I- 0 if and only if N r > O. If, as in Spiridonov [2002a], we replace r + 1 by r in the upper limit of the product in (11.2.11) and proceed as in the derivation of the condition (11.2.14) for an r+lEr series to be elliptically balanced, we find that the series rG r is elliptically balanced (E-balanced) if and only if ala2··· ar
= bl b2 ··· br ·
(11.2.36)
If rG r is a well-poised series with (11.2.37) then the elliptic balancing condition (11.2.36) reduces to al2 a22 ... a 2r =
( al b)r 1 .
(11.2.38)
In view of (11.2.32), (11.2.34), and (5.1.7), it is consistent to call a well-poised rG r series well-poised-balanced (WP-balanced) when (ala2··· ar)z = - (
± (albd!f.
(11.2.39)
In our consideration of modular series in the next section we are led to consider additive forms of special cases of the rather general power series
(11.2.40)
310
Elliptic, modular and theta hypergeometric series
and the Laurent series
rCis(al, ... ,ar;b1, ... ,bs;q,p;~,z) _
-
~ (a1, ... ,ar;q,P)nA n ~ nZ , n=-oo (b1, ... ,bs;q,P)n
(11.2.41)
where ~ = {An} is an arbitrary sequence of complex numbers. Some special cases of these series with An = qcm(n-l)/2 or, more generally, with An = exp(aln+a2n2 +a3n3) are considered in Spiridonov [2002a, 2003bj. If An = 1 for all n, then we will suppress ~ from the left sides of (11.2.40) and (11.2.41). When we encounter series with more than one base or nome, such as in Ex. 11.25(i), Ex. 11.26 and in several of the formulas in §11.6, or multivariable formulas, such as the multivariable extension of the Frenkel and Thraev summation formula in §11. 7, we will write the series in terms of elliptic shifted factorials. To help keep the size of this book down we will not repeat the main elliptic identities and summation and transformation formulas in the appendices. Nevertheless, for the convenience of the readers we collect below some of the most useful identities involving ()(a;p), the q,p-shifted factorials, and the q, P - binomial coefficients.
()(a;p) = ()(pja;p) = -a ()(lja;p) ()(a 2;p2) = ()(a, -a;p),
=
-a()(ap;p),
(11.2.42) (11.2.43)
()(a;p) = ()(a, ap;p2) = ()(a~, -a~, (aq)~, -(aq)~ ;p),
(11.2.44)
()(a;p)
(11.2.45)
=
()(apn;p)( -a)npG) ,
(a; q,P)n()(aqn;p) = (a; q,P)n+l = ()(a;p)(aq; q,P)n, (a; q,P)n+k = (a; q,P)n(aqn; q,ph,
(11.2.46)
(a; q,P)n = (ql-n ja; q,P)n( -atqG),
(11.2.48)
(a; q,P)n (_ ~)k (~)-nk . ) _ (a,q,Pn-k-(l_nj.) , a,q,Pk a q q
(11.2.49)
(11.2.47)
(qja;q,P)n( - ~rq-G),
(11.2.50)
q,ph(qja; q,P)n -nk (aq -no,q,p)k -_ (a; (l-kj. ) q , a,q,Pn q
(11.2.51)
(aq-n;q,P)n
=
(a; q-l,P)n = (a-\ q,P)n( _a)nq-G), 1 (_qja)n (n) (a; q, p) -n = ( aq- n ;q,p ) n ( j q a;q,p) n q 2 , (a; q,P)n+k (aq n ;q,p )k = (a; q,p)k(aqk; q,P)n = , (a;q,P)n (a;q,P)n (a; q,P)n = (ap\ q,P)n( _a)nkpn(~)qkG), (a 2; q2,p2)n = (a, -a; q,P)n, (a; q,p 2hn = (a~ , -a~, (aq) ~, -(aq) ~; q, P)n,
(11.2.52) (11.2.53) (11.2.54) (11.2.55) (11.2.56) (11.2.57)
11.2 Elliptic and theta hypergeometric series
(a; q,phn (a; q,phn (a; q,P)kn
=
= =
(a, aq; q2,P)n, (a, aq, aq2; q3,P)n, (a, aq, ... , aqk-\ qk ,P)n'
Corresponding to the q-binomial coefficient
311 (11.2.58) (11.2.59) (11.2.60)
[~] q defined in Ex. 1.2, we define
the q, P - binomial coefficient (or elliptic binomial coefficient) by
(q;q,P)n
(11.2.61 )
for k = 0, 1, ... ,n. Then
[ ~]
=
[n ~ k]
q,p
= (q-~; q,ph (_qn)kq-(~). q,p
(q, q,p)k
(11.2.62)
For complex a we can employ (11.2.62) to define
[a] k when k
= 0,1, ....
= (q-
q,p
As in Ex. 1.2 it follows that
[~]
(11.2.63)
satisfies the identities q,p
(11.2.64) (11.2.65) (11.2.66) Similarly, additional elliptic identities can be obtained by replacing each
(a; q)n by (a; q,P)n and making any necessary changes in the other identities of Appendix I that do not contain any infinite products. Elliptic analogues of (1.6), (1.41), and of some other identities containing infinite shifted factorials and/or q-gamma functions can be obtained by using the Jackson [1905d] and Ruijsenaars [1997, 2001] elliptic gamma function defined by 00
f(z; q,p)
=
II j,k=O
1 _ z-lqj+lpk+1 1_ z j k ' q P
where z, q, P are complex numbers and Iql, Ipi considered in Ex. 11.12. Since
(11.2.67)
< 1, whose main properties are
q,p) (z,. q,p )n -_ r(zqn; f( z;q,p ) ' n E
'71
~,
(11.2.68)
and (11.2.69)
312
Elliptic, modular and theta hypergeometric series
one can extend the definition of (z; q)cx in (1.6) by defining
( . ) _ r(zqcx; q,p) z, q, p cx - r( z;q,p ) ' and extend the definition of
[~] q
in (1.41) and of
(11.2.70)
[a] k
In (11.2.63) by q,p
defining (11.2.71) for Iql, Ipl < 1 and complex a and /3. The Felder and Varchenko [2003a] elliptic analogue of the Gauss multiplication formula (1.10.10) and of its q-analogue (1.10.11) is considered in Ex. 11.13.
11.3 Additive notations and modular series In what follows it will be convenient to employ the standard notation C for the set of all complex numbers, and Z for the set of all integers. In terms of the elliptic number [a; a, T] defined in (1.6.14) the (additive) elliptic shifted factorials (or a, T-shifted factorials) are defined by
I1~:~[a + k; a, T],
[a; a, T]n
= { 1,
1/ I1;;:~1 [a + n + k; a, TJ,
n = 1,2, ... , n =0, n=-1,-2, ... ,
(11.3.1)
where a, a, TEe and the modular parameters a, T are such that that Im(T) > o and a Z+TZ with Z+TZ = {m+Tn: m,n E Z}. Let [n;a,T]! = [l;a,T]n and
rt
m
[aI, a2,···, am; a, T]n =
II [ak; a, T]n.
(11.3.2)
k=l
Then we can formally define the additive forms of the r+IEr and series to be the series
r+l Vr
r+l er(al,a2, ... ,ar+l;bl , ... ,br ;a,T;z) =
f
[al,a2, ... ,ar+l;a,T]nzn, n=O [1, bl , ... , br ; a, T]n
(11.3.3)
and r+I V r(al;a6,a7, ... ,ar+l;a,T;z)
=
f
n=O
[al + 2n; a, T] [al;a,T]
[aI, a6, a7,···, ar+l; a, T]n n [1,1+al-a6,1+al-a7, ... ,1+al-ar+l;a,T]n z , (11.3.4)
respectively, where, as usual, it is assumed that the parameters are such that each term in these series is well-defined. When the argument z in the r+ I vr
11.3 Additive notations and modular series
313
series equals 1 we suppress it and denote the series in (11.3.4) by the simpler notation r+lVr(al; a6, a7, ... , ar+l; a, T). Since, by (11.2.3) and (11.3.1), [a; a, T]n = (qa; q,P)n qn(b-a l /2 (qb;q,P)n [b;a,T]n
(11.3.5)
and (11.3.6) we have that r+ler(al, ... ,ar+l;b1, ... ,br;a,T;Z) . q p. zq(1+b 1 +·+br-(al +.+ar+ dl /2) = r+l E r (qa 1 , ... , qar+ 1 "• qb 1 ... ,qbr '"
(11.3.7) and
(11.3.8) From (11.2.4) and the multiplicative form of the elliptic balancing condition in (11.2.14), it follows that r+ler is an elliptic hypergeometric series when (11.3.9) which is the additive form of the elliptic balancing condition. Hence, r+ler is called E-balanced when (11.3.9) holds, and it follows that
(11.3.10) holds with the plus sign of the ± when a[al + a2 + ... + ar+l - (1 + b1 + b2 + ... + br )] is an even integer and the minus sign when it is an odd integer. Using (11.2.4) we find that the additive form of the multiplicative well-poised condition (11.2.15) is (11.3.11) in which case r+ler is called well-poised, and the elliptic balancing condition (11.3.9) reduces to (11.3.12) Note that (11.3.9) and (11.3.11) are not equivalent to the additive forms of the "balanced" and "well-poised" constraints in Spiridonov [2002a, (20) and (26)], respectively, which correspond to requiring that the expressions inside the square brackets in (11.3.9) and (11.3.11) are equal to zero.
314
Elliptic, modular and theta hypergeometric series
If r+ler is a well-poised series that can be written in the very-well-poised form (11.3.4), e.g., when, via (11.2.17) and (11.3.6), a5 =
al 1 +T 1+-+-2 2IT' (11.3.13)
then the elliptic balancing condition reduces to r - 5 ] 2IT [ 1 + a6 + a7 + ... + ar+l - -2-(a1 + 1) EZ,
(11.3.14)
which is the condition for the r+lVr series in (11.3.4) to be elliptically balanced (E-balanced) and hence an elliptic hypergeometric series. Since e27ri (1'fl)/4 = ±1, it follows by setting z = qW that the additive form of the very-well-poised balancing condition in (11.2.29) is
r-5 ] 1=f1 IT [W + 1 + a6 + a7 + ... + ar+l - -2-(a1 + 1) + -4-(r + 1) E Z, (11.3.15) which, via (11.3.8), is the condition for the series
(11.3.16) to be very-well-poised-balanced (VWP-balanced) , where, as elsewhere, either both of the upper signs or both of the lower signs in =f1 and ±1 are used simultaneously. In particular, setting w = 0, it follows that if r is odd, then r+l vr(al; a6, a7,···, ar+l; IT, T) is VWP-balanced when
r- 5 IT [ 1 + a6 + a7 + ... + ar+l - -2-(a1 + 1) ] E Z,
(11.3.17)
and if r is even, then r+lVr(al;a6,a7, ... ,ar+l;IT,T;±1) is VWP-balanced when
r-5] 1=f1 IT [ 1+a6+a7+···+ar+1--2-(al+1) +-4- EZ.
(11.3.18)
Thus, by replacing the parameters a, b, c, d, e in (11.2.25) by qa, qb, qC, qd, qe, respectively, and applying (11.3.8), we obtain the additive form of the Frenkel and Turaev summation formula:
lOV9(a; b, c, d, e, -n; IT, T) [a + 1, a + 1 - b - c, a + 1 - b - d, a + 1 - c - d; IT, T]n [a + 1 - b, a + 1 - C, a + 1 - d, a + 1 - b - C - d; IT, T]n for n
= 0, 1, ... , provided that IT[b +
C
(11.3.19)
the series is VWP-balanced, i.e.,
+ d + e - n - 2a - 1] E Z.
(11.3.20)
Similarly, it follows from (11.2.27), (11.2.28), and (11.3.8) that
9V8(a; b, c, d, -n; IT, T; ±1) [a + 1, a + 1 - b - c, a + 1 - b - d, a + 1 - c - d; IT, T]n [a + 1 - b, a + 1 - c, a + 1 - d, a + 1 - b - c - d; IT, T]n
(11.3.21 )
11.3 Additive notations and modular series
315
for n = 0,1, ... , provided that the series is VWP-balanced, i.e., 3 - -1] a [b + c + d - n - -a 2 2
1 =f-1 +4
E Z.
(11.3.22)
This formula also follows by setting e = (a+1)/2+k/2a for k = 0,1 in (11.3.19) and using the fact that [(a+1)/2+k/2a; a, T]n = (-1)k[(a+1)/2-k/2a; a, T]n. In addition, the Frenkel and Turaev transformation formula (11.2.23) has the additive form b, c, d, e, f, A + a + n + 1 - e - f, -n; a, T) + 1, a + 1 - e - f, A + 1 - e, A + 1 - f; a, T]n [a + 1 - e, a + 1- f, A + 1 - e - f, A + 1; a, T]n 12Vl1 (A; A + b - a, A + c - a, A + d - a, e, f, A + a + n + 1 -
12Vl1 (a;
[a
X
e - f, -n; a, T)
(11.3.23)
for n = 0,1, ... , provided that the series is VWP-balanced, i.e., a[b + c + d + A - 2a - 1] E Z.
(11.3.24)
The summation formula (11.3.19) follows from (11.3.23) by setting A = a-d. Frenkel and Turaev [1997] used a slightly different notation for the z = 1 case of the series in (11.3.4) (with r-IWr-2 denoting an r+IVr series when z = 1) and assumed that the series terminates (to avoid the problem of convergence) and satisfies the balancing condition r-5 1 + (a6 + a7 + ... + ar+l) = -2-(a l + 1). (11.3.25) They called the function that is the sum of their terminating series a modular hypergeometric function and justified the name "modular" by showing that if the series r+IVr(al; a6, a7,"" ar+l; a, T) terminates and (11.3.25) holds, then these functions are invariant under the natural action of 8L(2, Z) on the a and T variables, i.e., if aI, a6, a7, ... , ar+l E C, then
b)
aT + a r+IVr ( al; a6, a7,"" ar+l; - - d ' - - d CT+ CT+ = r+I Vr(al;a6,a7, ... ,ar+l;a,T)
for all
(~ ~)
E 8L(2, Z), where 8L(2, Z) is the modular group of
(11.3.26)
2 x 2 ma-
trices such that a, b, c, d E Z and ad - bc = 1. Therefore, we call (11.3.25) the modular balancing condition, and when it holds we say that the series r+IVr is M-balanced. Note that if the series r+IVr(al;a6,a7, ... ,ar+l;a,T) is M-balanced, then it is E-balanced and VWP-balanced, but if it is E-balanced or VWP-balanced then it is not necessarily M-balanced. Frenkel and Turaev also showed that if aI, a6, a7, ... , ar+l E Z, (11.3.25) holds, and the series r+l vr(al; a6, a7, ... ,ar+l; a, T) terminates, then the sum of the series is invariant under the natural action of Z2 = {(m, n): m, n E Z} on the variable a, i.e., r+l vr(al;
a6,···, ar+l; a
+ m + nT, T)
= r+I Vr(al;a6, ... ,ar+l;a,T)
(11.3.27)
316
Elliptic, modular and theta hypergeometric series
for all (m, n) E Z2, and hence is elliptic in a with periods 1 and T. A formal unilateral or bilateral series of the form Ln en (a, T) is called a modular series (or modular invariant) if it is invariant under the natural action of SL(2, Z) on the a and T variables, i.e., (11.3.28)
for all
(~ ~)
E SL(2, Z), where, as usual, two formal series are considered
to be equal if their corresponding nth terms are equal. Spiridonov [2002a] extended (11.3.26) by proving that the unilateral theta hypergeometrie series
(11.3.29) is modular if
r
= s
+ 1,
(11.3.30a) (11.3.30b)
and ai
+ a~ + ... + a;+l = 1 + bi + b~ + ... + b;,
(11.3.30e)
and, more generally, that the bilateral theta hypergeometrie series
(11.3.31 ) is modular if
r = s,
(11.3.32a) (11.3.32b)
and a 21
+ a 22 + ... + a r2 = b21 + b22 + ... + b2r'
(11.3.32e)
where it is assumed that z and the a's and b's are complex numbers such that the series are well-defined. Hence, res and rgs will be called M-balaneed when (11.3.30a)-(I1.3.30c) and (11.3.32a)-(I1.3.32c), respectively, hold. Notice that if rgr is well-poised with k
then
=
1, ...
,r,
(11.3.33)
11.3 Additive notations and modular series
317
and so this well-poised rgr series is modular when the modular balancing condition r (11.3.35) al + a2 + ... + ar = "2 (a + 1) holds. Similarly, a well-poised r+ler series with
k = 1, .. . ,r,
(11.3.36)
is modular when the modular balancing condition al
+ a2 + ... + ar+l =
r+1
-2-(a1 + 1)
(11.3.37)
holds, which reduces to (11.3.25) if (11.3.13) also holds. In order to prove the Frenkel and Turaev, and the Spiridonov modularity results, (11.3.27), and some more general results, we start by observing that, by (1.6.9) and Whittaker and Watson [1965, §21.11 and §21.5]' 'l9 1(x, e7riT) satisfies the modular symmetry relations (11.3.38) (11.3.39) where the square root of -iT is chosen so that its real part is positive, and the quasi-periodicity relations 'l9 1(X 'l9 1(x
+ 7f, e7riT ) = -'l9 1(x, e7riT ) ,
+ 7fT, e7riT ) =
_e-7riT-2ix'l91(X, e7riT ).
(11.3.40) (11.3.41)
Hence, by (1.6.14), the elliptic number [a; a, T] has the modular symmetries
[a; a, T + 1]
=
[a; a, T],
(11.3.42)
[a; a IT, -liT]
=
e7ri (a 2-1)0"2 /T [a; a, T],
(11.3.43)
and the quasi-periodicity relations
+ 1, T] = [k; a + T, T] = [k; a
(_l)k+l [k; a, T],
(11.3.44)
(-1 )k+l e7ri(1-k 2 )(20"+T) [k; a, T],
(11.3.45)
where k E Z. See, e.g., van Diejen and Spiridonov [2000, 2001b], but note that (11.3.43) corrects a typo in both of these papers. Let us first consider when the bilateral series rgs is modular. Since the modular group 8L(2, Z) is generated by the two matrices
(0 -1)
10'
(11.3.46)
which, respectively, correspond to the two transformations (a, T)
----+
(a, T + 1),
(11.3.47)
it suffices to consider when each term (11.3.48)
318
Elliptic, modular and theta hypergeometric series
ofthe series in (11.3.31) is invariant under these transformations. From (11.3.1) and (11.3.42) it is clear that [a; u, T + l]n = [a; u, T]n for nEZ, and hence n E Z.
(11.3.49)
For the second transformation in (11.3.47) observe that from (11.3.1) and (11.3.43) it follows that
nEZ, with vn(a)
(11.3.50)
= n[a 2 + a(n - 1) + (n - 1)(2n - 1)/6 - 1], and thus (11.3.51)
nEZ, with
'\n =n[ai + a~ + ... + a; - (bi + b~ + ... + b;)] + n(n - 1)[al + a2 + ... + ar - (b 1 + b2 + ... + bs )] + n(r - s)[(n - 1)(2n - 1)/6 - 1]. (11.3.52) Consequently, (11.3.53) if and only if (11.3.54) whenever cn(u, T) -I=- O. Obviously, (11.3.54) holds for n = 0 since we have '\0 = 0, and in order for it to hold for any n -I=- 0 with (u, T) replaced by
(u/(CT+d),(aT+b)/(CT+d)) for (:
~)
E
8L(2,Z), it is necessary and
sufficient that '\n = O. Setting'\l = 0 in (11.3.52) gives
a21 + a22 + ... + ar2 = b21 + b22 + ... + b2s + r - s, which, with al series
=
-1, b1
=
(11.3.55)
1, is a necessary and sufficient condition for the
to be modular when ak -I=- 0 for k (11.3.55) and
=
2, ... , r. Setting'\l
=
'\2 = 0 gives
(11.3.56) which, with al = -2, b1 = 1, are necessary and sufficient for the series
to be modular when ak -I=- 0, -1 for k = 2, ... , r. Similarly, if Nand Mare nonnegative integers, then
rgs( -N, a2, a3, ... , ar; M + 1, b2, b3, ... , bs; u, T; z) N
=2:
(11.3.57)
11.3 Additive notations and modular series
319
is modular for arbitrary values of the a's and b's if and only if
An = 0,
-M::::;n::::;N,
(11.3.58)
which, since An is a cubic polynomial in powers of n, is equivalent to (l1.3.32a)(l1.3.32c) if and only if N + M 2:: 3. In particular, when N + M < 3 the restriction r = 8 is not needed for the series in (11.3.57) to be modular. It follows from the above observations that the rgs series in (11.3.31) is modular for arbitrary values of its parameters if and only if (l1.3.32a)-(l1.3.32c) hold. Replacing 8 by 8 + 1 in rgs and then setting bS +1 = 1 yields that the res series in (11.3.29) is modular for arbitrary values of its parameters if and only if (l1.3.30a)-(l1.3.30c) hold. It should be noted that, since the modularity of the above res and rgs series hold termwise under the stated conditions, when we extend res and rgs to
(11.3.59) and
(11.3.60) respectively, where A = {An} is an arbitrary sequence of complex numbers and aI, a2, ... , ar+l, Z E
Tn
rgs(al, ... ,ar;bl, ... ,bs;a+~+nT,T)
= rgs(al, ... ,ar;bs, ... ,bs;a,T)
(11.3.61 )
for all (~, n) E Z2, and hence is doubly periodic in a with periods 1 and Let al, ... , ar, bl , ... , bs E Z. Using (11.3.1) and (11.3.44), we find that cn(a
+ 1, T) =
(-l)'Ync n (a, T)
nEZ,
T.
(11.3.62)
with "In
= n[al + ... + a r - (b l + ... + bs ) + (n + l)(r - 8)/2]
(11.3.63)
and cn(a, T) defined as in (11.3.48). So, cn(a
+ 1, T)
= cn(a, T)
(11.3.64)
if and only if "In E 2Z
(11.3.65)
320
Elliptic, modular and theta hypergeometric series
whenever cn(o-, T) -=I=- 0, where 2Z Via (11.3.1) and (11.3.45),
=
{2m: mE Z} is the set of all even integers.
k, nEZ, (11.3.66) with f-ln(k)
= -n[k 2 + k(n - 1) + (n - 1)(2n - 1)/6 - 1], and hence cn(o- + T, T) = (_1)"'ne-7riAn(2u+T)cn(0-, T) (11.3.67)
for n E Z with An as defined in (11.3.52). Thus, if (11.3.65) holds, then (11.3.68) if and only if (11.3.69) when cn(o-, T) -=I=- O. Clearly, (11.3.69) holds for n = 0 since Ao = 0, and in order for it to hold for any n -=I=- 0 with 0- replaced by 0- + m + mT for (m, n) E Z2, it is necessary and sufficient that An = o. As in the modular case, it follows that if Nand M are nonnegative integers, then rgs( -N, a2, a3, ... , ar; M
+ 1, b2, b3, ... , bs ; 0-, T; z)
N
=2:
(11.3.70)
is invariant under the natural action of Z2 on the variable 0- for arbitrary values of the a's and b's if and only if
An
= 0,
"In E 2Z,
-M:=::;n:=::;N,
(11.3.71)
which is equivalent to (l1.3.32a)-(l1.3.32c) when N + M ~ 3. Consequently, the rgs series in (11.3.31) is invariant under the natural action of Z2 on the variable 0- for arbitrary values of its parameters if and only if (l1.3.32a)(l1.3.32c) hold. Replacing s by s + 1 in rgs and then setting bs+1 = 1 yields that the res series in (11.3.29) is invariant under the natural action of Z2 on the variable 0- for arbitrary values of its parameters if and only if (l1.3.30a)(l1.3.30c) hold. In addition, since the invariance under the natural action of Z2 on the variable 0- in the above res and rgs series hold termwise under the stated conditions, it follows that the series in (11.3.59) and (11.3.60) are invariant under the natural action of Z2 on the variable 0- when (l1.3.30a)(l1.3.30c) and (l1.3.32a)-(l1.3.32c), respectively, hold. Spiridonov [2002a] called a (unilateral or bilateral) theta hypergeometric series L Cn totally elliptic if g( n) = cn+d Cn is an elliptic function of all free parameters entering it (except for z) with equal periods of double periodicity. He proved that the most general (in the sense of a maximal number of independent free parameters) totally elliptic theta hypergeometric series coincide with the terminating well-poised M-balanced rer-l (in the unilateral case) and rgr (in the bilateral case) for r > 2.
11.4 Elliptic analogue of Jackson's
8¢7
11.4 Elliptic analogue of Jackson's
summation formula
8¢7
321
summation formula
Recall from (11.2.25) that the multiplicative form of the Frenkel and Thraev summation formula (11.3.19) is
v; ( . b
d -no ) _ (aq, aq/be, aq/bd, aq/ed; q,P)n a, ,e, ,e, q ,q,p - (aq /b ,aq / e,aq /d ,aq /b ed ;q,p )n
10 9
(11.4.1)
with a 2qn+l = bede and Ipi < 1. When p = 0 it clearly reduces to Jackson's summation formula (2.6.2). We will give a simple inductive proof of (11.4.1), which is a modification ofthose found independently by Rosengren, Spiridonov, and Warnaar. When n = 0 both sides of (11.4.1) are equal to 1, and for n = 1 it becomes 1 ()(aq2;p)(a, b, e, d, a 2q2/bed, q-\ q,ph + ()(a;p)(q, aq/b, aq/e, aq/d, bed/aq,aq2; q,p)l q (aq, aq/be, aq/bd,aq/ed;q,p)l (aq/b,aq/e,aq/d,aq/bed;q,p)l·
(11.4.2)
By using (11.2.5) and (11.2.42), (11.4.2) can be rewritten as ()(aq/b, aq/e, aq/d, aq/bed;p) + ()(b, e, d, a 2q2 /bed; p)aq/bed
= ()(aq, aq/be, aq/bd, aq/ed;p).
(11.4.3)
However, (11.4.3) is equivalent to the identity in Ex. 2.16(i) if we replace x, A, /-l and v by aq/(be)!, (be)!, aq/d(be)! and (e/b)!, respectively. So (11.4.1) is true for n = 1. Let us now assume that (11.4.1) is true when n = m. We need to prove that m+l
~
()(aq2k;p)(a, b, e, d, a 2qm+2/bed, q-m-\ q,ph k ()(a;p)(q, aq/b, aq/e, aq/d, bedq-m-l la, aqm+2; q,p)k q
(aq, aq/be, aq/bd, aq/ed; q,P)m+l (aq/b, aq/e, aq/d, aq/bed; q,P)m+l .
(11.4.4)
Observe that, by (11.2.43), (a 2qm+2/bed, q-m-\ q,ph (bedq-m-l/a, aqm+2; q,p)k (a 2qm+l /bed, q-m; q,ph (bedq-m / a, aqm+l; q, ph
x [()(a2qm+k+l/bed,bedqk-m-l/a,aqm+l,q-m-\p)] . ()(aqm+k+l, qk-m-l, a2qm+l /bed, bedq-m-l /a;p)
(11.4.5)
However, by Ex. 2.16(i) the ratio of the ()-terms inside the brackets in (11.4.5) is equal to . 1 + bedq-m-l ()(aqk,qk,a/bed,a 2q2m+2/bed;p) a ()( aqm+k+l, qk-m-l, a2qm+l /bed, bedq-m-l / a; p) (11.4.6) When we multiply (11.4.6) by ()(aq2k;p)(a, b, e, d, a 2qm+l /bed, q-m; q,p)k k ()(a;p)(q, aq/b, aq/e, aq/d, bedq-m la, aqm+l; q,P)k q
322
Elliptic, modular and theta hypergeometric series
and sum over k from 0 to m+ 1 we find that the first term in (11.4.6) contributes a zero term at k = m + 1 because (q-m; q,P)m+l = 0, while the second term contributes a zero term at k = 0 because of the factor ()(qk;p). After some simplifications it follows that b, e, d, a 2qm+2/bed, q-m-l; q,p) 10 Vg(a; b, e, d, a 2qm+l /bed, q-m; q,p)
10 Vg(a;
=
bedq-m + _-=--_ a
x
(aq; q,ph()(b, e, d, a/bcd, a 2q2m+2/bed;p)
-,-_-'------=,---=c-"--'--_'-----,---,--'------'-----,--_,------::--,---------'--,--------'--''--'---------,--
(aqm+l, bedq-m-l fa; q,ph()(aq/b, aq/e, aq/d;p) bq, eq, dq, a 2qm+2 /bed, q-m; q,p). (11.4.7)
10 Vg(aq2;
However, by the induction assumption both 10 Vg series on the right side of (11.4.7) can be summed. Thus, from (11.4.7) and the n = m case of (11.4.1), 10 Vg(a;
b, e, d, a 2qm+2/bed, q-m-l; q,p)
(aq,aq/be, aq/bd,aq/ed; q,P)m (aq/b,aq/e,aq/d,aq/bed;q,p)m bedq-m
(aq; q,ph()(b, e, d, a/bcd, a 2q2m+2/bed;p)
a
(aqm+l, bedq-m-l fa; q,ph()(aq/b, aq/e, aq/d;p)
+ --=--- -;---'--:-:~"::"':--~:--;---'------'----,---::-c,----=-;:------',-;-----'--'c..:--:-----,(aq3,aq/be,aq/bd,aq/ed;q,p)m
x (aq2/b,aq2/e,aq2/d,a/bed;q,p)m (aq,aq/be,aq/bd,aq/ed;q,p)m (aq/b,aq/e,aq/d,aq/bed;q,p)m ()(b, e, d, a 2q2m+2 /bed; p)
x
[1 _
]
()(aqm+l/b, aqm+l /e, aqm+l /d, bedq-m-l /a;p)
,
(11.4.8)
by use of the identities (11.2.42) and (11.2.43). Applying Ex. 2.16(i) once again we find that
1_
()(b, e, d, a 2q2m+2 /bed; p) ()(aqm+l /b, aqm+l/e, aqm+l /d, bedq-m-l /a;p) ()( aqm+l, aqm+l /be, aqm+l /bd, aqm+l / cd; p) ()(aqm+l/b, aqm+l /e, aqm+l /d, aqm+l /bed;p)'
(11.4.9)
Substituting (11.4.9) into (11.4.8) and using (11.2.43) shows that (11.4.1) is true for n = m+ 1 whenever it is assumed to be true for n = m. This completes the proof of (11.4.1) by induction. Another proof of (11.4.1) by induction is indicated in Ex. 11.4. Note that by letting e ----+ aqn+l in (11.4.1) we obtain the summation formula for a truncated 8 V7 series
t
k=O
()(aq2k;p)
(a, b, e, a/be; q,ph
()(a;p)
(q, aq/b, aq/e, beq; q,P)k
(aq,bq,eq,aq/be;q,p)n (aq/b,aq/e,beq,q;q,p)n'
which is an elliptic analogue of Ex. 2.5.
qk
(11.4.10)
11.5 Elliptic analogue of Bailey's transformation formula
323
As alluded to in §11.2 the limit of the summation formula (11.4.1) as one of three parameters b, e, d tends to infinity does not exist when p -=I=- 0, even though these limits do exist when p = 0 yielding formula (2.4.2). Similarly, one cannot take the limit a ----7 0 after replacing d by aq/d in (11.4.1) to get an elliptic analogue of the q-Saalschiitz formula (1.7.2), as was done in the basic hypergeometric case in §2.7. The root cause of the nonexistence of these limits is the fact that lima--+o ()( a; p) and lima--+oo ()( a; p) do not exist in the extended complex plane when p -=I=- O. Nevertheless, one can obtain an elliptic analogue of the q-Saalschiitz formula (1.7.2) at the 10 Vg level by employing Schlosser's observation (see Warnaar [2003e]) that by making the simultaneous substitutions {a,e,d,e,p} ----7 {dp,dqp/c,a,ep,p2} in (11.4.1) we obtain the summation formula
-n 2) (c/a, c/b,dqp, dqp/ab; q,p2)n ( lO V9 dp;a,b,dqp / c,ep,q ;q,p = (c, c / ab, dqp / a, dqp /b ; q, P2) n
(11.4.11)
= abe, Ipi < 1, and n = 0, 1, ... , which tends to (1.7.2) as p
with cdqn
----7
O.
11.5 Elliptic analogue of Bailey's transformation formula for a terminating 1O¢9 series Recall from (11.2.23) that the multiplicative form of Frenkel and Turaev's transformation formula (11.3.23) is 12 V l1 (a;
b, c, d, e, j, Aaqn+1 /ej, q-n; q,p) (aq,aq/ej,Aq/e, Aq/j;q)n (aq/e,aq/j,Aq/ej,Aq;q)n x 12 Vl1 (A; Ab/a, Ac/a, Ad/a, e, j, Aaqn+1/ej, q-n; q,p)
(11.5.1)
with A = qa 2/bcd. This formula reduces to the Frenkel and Turaev summation formula (11.4.1) when aq = be, bd, or ed, and it reduces to Bailey's transformation formula (2.9.1) when p = O. It can be proved by proceeding as in (2.9.2) and (2.9.3) with the q-shifted factorials and the S¢7 and 1O¢9 series replaced by q,p-shifted factorials and lOV9 and 12Vl1 series, respectively. In view ofthe importance of this transformation formula and for the sake of completeness, we decided to present its proof here. First observe that by (11.4.1) 10 Vg(A;
Ab/ a, Ac/a, Ad/a, aqm, q-m; q,p) (Aq, b, c, d; q,P)m (a/A, aq/b, aq/c, aq/d; q,P)m'
and hence the 12Vl1 series on the left side of (11.5.1) equals ~ ()(aq2m;p)
~ m=O
()( a; p)
f
(a, e, j, Aaqn+l /ej, q-n; q,P)m (a/A; q,P)m m (q, aq/ e, aq/ j, ejq - n / a, aqn+l; q, P)m (Aq; q, P)m q
()(Aq2j;p) (A, Ab/a, Ac/a, Ad/a, aqm, q-m; q,p)j j x j=O ()(A;p) (q, aq/b, aq/c, aq/d, Aql-m la, Aqm+l; q,p)j q
(11.5.2)
324
Elliptic, modular and theta hypergeometric series
=
tf m=O j=O
()(aq2m.;p)()(~q2j;p) (a; q.'P)m+j(a/~; q,P)m-j qm ()(a, p)()(A,p) (Aq, q,P)m+j(q, q, P)m-j
(e, j, Aaqn+l/ej, q-n; q,P)m(A, Ab/a, Ac/a, Ad/a; q,p)j (a)j x (aq/e, aq/ j, ejq-n / A, aqn+1; q, P)m(q, aq/b, aq/c, aq/d; q,p)j >:
=
t
j=O
()(Aq2j;p) (A, Ab/a, Ac/a, Ad/a, e, j, Aaqn+l /ej, q-n; q,p)j ()(A;p) (q, aq/b, aq/c, aq/d, aq/e, aq/ j, ejq-n / A, aqn+l; q, p)j
2' .. + '+1 . (aq' q phj (aq)j x (A : ' ) .""\ lQVg(aq J;a/A,eqJ,jqJ,Aaqn J /ej,qJ-n;q,p). q, q,p 2J A (11.5.3)
However, by (11.4.1)
10Vg(aq2j; a/A,eqj, jqj, Aaqn+j+1/ej,qj-n;q,p) (aq2j +l,Aqj+l/e,Aqj+1/j,aq/ej;q,P)n_j (Aq2j +l,aqj+l/e,aqj+l/j,Aq/ej;q,P)n_j (aq,Aq/e,Aq/j,aq/ej;q,P)n (Aq,aq/e,aq/j,Aq/ej;q,P)n x (Aq;qhj(aq/e,aq/j,ejq=n/A;q)j (~)j. (aq;qhj(Aq/e,Aq/j,ejq n/a;q)j a
(11.5.4)
Substitution of (11.5.4) into (11.5.3) gives (11.5.1). Note that by applying the transformation formula (11.5.1) to the 12VU series on the right side of (11.5.1), keeping e, Ac/a, Ad/a and q-n unchanged, we obtain that
12Vu(a; b, c, d, e, j, Aaqn+1 /ej, q-n; q,p) (aq,aq/ce, aq/de,aq/ej,Aq/j,b; q,P)n (aq/c, aq/d,aq/e, aq/j,Aq/ej,b/e;q,P)n x 12 Vu(eq-n /b; e, Ac/a, Ad/a, aq/bj, eq-n la, ejq-n /Ab, q-n; q,p), (11.5.5) which is the elliptic analogue of Ex. 2.19. Setting j = qa 2 /bcde in (11.5.1) gives the transformation formula for a truncated 10 Vg series ~
()( aq2k;p)(a,b,c,d,e,qa 2/bcde;q,ph k q ~ ()(a;p)(q, aq/b,aq/c, aq/d, aq/e, bcde/a; q,P)k k=O (aq, bcd/a, a2q2/bcde,eq; q,P)n (aq/e,bcde/a, a2q2/bcd,q; q,P)n x 12 Vu (qa 2/bcd; aq/bc, aq/bd, aq/cd, e, qa 2/bcde, aqn+l, q-n; q,p) (11.5.6) in which the 12 Vu can be transformed via (11.5.1) and (11.5.5) yielding other 12 Vu series representations for the above truncated series.
11.6 Multibasic summation and transformation formulas
325
11.6 Multibasic summation and transformation formulas for theta hypergeometric series First observe that by replacing a in (11.4.1) by a/q it follows that the n case of (11.4.1) is equivalent to the identity
1_
B(a,a/bc,a/bd,a/cd;p) B(a/bcd,a/d,a/c,a/b;p)·
B(b,c,d,a 2 /bcd;p) B(a/b,a/c,a/d,bcd/a;p)
=
1
(11.6.1)
Next, corresponding to (3.6.12), define
m:::;n, = n+ 1, 2: n + 2,
m m
(11.6.2)
for n, m = 0, ±1, ±2, ... , and let
II
ll(b k, Ck, d k, a2/b k kCk d kiP )
n-l Un
U
= k=O B(ak/bk,ak/Ck,ak/dk,bkckdk/ak;p)'
(11.6.3)
for integer n, where it is assumed that the a's, b's, c's, d's are complex numbers such that Un is well defined for n = 0, ±1, ±2, .... Then, by using (11.6.1) with a, b, c, d replaced by ak, bk , Ck, dk , respectively, we obtain the indefinite summation formula n
U- m - Un+l
=
L
(Uk - Uk+l)
k=-m (11.6.4) for n, m = 0, ±1, ±2, .... Since Uo = 1 by (11.6.2), setting m = 0 in (11.6.4) gives the summation formula ~ B(ak,ak/bkck,ak/bkdk,ak/ckdk;p) ~ B(ak/bkckdk,ak/dk,ak/ck,ak/bk;p)
k=O
x
= for n p
IT
B(bj , Cj, dj , a;/bjcjdj;p) . j=O B(aj/bj , aj/cj, aj/dj , bjcjdj/aj,p)
1_
IT
B(bj , Cj, dj , a;/bjcjdj;p) j=O B(aj/bj , aj/cj, aj/dj , bjcjdj/aj;p)
(11.6.5)
= 0,1, ... , which is equivalent to formula (3.2) in Warnaar [2002bj. When
= 0 this formula reduces to a summation formula of Macdonald that was
first published in Bhatnagar and Milne [1997, Theorem 2.27]' and it contains the summation formulas in W. Chu [1993, Theorems A, B, C] as special cases. Notice that in (11.6.1), (11.6.3), (11.6.4) and (11.6.5) we have arranged the components of each quotient of products of theta functions so that the well-poised property of these quotients is clearly displayed; e.g., in (11.6.5)
326
Elliptic, modular and theta hypergeometric series
the quotients of the theta functions that depend on k are arranged so that each product of corresponding numerator and denominator parameters equals aVbkckdk, and each of the corresponding products that depend on j equals If we set
ak = ad(rst/q)k, bk = brk, Ck = cs k , d k = ad2t k /bc, then
Un
_
Un
reduces to
(a;rst/q2,P)n(b;r,P)n(c;s,P)n(ad 2/bc;t,P)n (dq;q,P)n(adst/bq;st/q,P)n(adrt/cq;rt/q,P)n(bcrs/dq;rs/q,P)n
= ~--~~~~~~~~~~~~~~~~~~--~~-
and it follows from (11.6.4) by applying (11.2.42), (11.2.43) and (11.2.49) that we have the Gasper and Schlosser [2003] indefinite multibasic theta hypergeometric summation formula
n
L
k=-m
O(ad(rst/q)k, brk/dqk, csk/dqk, adtk/bcqk;p) O(ad, bid, c/d, ad/bc;p)
(a; rst/q2 ,ph(b; r, p)k(C; s,ph(ad 2/bc; t,ph k x (dq; q,p)k(adst/bq; st/q,p)dadrt/cq; rt/q,p)k(bcrs/dq; rs/q,p)k q O(a,b,c,ad2/bc;p) dO(ad, bid, c/d, ad/bc;p) (arst/q2;rst/q2,P)n(br;r,P)n(cs;s,P)n(ad2t/bc;t,P)n { x (dq; q,P)n(adst/bq; st/q,P)n(adrt/cq; rt/q,P)n(bcrs/dq; rs/q,P)n _ (c/ad; rt/q,P)m+l (d/bc; rs/ q,P)m+l (l/d; q, P)m+l (b/ad; st/q, P)m+l } (l/c; s, P)m+l (bc/ad 2; t, P)m+l (l/a; rst/q2, P)m+l (l/b; r, P)m+l (11.6.6) for n, m = 0, ±1, ±2, . .. . Formula (3.6.13) follows from (11.6.6) by setting P = 0 and then setting r = P and s = t = q. If P = 0 and
max(lql, Irl, lsi, Itl, Irs/ql, Irt/ql, Ist/ql, Irst/q 21) < 1, then we can let n or m in (11.6.6) tend to infinity to obtain that this special case of (11.6.6) also holds with nand/or m replaced by 00, just as in the special case (3.6.14). Even though one cannot let n ---; 00 or m ---; 00 in (11.6.6) when P -I=- 0 to obtain summation formulas for nonterminating theta hypergeometric series since lima-->o O( a; p) does not exist when P -I=- 0, it is possible to let n ---; 00 or m ---; 00 in (11.6.4) to obtain summation formulas for nonterminating series containing products of certain theta functions. For example, if we let O(bk,ck,dk,a~/bkckdk;P)
Z k = -:-;-----,::-'-c.:.:.....:.,.:---'--'-'----':''':--'-':--''--'-'-:--=--';__---,-
O(ak/bk,ak/ck,ak/dk,bkckdk/ak;p)
denote the kth factor in the product representation (11.6.3) for that when a is not an integer power of P lim
1
b ..... a'l
O(b;p) O(a/b;p)
=
1
,
Ipi < 1,
Un
and observe
11.6 Multibasic summation and transformation formulas
327
then it is clear that there exist bilateral sequences of the a's, b's, c's, and d's in (11.6.4) such that Re Zk > 0 for integer k and the series 00
L
log Zk
converges,
(11.6.7)
k=-oo
where log Zk is the principal branch of the logarithm (take, e.g., bk , Ck, and 1
af,
that Ilogzkl < l/k 2 for k = ±1,±2, ... ). Then limn---> 00 Un and limm--->oo U- m exist, and we have the Gasper and Schlosser [2003] bilateral summation formula d k so close to
provided that (11.6.7) holds. However, such bilateral sums do not appear to be particularly useful. It is more useful to use the m = 0 case of (11.6.6) in the form
n ()(ad(rst/q)k, br k/dqk, cs k/dqk, adt k/bcqk;p) ()(ad, bid, e/d, ad/be;p) k=O
L
(a;rst/q2,p)k(b;r,p)k(e;s,p)k(ad 2/be;t,p)k k x (dq;q,p)k(adst/bq;st/q,p)k(adrt/eq;rt/q,p)k(bers/dq;rs/q,p)k q ()(a,b,e,ad 2/be;p) d ()(ad, bid, e/d, ad/be; p) (arst/q2;rst/q2,P)n(br;r,P)n(es;s,P)n(ad 2t/bc;t,P)n x ~--~~~-=~~~~~~~--~~~~~--~~(dq;q,P)n(adst/bq;st/q,P)n(adrt/eq;rt/q,P)n(bers/dq;rs/q,P)n ()(d,ad/b,ad/c,be/d;p) (11.6.9) d ()(ad, b/ d, e/d, ad/be; p) , its e = s-n special case
n ()(ad(rst/q)k, br k/dqk, sk-n /dqk, adsnt k/bqk;p) ()(ad,b/d,s-n/d,adsn/b;p) k=O
L
(a; rst/q2,p)k(b; r,p)k(S-n; s,p)k(ad2sn /b; t,p)k k q (dq;q,p)k(adst/bq;st/q,p)k(adsnrt/q;rt/q,p)k(brs1-n/dq;rs/q,p)k ()(d,ad/b,adsn,dsn/b;p) (11.6.10) ()(ad,d/b,dsn,adsn/b;p) ,
x
328
Elliptic, modular and theta hypergeometric series
and the d
----t
1 limit case of (11.6.10)
n ()(a(rst/q)k, brk /qk, sk-n /qk, asnt k /bqk;p) ()(a b s-n asn/b'p) k=O "" (a; rst/q2 ,ph(b; r, p)k(S-n; s,p)k(as n /b; t, P)k k X (q; q, p)k(ast/bq; st/q, p)k(asnrt/q; rt/q, p)k(brs 1- n /q; rs/q,p)k q = on,O, (11.6.11)
L
where n = 0,1, ... , which are generalizations of (3.6.15), (3.6.16), and (3.6.17), respectively. In particular, replacing n, a, b, and k in the s = t = q case of (11.6.11) by n - m, armqm, brmq-m, and j - m, respectively, gives the orthogonality relation n
L
anjbjm
=
(11.6.12)
on,m
j=m with
(11.6.13) (11.6.14) which shows that the triangular matrix A = (anj) is inverse to the triangular matrix B = (b jm ), and gives a theta hypergeometric analogue of (3.6.18)(3.6.20). Proceeding as in the derivation of (3.6.22), it follows that (3.6.22) extends to the bibasic theta hypergeometric summation formula
()(a/r, b/r;p)
t
(aqk, bq-k; r'P)n_1()~aq2k /b;p) (_l)kq(~) k=O (q; q,ph(q; q,P)n-k(aq /b; q,P)n+1
= On 0 ' (11.6.15)
for n = 0, 1, ... , which reduces to
8V7(a/b;q/b,aqn-l,q-n;q,p)
=
on,O
when r = q. The summation formula (11.6.10) and the argument in §3.8 can be employed to extend (3.8.14) and (3.8.15) to the quadratic theta hypergeometric transformation formulas n ()( acq3k; p) (a, b, cq/b; q,P)k(f, a 2c2q2n+1 / f, q-2n; q2,P)k k
{; ()(ac;p)
(cq2, acq2/b, abq; q2,p)k(acq/ f, f / acq2n, acq2n+1; q,p)k q
(acq;q,P)2n(ac 2q2/bf,abq/f;q2,P)n (acq/f;q'P)2n(abq,ac 2q2/b;q2,P)n x 12 Vn (ac 2/b; f, ac/b, c, cq/b, cq2/b, a2c2q2n+1 / f, q-2n; q2,p)
(11.6.16)
and
2n ()(acq3k; p) (d, f, a2c2q/ df; q2 ,ph(a, cq2n+1, q-2n; q,P)k k {; ()(ac;p) (acq/d, acq/ f, df lac; q,p)k(cq2, aql-2n, acq2n+2; q2,p)k q
11.6 Multibasic summation and transformation formulas
329
(acq,acq/d!;q,P)n(acq1-n/d,acq1-n/!;q2,P)n (acq/d,acq/!;q,P)n(acq1-n,acq1-n/d!;q2,P)n x 12 Vu (acq-2n-\ c, d,!, a2c2q/d!, aq-2n-\ ql-2n, q-2n; q2, p) (11.6.17) for n = 0,1, ... ; see Warnaar [2002b, Theorems 4.2 and 4.7]. As in the derivation in Gasper [1989a] of the quadbasic transformation formula in Ex. 3.21, indefinite summation formulas such as in (11.6.5) and (11.6.9) can be extended to transformation formulas by using the identity
n n-k n n-k LAkLAj = LAkLAj, k=O j=O k=O j=O
(11.6.18)
which follows by a change in order of summation. In particular taking Ak to be the kth term in the series in (11.6.5) and Ak to be this term with ak, bk, Ck, d k , and P replaced by A k , Bk, Ck, Dk, and P, respectively, yields the rather general transformation formula
~ B(ak,ak/bkck,ak/bkdk,ak/ckdk;p) ~ B(ak/bkckdk,ak/dk,ak/ck,ak/bk;p) k=O x
IT
x
{I-IT
B(bj , Cj, dj , a;/bjcjdj;p) . j=O B(aj /bj , aj /Cj, aj /dj , bjcjdj /aj ,p) B(Bj,Cj,Dj,A;/BjCjDj;P) } j=O B(Aj/Bj , Aj/Cj , Aj/Dj , BjCjDj/Aj ; P)
= ~ B(Ak' A k/ BkCk, A k/ BkDk, Ak/CkDk; P) ~ B(Ak/ BkCkDk, A k/ Dk, Ak/Ck, A k/ B k; P) x
IT
B(Bj,Cj,Dj,A;/BjCjDj;P). j=O B(Aj / B j , Aj/Cj , Aj/Dj , BjCjDj/Aj , P)
x { 1-
nrr-k B(bj , Cj, dj , a;/bjcjdj;p) } . j=O B(aj/bj , aj/cj, aj/dj , bjcjdj/aj;p) (11.6.19)
The special case of (11.6.19) corresponding to using (11.6.9) instead of (11.6.5) contains the parameters a, b, c, d, A, B, C, D, and the bases r, s, t, R, S, T, the nomes P and P, and it contains the quadbasic transformation formula in Ex. 3.21 as a special case (see Ex. 11.26). By proceeding as in Gasper and Schlosser [2003] we can use (11.6.11) to derive multibasic extensions of the Fields and Wimp, Verma, and Gasper expansion formulas in (3.7.1)-(3.7.3), (3.7.6)-(3.7.9), and multibasic theta hypergeometric extensions of (3.7.6)-(3.7.8). Set a = 'Y(rst/q)j and b = a(r/q)j in (11.6.11). For j, n = 0,1, ... , let Bn(P) and Cj,n be complex numbers such that Cj,o = 1 and the sequence {Bn (p)} has finite support when P -=I=- O. Then,
Elliptic, modular and theta hypergeometric series
330
as in (3.7.5), it follows that Bj(p)x j n
00
= {; ~
,a-
,a-
1 snHtj, 1 (st)nH, s-kqj-n;p) ()(f(rst/q)n, a(r /q)n, ()(sj-n-k;p)(q; q,P)n(frtsn+k /q; rt/q,P)n(ars 1- n - k /q; rs/q,P)n
x (fa-1(stt+lqj-n-\ st/q, p)k-1 (fa- 1snHtJ+\ t, P)n-j-l x (arq-j; r, P)n-1 (frstqj-2; rst/q2 ,P)n-1 (q-n; q,p)j
x (_l)n Bn+k(p)Cj,nH_jXnHqn(l+j-n-kHG)
(11.6.20)
for j = 0,1,00 .. Now multiply both sides of (11.6.20) by Ajwj/(q;q,p)j and sum from n = 0 to 00 to obtain the Gasper and Schlosser multibasic expansion formula
n ()( s -k qj-n ;P) (,rst qj-2 ;rst/ q2) (arq ,p L . k j=O ()(sJ-n- ;p)(q; q,p)j n-1
X
- j ;r,p) n-1
x ()(fa-1snHtj; p)(fa- 1(st)n+1qj-n-\ st/q,p)k-1
x (fa- 1sn+ktJ+\ t,P)n-j-1AjCj,n+k-jW j qn(j-n-k) ,
(11.6.21 )
which reduces to (3.7.6) by setting P = 0 and then letting r = P and s = t = q. Setting r = s = t = q in (11.6.21) yields an expansion formula that is equivalent to the following extension of (3.7.7) fAnBn(P) (xw)n n=O (q; q,P)n 00
X '"'
~
x~
()(
,q
=
f
n=O
(::f
(a"qn+1/ a , a,(3; q,P)n (q, ,qn; q,P)n a
2n+2k/. )( n/ -1 n (3 n. ) a,p ,q a, a ,aq, q ,q,p k Bn+k(P)Xk ()(fq2n /a;p)(q, ,q2n+1; q,p)k (q-n"qn;q,p)j
~ (q, ,qn+1 la, q1-n la, a, (3; q,p)j
k(wq)j J
,
(11.6.22)
where, as above, {Bn(P)} has finite support when P -=I=- O. Of course, one cannot let a ----+ 00 in (11.6.22) to get an extension of (3.7.3) that holds for any P -=I=- O. Analogous to the q-extension of the Fields and Wimp expansion formula (3.7.1) displayed in (3.7.8), from (11.6.22) one easily obtains the rather general theta hypergeometric expansion formula
f
(aR' CT; q,P)n AnBn(p)(xwt n=O (q,bs,dU;q,P)n
11.7 Rosengren's elliptic extension of Milne's fundamental theorem
n (-n x" q ,,,{qn ,aR, f M;q,P )j A.(wq)j f;:o (q,,,{qn+1/ u ,q1-n/u ,bs ,eK;q,p)j J
331
(11.6.23) ,
where we used a contracted notation analogous to that used in (3.7.1) and (3.7.8), and to avoid convergence problems it is assumed that {En} has finite support when Pi=- O. Additional formulas are given in the exercises.
11. 7 Rosengren's elliptic extension of Milne's fundamental theorem
Milne's [1985a] fundamental theorem states that
IT
~(zqk) (arzs/zr ; qhs = (a1 ... an; q)N ~(z) r,s=1 (qzs/zr;qk (q;q)N'
(11.7.1)
and the a's and z's are fixed parameters. This is the identity that played a fundamental role in Milne's derivation of the Macdonald identities for the affine Lie algebra A~1), as well as his general approach to hypergeometric series on An or U(n). An important tool for proving (11.7.1) is an easily verifiable identity n
I1(b j - a k) " j=1 ~ ak I1 (aj - ak) k=1 #k n
=
b1 ... bn a1 ... an - 1.
(11. 7.3)
To derive an elliptic extension of Milne's identity, Rosengren [2003c] used the following elliptic extension of (11.7.3) n
n
I1
B(ak/bj;p) j=1 _ 0 " k~1 I1 B(ak/aj;p) - , -
(11. 7.4)
jf.k
where it is assumed that the balancing condition a1 ... an = b1 ... bn holds. Note that (11.7.4) is the same identity as in Ex. 5.23. Slater [1966] gave a proof of it by using special relationships between the parameters in the general transformation formula (5.4.3). Also see Tannery and Molk [1898] for a simple proof via residues. However, since Rosengren [2003c] gave a rather elegant yet
332
Elliptic, modular and theta hypergeometric series
elementary proof of (11.7.4) that is similar in spirit to the way Milne proved (11.7.3), we will present his proof. First, it is easy to see that Ex. 2.16(i) is equivalent to the n = 3 case of (11.7.4). Assume that (11.7.4) is true for n = m. Then, by separating the (m + 1)-th term from the series on the left side of (11.7.4), we can rewrite it in the form m-l m-l ()(am/bm;p) IT ()(am/bj;p) m-l ()(ak/bm;P) IT ()(ak/bj;p) ()(ak/am;P) ji11 ()(ak/aj;p) = m-l j=1 , (11.7.5) k=1 #k IT ()(am/aj;p) j=1
L
where the a's and b's are always assumed to satisfy the balancing condition. Considered as a function of am the expression on the left side of (11.7.5) resembles a partial fraction expansion of the product on the right side. When n = m + 1, we write am +l = t, say, and seek an expansion of the form
f
IT
()(t/bj;p) = Ok ()(b1 ··· bma k/ a l··· amt;p) j=1 ()(t/aj;p) k=1 ()(ak/t;p) .
(11. 7.6)
That such an expansion exists follows by using induction on m and the fact that the m = 2 case is equivalent to (11.4.3). Multiplying both sides of (11.7.6) by ()(t/ak;p) = -()(ak/t;p)t/ak and setting t = ak, we find that m
IT Ok
= _
()(ak/bj;p)
...,.-;-::_----:_-:--J_. =_1_ _~=-_-:-;-----;-_---:-
()(b1 ··· bm /al ... am;p) IT#k ()(ak/aj;p)
()( am+dbm+1; p) with al··· am+l = a m +l to get
=
(11.7.7)
IT ()( ak/ aj; p)
#k b1 ··· bm+1. Now substitute (11.7.7) into (11.7.6) and set
t
m
m ()(ak/bm+1;p)
L
k=1
IT
m
()(ak/bj;p)
1 ()(ak/am+l;P) ji1 ()(ak/aj;p) #k
()(am+dbm+1;p) = -
IT
()(am+dbj;p)
fi () (am+lj=~ aj;p )
j=1
which is the same as (11.7.5) with m replaced by m + 1. This completes the proof of (11.7.4). An elliptic extension of (11.7.1) given in Rosengren [2003c, Theorem 5.1] states that
(b/al, ... , b/an +l; q,p)N (q,bz 1, ... ,bzn ;q,p)N '
(11. 7.8)
11.7 Rosengren's elliptic extension of Milne's fundamental theorem
where
II
~(z;p) = ~n(Z;P) =
zrB(zs/zr;P)
333
(11. 7.9)
l:S;r<s:S;n is the elliptic analogue of the Vandermonde determinant in (11.7.2) and the parameters satisfy the balancing condition (11.7.10) If we set P = 0 and take the limits b ----+ 0, a n+1 ----+ 0 such that b/an+1 ----+ a1 ... an Zl ... Zn, then it is easy to see that (11.7.8) approaches Milne's limit (11.7.1). When P = 0, (11.7.8) reduces to Milne's [1988a, Theorem 6.17] An Jackson summation formula, which is a multivariable extension of Jackson's sum (2.6.2). To prove (11.7.8) by induction, observe that when N = 1 we have that k i = Dij for some j, and thus j can be used as the summation index. Then, after a bit of manipulation, 1 n
~(z;p)
TI
(qzr!zs;q,phr
r,s=l
and formula (11.7.8) becomes
n+1
L
TI
n+1
TI B(b/as;p) s=l = -=-s=--=,-l_ _ __ (11.7.11) "_ B(bzj;p) TI B(zj/zs;p) TIn B(b . ) , )-1 sopj Zs,p s=l which is the m = n + 1 case of (11. 7.5) with a different set of parameters. So (11.7.8) is true when N = 1. Assume that it is true for a fixed N. We will show that it is also true for N + 1. Denoting the right side of (11. 7.8) by RN, we have that n
B(aszj;p)
_ (b/a1,"" b/an+1; q,p)N+1 R N+1(q, bz1, ... ,bzn ; q, p)N+1
B(q;p) B(bqN /a1,"" bqN /an+1;p) R B(qN+1;p) B(q, bqN Zl, ... ,bqN Zn; p) N·
(11. 7.12)
However, by the induction hypothesis we can replace RN by the series on the left side of (11.7.8) and replace the ratio B(bqN /a1,"" bqN /a n+1;p) /B(q, bqN Zl, ... , bqN zn;P) by the N = 1 case of the series in (11.7.8) with Zm and b replaced by zmqkTn and bqN, respectively. Thus,
kl,""",kn?O k1+"""+kn=N
334
Elliptic, modular and theta hypergeometric series
(11. 7.13)
In the above summation, replace k by k - j. Since jr E {a, I}, we have
(}(bzrqN+kr-jr; p) (bzrqN+kr-jr; q,P)jr
(}(bzrqN +kr; p) (bzrqN+kr; q,P)jr'
(b ) Zr; q,p kr-jr
(bzr ; q,phr (b zrq kr-l.,q,p).' Jr
=
and n
II (Zrq1+kr-ks-jr+js /Zs; q,P)jr = (}(q;p) II (Zrqkr-ks /Zs; q,P)jr· r,s=l
Thus
kl, ... ,kn?':O
kl +·+kn=N +1
n
(}(bzrqN+kr;p)
n
x
g
TI (aszr;q,P)k r
(}(bzrqN;p)(bz r ; q,P) ksr=[e=1 (Zrq/Zs; q,p)k r
(11.7.14) The sum over j can be rewritten in the form n
n
~1
(}(bzmqk",-\p) (}(bzmqN+krn;p)
m-
TI
(}(Zmqk",/Zs;p)
rr=~(Zmqkrn-k8/Zs;p)
s#m
r=1 n
TI
(}(bzrqN+kr;p)
r=1 by (11.7.11). From (11.7.14) and (11.7.15) it follows that
RN+1 =
L
kl, ... ,kn?':O kl +·+kn=N+1
(11. 7.15)
11.7 Rosengren's elliptic extension of Milne's fundamental theorem
335
which completes the proof of (11.7.8). Formula (11.7.8) may be regarded as a multivariable extension of Frenkel and Thraev's summation formula (11.4.1) which, as we have seen before, is an elliptic analogue of Jackson's summation formula (2.6.2) for a terminating very-well-poised S
b. n +1(Zn+1qkn+l ;p) b. n+1(Zn+1; p)
qkrB(Zsqks-kr/zr;P)
II
B(zs/zr;P)
II
qkrB(Zsqks-kr/zr;P) B(zs/zr;P)
(11. 7.18)
n+1n+2
n n+2
n+2
II II (aszr ; q,phr II II (bszr ; q,phr II (a- 1bsq-N; q,p)N-lkl, =
r=ls=l
r=ls=l
~1
s=l n
II
II
(bzr ; q,p)k r = (q1-N /e; q,p)N-lkl (aqzr/ e; q,p)k r , r=l r=l n+1 n n (qzr/zs;q,phr = (q;q,p)N-lkl II(azrqN+\q,P)kr (qzr/zs;q,phr· r,s=l r,s=l r=l
II
II
So, n+2
I1 (aszr ; q,phr ____ II n+1
n+1
s_=_l~--------
r=l (bzr ; q, P)kr
I1 (qzr/ Zs; q,phr
s=l
Elliptic, modular and theta hypergeometric series
336
Using the identities (11.2.49) and (11.2.50), we can simplify the last expression above and obtain the following well-poised form of (11.7.8)
~(zqk;P) [ kl,.~n20 ~(z;p) 11
n O(azrqlkl+kr;p) S:Q:1 (az s ; q,P)lkl O(azr;p)
Ikl::;N
IT (qzr!zs;q,phr
s=l
III
n+2 (bsZr ; q,p)kr(e, q-N; q,p)lkl
X
qlkl
1
n+2
III (aq/bs;q,p)lkl (aqzr/e, azrqN+1; q,p)kr
= eN
IT
IT
(aqzr; q,p)N (aq/ebs; q,P)N. r=l (aqzr/e; q,p)N s=l (aq/bs; q,p)N
(11. 7.19)
For some elliptic multivariable transformation formulas and additional summation formulas, see the exercises and notes.
Exercises
11.1 Verify the O(a;p) and q,p-shifted factorial identities in (11.2.42)-(11.2.60). 11.2 Verify the q,p-binomial coefficient identities in (11.2.62) and (11.2.64)(11.2.66). 11.3 Show that
O(aq2n; p) O(a;p)
(qa!, -qa!; q,P!)n 1 1 1 (a"2,-a-"2;q,p"2)n (qa!, -qa!, q(ap)!, -q(ap)!; q,P)n (a!, -a!, (ap)!, -(ap)!; q,P)n =
(qa!,-qa!,q(a/p)!,-q(ap)!;q,P)n(_ )-n 1 1 1 1 q (a"2, -a"2, (ap) "2 ,-(a/p)"2; q,P)n
and convert these identities to the [a; u, Tln notation.
11.4 Prove (11.4.1) via induction by starting with the identity
qkO(aq2k, q-n-1, e, aqn+1 /e;p) =
O(aqk, qk-n-l, eqk, aqn+k+1 /e;p) - O(aqn+k+l, qk, el-n-l, al/e;p),
Exercises
337
which is equivalent to (11.4.3). (Rosengren: June 13, 2002, e-mail message) 11.5 Verify that
t
k=O
for n
()(aq2k;p) (a, b, e, a/be; q,ph qk ()(a;p) (q,aq/b,aq/e,beq;q,p)k
=
(aq, bq, eq, aq/be; q,P)n (q,aq/b,aq/e,bcq;q,p)n
= 0,1, ... , which is an elliptic analogue of Ex. 2.5.
11.6 Derive the following transformation formula for a well-poised 4 E 3(a,
4E3
series
b, aqn+2/b, q-n; aq/b, bq-n-1, aqn+\ q,p; -1)
(aq, q2(ap)~ /b, -q2(a/p)~ /b, -q; q,P)n (q2 /b, q(a/p)~, -q(ap)~, - aq2/b; q,P)n X 12 Vu (-aq/b;
where n
1
1
1
1
(ap) 2, -(alp) 2, qa 2 /b, -qa 2 /b, -q, aqn
+2
/b, q-n; q,p),
= 0,1, ....
11.7 Extend the terminating case of the expansion formula in (2.8.2) to
f
(a, b, e; q,P)n An n=O (q,aq/b, aq/e; q,P)n =
f
()(>. q2n;p)(A, Ab/a, Ae/a, aq/be; q,P)n(a; q,phn (9:)n n=O ()(A; p)(q, aq/b, aq/e, a 2q/ Abc; q,P)n(Aq; q,phn A ~
X
(aq2n,a/A,Abeqn/a;q,p)k A (q,Aq2n+l,a2qn+1/Abe;q,p)k n+k,
6
where the sequence {An} has finite support and A is an arbitrary parameter. Use this formula to derive (11.5.1). 11.8 Show that
t
(~,qa~;-qa~,b,c,q-n;q,p)k
(_I)k
k=O (q, a 2 , -a 2 , aq/b, aq/e, aqn+1; q,p)k
(aq,aq/be,qb-1(a/p)~,qc-1(a/p)~;q,p)n (aq/b,aq/e,q(a/p)~,ab-1e-1(a/p)~;q,p)n'
where be = _ aqn+1, n = 0,1, .... 11.9 (i) Extend Ex. 1.4(i) to the inversion formula
f n=O
(a1, ... ,ar ;q,P)n ((_ltqG))1+s-rAnzn (q,b1, ... ,bs;q,P)n 00
= '"""'
(-1
-1.
-1
)
a1 , ... ,ar ,q ,p n A (a1··· arz) 1 ~ ( q -1 ' 1 b- 1, ... , bn=O s ; q -1 ,p) n n qb1··· bs
when the sequence {An} has finite support.
n
Elliptic, modular and theta hypergeometric series
338
(ii) Extend Ex. 1.4(ii) to the reverse in order of summation formula
when n
=
0,1, ... , and {An} is an arbitrary sequence of complex numbers.
11.10 Prove that ~ O(aq2k;p)
~ O(a;p)
k=O
k (a,b,c,d,e,a 2q/bcde;q,p)k (q,aq/b, aq/c, aq/d, aq/e, bcde/a;q,p)k q
(aq, bcd/a,a2q2/bcde, eq; q,P)n (q,a 2q2/bcd,bcde/a, aq/e; q,P)n x 12 Vu (a 2q/bcd; aq/bc, aq/bd, aq/cd, e, a2q/bcde, aqn+1, q-n; q,p), where n = 0,1, .... 11.11 Verify the transformation formulas (11.6.16) and (11.6.17). 11.12 As in (11.2.67), the r( z; q, p) elliptic gamma function is defined by 00
r(z; q,p) =
II j,k=O
1 _ z-l qj +1 p k+1 1_ z j k
where z, q, p are complex numbers and
(i) (ii) (iii) (iv)
(v)
'
qP
Iql, Ipl < 1.
Show that
r(z; q, 0) = eq(z), r(zq; q,p) = O(z;p)r(z; q,p), r(zp; q,p) = O(z; q)r(z; q,p), r(zqn; q,p) = (z; q,P)nr(z; q,p), r q(z) = (1 - q)l-z(q; q)oor(qZ; q, 0), 0 < q < 1,
and that r(z; q,p) and r(qZ; q,p) are meromorphic functions of z, which are not doubly periodic. (See Jackson [1905d], Ruijsenaars [1997, 2001], Felder and Varchenko [2000], and Spiridonov [2003b].) 11.13 Define the
f\z; q,p) elliptic gamma function
by
_ (q; q)oo 1-z 00 1 _ qj+1-z p k+1 r(z; q,p) = ( . ) (O(q;p)) 1 _ j+z k ' p,p 00 j,k=O q P
II
Exercises
339
where z, q, P are complex numbers and Iql, Ipl < 1. Extend the Gauss multiplication formula (1.10.10) and its q-analogue (1.10.11) to
-
-(1
r(nz;q,p)r ;,;r,p =
)r-(2;,;r,p) ···r-(n-n-;r,p -1 )
()(r;p)]nZ-1_ _( [ ()(q;p) r(z; r,p)r z
1
+ ;,; r,p
)
_(
... r z
+
n-l ) -n-; r,p
with r = qn, and show that r(z;q,O)
= rq(z).
(Felder and Varchenko [2003a]) 11.14 (i) Show that if n is a nonnegative integer and k is a positive integer, then -kn. 2k+8 TT V2k+7 (b. a ,C, a b/ c, bq, bq 2 , ... , bq k ,aqn ,aqn+1 , ... ,aqn+k-1 , q , q k ,p ) (a/c,c/b;q,P)n(qk,abqk;qk,P)n (cqk,abqk/c;qk,P)n(a,l/b;q,P)n·
(ii) Show that
and that this formula tends to (lU7) as p ----; O. (iii) Show that 12 Vu(cp; C,
-c, -cp, epic, cqp/e, qn+1, q-n; q,p2) (cqp,e/c 2;q,p2)n(eq-n;q2,p2)n
(qp/c, e; q,p2)n(eq-n /c 2; q2,p2)n
and that this formula tends to (lU9) as p ----; O. (See Warnaar [2002b] for part (i) and Warnaar [2003f] for parts (ii) and (iii). ) 11.15 Prove that if a, bk, Ck are complex numbers such that Cj 1 for integer j and k, then n
L k=m
with
and
inkgkm = Dn,m
i=- Ck and aCjCk i=-
340
Elliptic, modular and theta hypergeometric series (Warnaar [2002b])
11.16 Prove that
n e(a2q5k;p) (a2; q4,p)k(a, aq, aq2; q3,ph(aqn+1, q-n; q,ph k e(a 2;p) (q; q,p)k(a, aq, aq2; q2,ph(aq3-n, a 2qn+4; q4,p)k q
~
_{ (~,q2'l,a:q4;q~'f)n/4 , n:::=0(mod4), (aq ,aq ,aq ,q/a,q ,P)n/4 0, n =I=- 0 (mod 4). (Warnaar [2002b]) 11.17 Show that if be = a 2qn+1 and d = qn+l, then
e (4k) aq ;P (b,e;q3) ,p k (a,d;q,p )k (-n q ;q,p )2k k ~ e(a;p)(aq/b, aq/e; q,ph(aq3/d, q3; q3,p)k(aqn+1; q,phk q
[n/2] ""'
(aq; q,P)n(aq2-n /b; q3,P)n (aq/b;q,P)n(aq2-n;q3,P)n· (Warnaar [2002b]) 11.18 For be = aq and cd ~
= aqn+1,
show that
e(aq4k;p)(a, b; q3,ph(d, q-n; q,p)k(e; q,phk
k
~ e(a;p)(q, aq/b; q,p)k(aq3/d, aqn+3; q3 ,ph(aq/e; q,phk q _ { (q,q22,aq3,b2/3a;~3'f)n/3 , n:::= 0 (mod 3), (bq,bq ,b/a,aq /b,q ,P)n/3 0, n =I=- 0 (mod 3). (Warnaar [2002b]) 11.19 Extend the e = bq-n-1 case of (3.8.19) to
[n/2] ( 4k ) ( 3 ) ( ) (-n ;q,P2k ) ,,",eaq;p b,e;q,Pkd,e;q,Pkq k ~ e(a;p) (aq/b, aq/e; q,ph(aq3/d, aq3/e; q3,p)k(aqn+1; q,phk q (aq;q,P)n( aq2-n/b;q3,P)n (aq/b;q,P)n( aq2-n;q3,P)n x 12 Vu (a 2Ide; b, e, aid, ale, q2-n, q1-n, q-n; q3,p), where be = a2qn+1 and de = aqn+1. (Warnaar [2002b]) 11.20 Show that if be = aq and de ~ e(aq3k;p)
~
e(a;p)
= aqn+1,
then
(a,b,e;q2,p)k(d,e,q-n;q,P)k k (q, aq/b, aq/e; q, p)k(aq2/d, aq2/e, aqn+2; q2 ,p)k q
(aq2,aq2/be,aq2/bd,aq2/ed;q2,p)n/2 -_ { (aq 2/ b, aq 2/ e, aq 2/ d, aq 2/ bed; q2) ,p n/2 , n even 0, n odd.
Exercises
341
(Warnaar [2002b]) 11.21 Show that if bed = aq, ej ~ ()( aq3k;p)
6
()(a;p)
=
a 2q2n+l, and either b = a, or e
a, then
=
(b,e,d;q,ph(e,j,q-2n;q2,p)k k (aq2/b, aq2/e, aq2/d; q2,ph(aq/e, aq/ j, aq2n+l; q,P)n q
(aq2,a2q2/bee,a2q2/bde,aq2/ed;q2,p)n (a2q2/be,aq2/e,aq2/d,a2q2/bede;q2,p)n· (Warnaar [2002b]) 11.22 Show that if bed = aq, de
=
~ ()(aq4k;p)
6
()(a; p)
a 2q3n+l, and either b = a or e
=
a, then
(b, e; q,ph(d; q,phk (aq3/b, aq3/e; q3 ,p)k(aq/d; q,phk
(e,q-3n;q3,ph k q (aq/e, aq3n+l; q, p)k (aq3,a2q3/bee,a2q3/bde,aq3/ed;q3,p)n (a2q3/be,aq3/e,aq3/d,a2q3/bede;q3,p)n·
x
(Warnaar [2002b]) 11.23 (i) Verify the quadratic elliptic transformation formula 14 V I3 (a; b, -b, e, -e, dqn, _dqn, q-n, _q-n, qa 2/)..2; q,p)
(a2q2,d2/)..2,b2d2/a2,e2d2/a2;q2,p2)n (d2/a2,q2)..2,a2q2/b2,a2q2/e2;q2,p2)n x 14 VI3()..2; b2, e2 , d2q2n, a 2q2 /)..2, q-2n, _)..2/ a , _)..2 q/ a , _)..2/ap, _)..2 qp/ a ; q2,p 2), where)..
=
bed/aq and n
=
0,1, ....
(ii) Prove that the P ----) 0 limit of the above transformation is
12 W ll(a; b, -b, e, -e, dqn, _dqn, q-n, _q-n, qa 2/)..2; q, q) (a2q2,d2/)..2,b2d2/a2,e2d2/a2;q2)n (d2/a2,q2)..2,a2q2/b2,a2q2/e2;q2)n x 10 Wg ()..2; b2, e2 , d2q2n, a 2q2 /)..2, q-2n, _)..2/ a , _)..2 q/ a ; q2, )..2q/a2) and that the 12 Wll series is VWP-balanced and balanced, while the series is VWP-balanced but not balanced.
10 W g
(See (Spiridonov [2002a]) for part (i), and Nassrallah and Rahman [1981] for part (ii). Also see (3.10.15), Andrews and Berkovich [2002] and Warnaar [2003c,e].) 11.24 (i) Verify the quadratic elliptic transformation formula 14VI3(a;a 2/)..2,b,bq,e,eq,dqn,dqn+l,q-n,ql-n;q2,p)
(aq,d/)..,)..q/b,)..q/e;q,p)n ()..q,d/a,aq/b,aq/e;q,p)n x
14 V I3 ()..;
1
1
a/).., IL, -IL, ILP'l, -ILP-'l, b, e, dqn, dq-n; q,p),
342
Elliptic, modular and theta hypergeometric series where>..
=
bed/aq,
J1, =
±>,,(q/a)~, and n
=
0,1, ....
(ii) Prove that the P ----) 0 limit of the above identity is
12VVll (a; a2/>..2, b, bq, e, eq, dqn, dqn+l, ql-n, q-n;q2,q2) (aq,d/>..,>"q/b,>"q/e;q)n (>"q,d/a,aq/b,aq/e;q)n x 10 VVg (>..; a/ >.., J1" -J1" b, e, dqn, q-n; q, _q>..2 / a) and that the 12 VVl1 series is VWP-balanced and balanced, while the series is VWP-balanced but not balanced.
10 VVg
(See Warnaar [2002b] for part (i), and Rahman and Verma [1993] for the transformation in part (ii). Also see Andrews and Berkovich [2002] and Warnaar [2003c,e].) 11.25 (i) Extend the quadbasic transformation formula in Ex. 3.21 to
t
()(arkqk, brkq-k;p) (a, b; r,p)de, a/be; q,ph k=O ()(a,b;p) (q,aq/b; q,p)k(ar/e, ber;r,p)k (CR-n /A, R-n / BC; R, p)k(Q-n, BQ-n /A; Q, P)k k X (Q-n/C,BCQ-n/A;Q,Ph(R-n/A,R-n/B;R,Ph q (ar, br; r,P)n(eq, aq/be; q,P)n(Q, AQ/ B; Q, P)n(AR/C, BCR; R, P)n (q, aq/b; q,P)n(are, be/r; r,P)n(AR, BR; R, P)n(CQ, AQ/ BC; Q, P)n
x
t
()(ARkQk,BRkQ-k;p) (A,B;R,Ph(C,A/BC;Q,P)k ()(A, B; P) (Q, AQ/ B; Q, P)dAR/C, BCR; R, P)k k=O x (er-n la, r- n /be; r,p)k(q-n, bq-n fa; q,P)k Qk (q-n /e, beq-n fa; q,p)k(r- n la, r- n /b; r,p)k for n = 0,1, .... (ii) Deduce the following transformation formula for a "split-poised" theta hypergeometric series
t
()(aq2k;p) (a,b,e,a/be;q,p)k k=O ()(a;p) (q, aq/b, aq/e, beq; q,p)k (q-n, B/Aqn, C/Aqn, I/BCqn; q,ph k (I/Aqn, I/Bqn, l/cqn, BC/Aqn; q,P)k q (aq, bq, eq, aq/be, Aq/ B, Aq/C, BCq; q,P)n (Aq, Bq, Cq, Aq/ BC, aq/b, aq/e, beq; q,P)n
X
x
X
t
()(Aq2k;p) (A,B,C,A/BC;q,p)k k=O ()(A;p) (q, Aq/ B, Aq/C, BCq; q,ph (q-n,b/aqn,e/aqn,l/beqn;q,p)k k q (l/aqn,l/bqn, l/eqn, be/aqn; q,p)k
for n = 0,1, ... , which is an extension of the transformation formula for a split-poised 1O¢9 series given in Ex. 3.21. Write this formula as a transformation formula for a split-poised 12En series.
Exercises
343
(Gasper and Schlosser [2003])
11.26 Extend Ex. l1.25(i) to
t
()(a(rst/q)k, brkq-k, cskq-k, at k /bcqk;p) k=O ()(a, b, e, a/bc;p) (a; rst/q2,p)db; r,p)k(c; s,ph(a/bc; t,P)k (q;q,p)k(ast/bq;st/q,p)k(art/cq;rt/q,p)k(bers/q;rs/q,p)k (Q-n; Q, Ph(B(Q/ST)n /A; ST/Q, P)k(C(Q/ RT)n /A; RT/Q, P)k x ((Q2 /RST)n/A; RST/Q2, p)k(R-n / B; R, p)k(s-n /C; S, Ph ((Q/Rs)n/BC;RS/Q,P)k k x (BC/ATn;T,p)k q (arst/q2;rst/q2,P)n(br;r,P)n(cs;s,P)n(at/bc;t,P)n (q;q,P)n(ast/bq;st/q,P)n(art/cq;rt/q,P)n(bers/q;rs/q,p)n (Q; Q,P)n(AST/BQ; ST/Q,P)n X..,....,...'--'--"":""'--:'-""""'-------'--::--'-:-"""":--'----'--....,..(ARST /Q2; RST/Q 2, P)n(BR; R, P)n (ART /CQ; RT /Q, P)n(BCRS/Q; RS/Q, P)n x-'------'-----;-==-=---=-~--;--;-::::-:-=-=:--::::-_=_=_----'-....::...:....--'-(CS; S,P)n(AT/BC;T, P)n x ()(A(RST/Q)k, BRkQ-k, CSkQ-k, ATk/BCQk;p) ()(A, B, c, A/ BC; P) k=O (A; RST/Q2, P)k(B; R, P)k x~'--='--=-:-~-===-:-:'::-'-:'---,==-':-::-'--'::-: (Q; Q, P)k(AST/BQ; ST/Q, P)k (C; S, P)k(A/BC; T, P)k x (ART/CQ; RT/Q, P)k(BCRS/Q; RS/Q, P)k (q-n; q,p)k(b(q/ st)n fa; st/q,p)k x-'-=--------;.-;-:~-"--'-:-=:-----'-----'--;--';::_____'_,-=-:-=---'-'-' ((q2 /rst)n fa; rst/q2 ,ph x (c(q/rt)n fa; rt/q,ph((q/rs)n /bc; rs/q,ph Qk (r- n /b; r,p)k(S-n /e; s,p)k(bc/atn ; t,p)k x~~~~~~~~~~~~~~~~~~
t
for n = 0,1, .... Use (11.6.9) and (11.6.18) to extend this formula to a transformation formula containing the two additional parameters d and D. (Gasper and Schlosser [2003]) 11.27 Show that if
344
Elliptic, modular and theta hypergeometric series and a~ = bkckdkekfkgk for k = 0, ±1, ±2, ... , then
~ ()(aVbkckdkh, aVbkckdkek, ak, ak/ekfk;P) U k
f:::n ()(ak/ek, ak/ fk' aVbkCkdkekfk, aVbkCkdk;p) x =
[
()(ak/ckdk,ak/bkdk,ak/bkck,ek,fk,gk;p)
1
- ()(ak/bk,ak/ck,ak/dk,a~/bkckdkek,a~/bkckdkfk,a~/bkckdkgk;P) Urn - Un+1
for n,m
=
]
0, ±1, ±2, ....
(Gasper and Schlosser [2003]) 11.28 Extend the indefinite multibasic theta hypergeometric summation formula in (11.6.9) to
k a (V ) k a 2 ( v 2 ) k
a2 (V2 ) k a2
)
~ () (av 'J[j Ui 'TiCrJ! qrsl 'vr;cag qrsu 'DCa; P ~ ( a a2 a 2 a 2 ( v 2 ) k. ) () a, J[j' bcd] , bcdg' bcd qr s
k=O
,p
3
3
(b; q,p)k(C; r,ph(d; s, p)k(f; t, p)k(g; u, p)k (bcdfg; q;!stu ,p) k (qrstu/v 2)k x
--------------------------------~~~~--~--~----
a v a v a v av v av v bcdfgqrstu qrstu (-b;-,P)k(-;-,P)k(-d;-,P)k(ft;-t,P)k(-;-,P)k( 22 ; - 2 ,P)k c r s gu u a v v q
1
()(cdU!s)k, bd(~s)k, bc(~r)k,ftk,guk, bCdfg(q;!s3tu )k;p) x [ 1------------------------.;-------;o;-------,;---~---=-------;;-------a (V)k a (V)k a (V)k a 2 ( V2 )k a 2 ( v 2 )k fg (tU)k. ) () ( 7i q 'c r 'd S 'TiCrJ! qr st 'Ocag qr su 'a v ' P ()(a/f,a/g,a 2/bcdfg,a 2/bcd;p) ()(a 2/bcdg,a 2/bcdf,a,a/fg;p) x
[1 _
(b; q,P)n+l(C; r,P)n+l(d; S,P)n+l(f; t,P)n+l(g; U,P)n+l
(~; ~'P)k+l (%; ¥,P)n+l (a; ~,P)n+l Uj-; ¥,P)n+l (~; *,P)n+l a3
1
V3
x (-oCilfii; 7jFSfU' p) n+ 1 , ( bcdtg. qrstu) a
'
----;;r-' P
n+l
where n = 0,1, .... (Gasper and Schlosser [2003]) 11.29 Define the elliptic beta function BE(t;q,p) via its elliptic beta contour integral representation
BE(t; q,p)
=
rllE(Z;
IJJ'
t; q,p) dz,
z
where l' is the positively oriented unit circle, 4
I1
r(ztk,tk/Z;q,p) k=O llE z;t;q,p = -2 .r( Z, 2 1/ 2 7rZ z, zA, A/ z; q,p)' (
)
1
345
Exercises 4
t =
(to,t1,t2,t3,t4), max(ltol, It11, It21, It31, It41, Iql, Ipl) < 1, A
I1
=
tk,
k=l
Iqpl < A, and n
r(Zl,"" Zn; q,p)
=
II r(Zk; q,p),
n
=
1,2, ... ,
k=l
with f(z; q,p) as defined in Ex. 11.12. Prove that 2
BE(t; q,p)
=
I1 O~j
(q; q)oo(p;p)oo
r(tjtk; q,p) 4
I1
k=O
r(A/tk; q,p)
(Spiridonov [200la]) ~E(Z; t) = ~E(Z; t; q,p) and BE(t) = BE(t; q,p), with ~E(Z; t; q,p), BE (t; q, p) and A defined as in the previous exercise, and let
11.30 Let
Rm,j (z)
=
t3 q q q t3 -m Aqm-1 ) 12 V11 ( t4; tOt4 ' t1t4 ' t2t4 ,t3 Z, ---;' q , -t-4-; q,p V; (t3 P P P t3 _j Apd-1 ) X1211 t4;tOt4't1t4't2t4,t3z,---;,p ,~;p,q
for j, k, m, n = 0,1, .... Note that the base and nome in the second 12 V11 factors are p and q, respectively. Prove that Rm,j(z) and Tn,k(Z) satisfy the biorthogonality relation
where hn k =
,
()(A/qt4;p)(q, qt3/t4, tOt1' tOt2' t1t2, At3; q,P)nq-n ()(Aq2n /qt4;p)(1/t3 t4' tOt3, t1 t3, t2 t3, A/qt3, A/qt4; q,P)n ()(A/pt4; q)(p,pt3/t4, tOt1, tOt2, t1t2, At3;P, q)np-k x ()(Ap2k /pt4; q)(1/t3t4' tOt3, t1t3, t2t3, A/pt3, A/pt4;p, qh
~~~~~~~~-----~~~~~~~
and Cj,k,m,n is a closed positively oriented contour separating the points Z = t o,1,2, 3pTqs, t 4pT-j qs-m, A- 1pT+1-k qs+l-n with r, S = 0,1, ... , from their inverses. (Spiridonov [2003b]) 11.31 Show that if
346
Elliptic, modular and theta hypergeometric series then
kl, ... ,krn?D k1+···+krn=N
which is an extension of (11.5.5), (11.7.7), and (11.7.10). (Kajihara and Noumi [2003] and Rosengren [2003c]) 11.32 Let Xl, ... , X n , A 2, ... , An and C be indeterminates. Prove that if, for m = 0, ... , n -1, Pm is a Laurent polynomial of degree less than or equal to m such that Pm(C/X) = Pm(X), then
II
n
A j X j (l- Xd X j)(l- C/XiXj )
II Pi-
1 (1/Ai ),
i=l
l~i<j~n
N
where the degree of the Laurent polynomial P(x) =
l:
aixi, aN
of. 0,
is
i=M
defined to be N. (Krattenthaler [1995a])
11.33 Prove the following elliptic extension of the identity in the above exercise
II
n
AjX/J(XdXj;p)()(C/XiXj;P)
II Pi-
1 (1/Ai ), i=l where Xl, ... , X m , A 2, ... , An, and C are indeterminates, and Pj (x) is analytic in 0 < Ixl < 00 for j = 0, ... , (n - 1) with periodicity Pj(px) = (C/pX2)j Pj(x) and symmetry Pj(C/x) = Pj(x). (Warnaar [2002b]) l~i<j~n
11.34 Deduce from the above exercise that d et (AXi,AC/Xi;q,P)n-j l~i,j~n (BXi' BC/ Xi; q,P)n-j
= AG)qG)
II
XjB(XdXj;p)()(C/XiXj;p)
347
Exercises (Warnaar [2002b])
11.35 Let Zl, ... , Zn, a, b, c, d, and e be indeterminates and N a nonnegative integer such that a 2qN -n+2 = bcde. Show that
II
B( aZrzsqkrHs; p) B(azrzs;p)
IT x = IT
B(az;'q2kr;p) (az;', bzr, cZr, dzr, ezr, q-N; q,p)k s Ikl r=l B(az;;p) (q, aqzr/b, aqzr/e, aqzr/d, aqzr/e, aqN+l z;; q,pk q
(aqz;',aq2-r/bc,aq2-r/bd,aq2-r/cd;q,p)N r=l (aq2-n /bcdzr , aqzr/b, aqzr/c, aqzr/d; q,p)N'
which is an elliptic extension of Schlosser's [2000a]
en Jackson sum.
(Warnaar [2002b]) 11.36 Prove the following elliptic extension of Gasper's summation formula in Ex. 2.33(i):
t
B(aq2k;p) (a, b, alb, q-N; q,ph kIT (Cjqm j , aq/Cj; q,ph k=O B(a;p) (q, aq/b, bq, aqN+l; q,p)k q j=l (aql-mj /Cj, Cj; q,ph =
IT
(q, aq; q,p)N (Cj/b, cjb/a; q,P)mj (bq, aq/b; q,p)N j=l (Cj, Cj/a; q,P)mj
with ml
+ ... + mr = N, where ml, ... ,mr are nonnegative integers.
(Rosengren and Schlosser [2003b]) 11.37 Prove the following multidimensional extension of Ex. 11.29:
I1
n·
2n n! (p;p)~(q; q)~
where B = t 2n - 2
I1 O:S;k:S;4
II r(t ·-1 J-
J;
rW-ltrt s ; q,p)
q, p) --'O:s;=:r=<:-s-=:s;=--4...,.....,...,.....,----;;--_---:-
r(t; q,p)
I1
r(tl-h;l B; q,p) ,
O:S;r:S;4
tk, max(ltl, Itol, Itll, It21, It 31, It41, Ipl,lql)
Ipql < B, and ,][,n is the n-dimensional unit torus. (van Diejen and Spiridonov [2003, (29)] and Rains [2003b])
< 1,
348
Elliptic, modular and theta hypergeometric series
11.38 Prove Rosengren's [2003c] elliptic Dn Jackson summation formula
n-1
I1 (bas, bias; q,p)N
= (_qN -1 b) N _----=-s_==-l----=n:------(q; q,p)N
I1 (bzs, b/zs; q,p)N
s=l
11.39 From the previous exercise deduce that
x x
II
(zsbr , zs/br ; q,pk l:<:;r<s:<:;n (zrzs;q,phr+ks r,s=l (qzs/zr;q,P)k s
IT
r=l x
IT
n
1
(azr ; q,p)lkl (aq/ Zr; q,p)lkl-k r (aqbr, aq/br; q,P)lkl (q-N,c,a 2 qN+1/ c;q,p)lkl
qlkl
}
I1 (aqzs/c, czs/aqN, az sqN+1; q,phs
s=l
=
IT
(aqzs, aq/zs, aqbs/c, aqbs/c; q,P)N, s=l (aqzs/c,aq/zsc, aqbs, aq/bs; q,p)N
where Ikl = k1 + ... + k n . (Rosengren [2003c]) 11.40 Show that
[IT
m1tm n {~(Zqk;p) qlkl B(azsqks+lkl;p)] kl, ... ,kn=O ~(z;p) s=l B(azs;p)
II
n (b, c, d; q,P)lkl (azr ; q,P)lkl azq x (aq/e, aq/ /, aq/g; q,P)lkl r=l ( mr+1; q,P)lkl
x
IT
IT
(ezs,/zs,gzs;q,pk (q-mrzs/zr;q,pk} s=l (aqzs/b, aqzs/c, aqzs/d; q,p)k s r,s=l (qzs/zr; q,p)k s
Notes
IT (aqzs, Aqzs/d; q,p)Tns
= (~) Iml (Aq/ j, Aq/g; q'P)lml A
(aq/ j, aq/g; q'P)lml s=l (Aqzs, aqzs/d; q,p)Tn s
Tn1,···,Tnn " X
349
~
k1, ... ,kn=O
{~(
zq k.) ,p Ikl ~(z;p) q
[nII ()('AZsqks+lkl.)] ,p s=l
()(AZs;p)
II
n (AZr ; q,p)lkl (Ab/a, Ac/a, d; q,p)lkl x (aq/e, Aq/ j, Aq/g; q,P)lkl r=l (Aq1+ Tnr zr; q,P)lkl
x
IT
IT
(Aezs/a,jzs,gzs;q,phs (q-Tnrzs/zr;q,P)ks}, s=l (aqzs/b, aqzs/c, Aqzs/d; q,p)k s r,s=l (qzs/zr; q,p)k s
a2q/bcd and a 2q2+lm l Iml =m1+···+ m n· (Rosengren [2003c])
where A
=
=
bcdejg with
Ikl
=
kl
+ ... + k n
and
Notes §11.2 and 11.3 Totally elliptic multiple hypergeometric series are defined and considered in Spiridonov [2002a, 2003a]. Spiridonov [2003a] showed that the elliptic Milne An, Jackson en and Bhatnagar-Schlosser Dn theta hypergeometric series in the left sides of his equations (21), (18), and (22), respectively, are totally elliptic and modular invariant. He used these results and other observations to motivate his conjecture that, as in the one-variable series case, every totally elliptic multiple hypergeometric series is modular invariant. §11.6 The P = 0, s = q special case of (11.6.21) is equivalent to a corrected version of the generalization of (3.7.6) given in Subbarao and Verma [1999]. §11.7 Rosengren's inductive proof of (11.7.8) is similar to Milne's [1985a, pp. 49-50] inductive proof of (11.7.1) based on the analysis in Milne [1980b, pp. 179-182, and 1985c, pp. 17-19]. For additional material on basic and elliptic summation and transformation formulas, see Bhatnagar [1998, 1999], Bhatnagar and Milne [1997], Bhatnagar and Schlosser [1998], van Diejen [1997b], van Diejen and Spiridonov [2000-2003], Gustafson [1987a-1994b], Gustafson and Krattenthaler [1997], Gustafson and Rakha [2000], Ito [2002]' Leininger and Milne [1999a,b], Lilly and Milne [1993], Milne [1980a-2002], Milne and Bhatnagar [1998], Milne and Lilly [1992, 1995], Milne and Schlosser [2002]' Rains [2003a,b], Rosengren [1999-2003f], Rosengren and Schlosser [2003a,b], Schlosser [1997-2003e], Spiridonov [1999-2003b], and Warnaar [1999-2003e]. Ex.11.15 This orthogonality relation is an elliptic analogue of Krattenthaler [1996, (1.5)]. Ex.11.17 This summation formula is an elliptic analogue of Gasper [1989a, (5.22) with b = qn+1]. Ex.11.18 This identity is an elliptic extension of W. Chu [1995, (4.6d)]. Ex.11.20 This formula is a generalization of Gessel and Stanton [1983, (6.14)].
350
Elliptic, modular and theta hypergeometric series
Ex. 11.21 When p = 0 and b = a this formula reduces to Gessel and Stanton [1983, (1.4)]. When p = 0 and e = a it reduces to W. Chu [1995, (5.1d)] and to Rahman [1993, (1.9) with b = q-2n]. Ex.11.22 When p = 0 and b = a this is an elliptic analogue of Gasper [1989a, (5.22) with c = q-3n]. Ex.11.29 Also see the material on elliptic integrals and elliptic gamma functions in Felder, Stevens and Varchenko [2003a,b], Felder and Varchenko [2000-2003b], Narukawa [2003], and Nishizawa [2002]. Ex.11.36 Also see the elliptic summation formula in Rosengren and Schlosser [2003b, Corollary 5.3]. Ex.11.38 The p = 0 case was independently discovered by Bhatnagar [1999] and Schlosser [1997]. Ex.11.39 When p = 0 this reduces to an identity in Schlosser [1997]. Ex.11.40 Equivalent forms of the p = 0 case were independently discovered by Denis and Gustafson [1992, Theorem 3.1], who derived it from a multivariable integral transformations via residues, and by Milne and Newcomb [1996, Theorem 3.1] via series manipulations. Also see Rosengren [2003a].
Appendix I IDENTITIES INVOLVING q-SHIFTED FACTORIALS, q-GAMMA FUNCTIONS AND q-BINOMIAL COEFFICIENTS
q-Shifted factorials: I, (a; q)n = { (1 - a)(l - aq)··· (1 - aqn-l), 1 [(1-aq-l)(1-aq-2) ... (1- aqn)r,
(a; q)-n
=
1
_
(aq n; q)n
=
n = 0, n = 1,2, ... , (I.1) n=-1,-2, ....
(-q/a)n (n) q 2 (q/a; q)n
(1.2)
and (a;q-l)n = (a-\q)n(-a)nq-G), where (~)
=
(1.3)
n(n - 1)/2. 00
(a; q)oo
=
II (1 -
(1.4)
al),
k=O
() a; q
n =
(
(a; q)oo n ) , aq ; q 00
(1.5)
and, for any complex number n, ()
(a;q)oo
(1.6)
a; q '" = ( aq '" ; q) 00 ,
where the principal value of q'" is taken and it is assumed that
Iql < 1. (1. 7) (1.8) (1.9) (I.10)
(1.11 ) (I.12) (I.13) 351
352
Appendix I
( aq-n;q) (
aq
(qja;q)n (qja; q)k
=
n-k
-2n. ) ,q n
(aq-kn;q)
q
(qja; qhn (_~)n -3G) (qja;q)n q2 q .
=
=
q)n (aq n.,q) k -_ (a; qh(aqk; () . a;q n
=
(1.17)
(1.19) (1.20)
k
_ (a; q)n(aqn; qh (aq 2k.) ,q n-k () . a; q 2k _ (a; q)n(aqn; q)(j-l)k ( jk.) aq ,qn-k(a;q)jk . (aI, a2,···, ak; q)n
(1.16)
(1.18)
(aq kn.,q)n -_ (a;( q)(k+l)n ) . a;q kn (a;q)n (aq k) ; q n-k = -(- ) . a; q
(1.14) (1.15)
(qja;qhn (_at qG)-kn 2 • (qja; q)(k-l)n (a; q)n(aqn; q)k.
=
n (a; q)n+k
(_~)n-k q(~)-G).
(1.21 ) (1.22) (1.23) (1.24) (1.25) (1.26)
(al; q)n(a2; q)n··· (ak; q)n.
(aI, a2,···, ak; q)oo
= (al; q)oo(a2; q)oo ... (ak; q)oo. (a; qhn = (a, aq; q2)n, (a; qhn = (a, aq, aq2; q3)n,
and, in general,
(a; q)kn = (a, aq, ... , aqk-\ qk)n. (a 2; q2)n = (a, -a; q)n, (a 3;q3)n = (a,aw,aw 2;q)n, W = e27ri / 3 ,
(1.27) (1.28) (1.29)
and, in general,
(a k.,qk) n -_ (a, aWk, ... ,awkk-l.,q) n, (qa~, -qa~; q)n
(aq2; q2)n (a;q2)n
(d,-d;q)n
(qai, qwai, qw2ai; q)n (ai,wai,w 2ai;q)n
Wk -_ e 27ri/k .
(1.30)
1 - aq2n I-a'
(aq 3; q3)
(1.31 )
1 _ aq3n
(a;q3)n
(1.32)
I-a'
and, in general, 1. 1. ( qak,qwkak, ( a 1. k , Wk ak1.
1. ) ... qwkk-l ak;q n
k-l
, ... , wk
1.
)
a k ;q n
(aqk; qk)n (a;qk)n
1- a
'
(1.33)
where W = e27ri / 3 and Wk = e27ri / k . lim ( Zq a;q)oo=(I_z)-a, q-->l-
(z; q)oo
Izl
(1.34)
Appendix 1
353
q-Gamma function:
0< q
< 1, (1.35)
q> 1.
(1.36) (1.37)
rq(nx)rr =
(~) rr (~) ... rr (n:
1)
f x - 1rr(x)rr (x+~) ... rr (x+ n:l),
(l+q+ ... +qn- 1
(1.38) with r
=
qn.
q-Binomial coefficient:
(1.39) and, for
Iql < 1 and complex a
and (3,
(qf3+ 1 , qcx-!3+1; q) 00 (q, qcx+l; q)oo rq(a+l)
(1.40) (1.41 ) (1.42) (1.43) (1.44)
[a;
1] q =
[~] q qk + [k ~ 1] q = [~] q + [k ~ 1] q qcx+l-k, [ + ]
~- ~
q
(qcx+\ q)n (q; q)n_k(qcx+l; q)k'
[~L_l = [~L qk -cxk, 2
(1.45) (1.46) (1.47)
where n, k are nonnegative integers. For elliptic analogues, see Chapter 11.
Appendix II SELECTED SUMMATION FORMULAS
Sums of basic hypergeometric series: The two q-exponential functions, 00
eq(z) =
zn
1
~ (q; q)n =
Izl < 1,
(z; q)oo'
~ q(;)zn Eq(z) = ~ -(-.-) = (-z; q)oo. n=O q, q n
(ILl )
(11.2)
The q-binomial theorem, l¢O
q)oo (a;-;q,z ) = (az; (z;q )00 ,
Izl < 1,
(11.3)
Of, when a = q-n, where, as elsewhere in this appendix, n denotes a nonnegative integer,
l¢O(q-n;_;q,z) = (zq-n;q)n. The sum of a
1 ¢1
(11.4)
series, 1¢1
(a;c;q,c I) a
=
(cia; q)oo (
c; q
)
(11.5)
.
00
The q-Vandermonde (q-Chu-Vandermonde) sums, rI.. ( - n ) (cla;q)n n 2'1-'1 a, q ; c; q, q = ( ) a c;q n
(11.6)
and, reversing the order of summation, rI.. ( - n 2'1-'1 a, q ; c; q, cq nl) a = (cia; ( )q)n . c;q n
(II. 7)
The q-Gauss sum, 2¢1
clb; q)oo (a, b; c; q, c Iab) = (cia, ( I b ) . c, c a ; q
(II.8)
00
The q-Kummer (Bailey-Daum) sum, rI.. ( b. lb. _ Ib) = (-q; q)oo(aq, aq2 Ib 2; q2)00 2'1-'1 a, ,aq ,q, q (_ qIb ,aqlb.) . ,qoo
(11.9)
A q-analogue of Bailey's 2F1 (-1) sum, 2¢2
bql a; q2)00 (a, qI a; -q, b; q, -b ) = (ab, (b.) . ,q
354
00
(ILlO)
355
Appendix II A q-analogue of Gauss' 2F1(-1) sum, 2¢2 (
(a 2q, b2q; q2)00 2 2 ! ! ) a ,b ;abq2,-abq2;q,-q = ( 2b 2 . 2) . q,a q,q 00
(11.11 )
The q-Saalschiitz (q- Pfaff-Saalschiitz) sum, A.
[
3'1-'2
a, b, q-n . ] _ (c/a, c/b; q)n c, abc- 1 q 1 -n ' q, q - ( C, c / a b; q)n .
(11.12)
The q- Dixon sum, [
4¢3
a,-qa!,b,c
qa!] q, -a'2, aq/b, aq/c bc 1
;
(aq,qb- 1a!,qc- 1a!,aq/bc;q)
=
1
1
00 ,
(aq/b, aq/c, qa'2, qa'2 /bc; q)oo
(11.13)
or, when c = q-n, 4¢3
a,-qa!,b,q-n. q1+ n a!]_ (aq,qa!/b;q)n 1 1+ ,q, b - (1 ) . -a'2, aq/b, aq n qa'2, aq/b; q n
[
(11.14)
Jackson's terminating q-analogue of Dixon's sum,
¢ [q-2n, 3
c . q2-n] _ (b, c; q)n(q, bc; qhn q1-2n /c' q, bc - (q, bc; q)n(b, c; qhn .
b, q1-2n /b,
2
(11.15)
A q-analogue of Watson's 3F2 sum, 8¢7
)" q)'!, -q),!, a, b, c, -c, ),q/c2
[
1
1
), '2, -), '2,
),q/ a, ),q/b, ),q/ c, -),q/ c, c
2;
),q] q, --b a
(),q, q)oo (aq, bq, c2q/ a, c2q/b; q2)00 (),q/a, ),q/b; q)oo(q, abq, c2q, c2q/ab; q2)00'
c2 /),;
where),
=
(11.16)
-c(ab/q)!; and Andrews' terminating q-analogue, q-n, a 2qn+1, c, -c ] 4¢3 [ 2 ; q, q aq, -aq,c 0, if n is odd, { = cn(q, a 2q2 /c 2; q2)n/2 if n is even. (a 2q2, c2q; q2)n/2 '
(11.17)
A q-analogue of Whipple's 3F2 sum, A.
8'1-'7
[-c,
q(-c)!, 1
(-C)'2,
-q(-c)!, 1
-(-c)'2,
a,
q/a,
c,
-d,
-cq/a,
-ac,
-q,
cq/d,
(-c, -cq; q)oo (acd, acq/d, cdq/a, cq2/ad; q2)00 (cd, cq/d, -ac, -cq/a; q)oo
-q/d ] ;q, c cd
(11.18)
(11.19)
Appendix II
356 The sum of a very-well-poised 1
'" [a,
series,
1
qa"2, 1 a"2,
6'1-'5
6¢5
-qa"2, 1 -a"2,
b, aq/b,
e,
d
aq/e,
aq/d
aq ] ;q, bed
(aq,aq/be,aq/bd,aq/ed;q)= (aq/b,aq/e,aq/d,aq/bed;q)=
or, when d ¢ 6
[
5
(11.20)
q-n,
=
a,qa !2 ,-qa !2 , b,e,q -n
.
Jackson's q-analogue of Dougall's 1
7F6
n+l] _ (aq,aq /b·) e,q n be - (aq/b, aq/e; q)n·
~
a!, -a!, aq/b, aq/e, aqn+l ,q,
sum,
1
qa"2, -qa"2, b, e, 1 1 a"2, -a"2, aq/b, aq/e, (aq,aq/be,aq/bd,aq/ed;q)n (aq/b,aq/e,aq/d,aq/bed;q)n'
d,
e,
aq/d,
aq/e,
'" [a, 8'1-'7
(11.21 )
(11.22)
where a 2q = bedeq-n. A nonterminating form of the q- Vandermonde sum, 2¢1
(
a, b; e; q, q
)
(q/e, a, b; q)=
+ (e / q, aq / e, bq/ e; q) = 2
x 2¢1(aq/e, bq/e; q /e; q, q)
(q/e, abq/e; q)=
(11.23)
= (aq / e, bq/ e; q) = .
A nonterminating form of the q-Saalschiitz sum,
e]
3
¢ [a, b, 2 e,j ;q,q
x
(q/e, a, b,
e, qj Ie; q)=
+ (e/q,aq/e, bq/e, eq/e, j;q)=
'" [aq/e, bq/e, eq/e. ] _ (q/e,j/a,j/b,j/e;q)= ' q, q - (aq / e, bq / e, eq / e, j ; q)= , q2/ e, qj / e
(11.24)
3'1-'2
where ej = abeq. Bailey's nonterminating extension of Jackson's 1
'" [a,
qa"2, 1 a"2,
8'1-'7
8¢7
sum,
1
-qa"2, 1 -a"2,
b, aq / b,
e, aq / e,
d, aq / d,
j ] aq/j;q,q
e, aq / e,
b (aq, e, d, e, j, bq/a, bq/e, bq/d, bq/e, bq/ j; q)= a (aq/b, aq/e, aq/d, aq/e, aq/j,be/a, bd/a, be/a, bj/a, b2q/a; q)
x
8¢7
[
b2/a, qba-!, -qba-!, b, be/a, bd/a, be/a, bj /a 1
1
ba-"2, -ba-"2, bq/a, bq/e, bq/d, bq/e, bq/ j
;
q, q
(aq, b/a, aq/ed, aq/ee, aq/ej,aq/de,aq/dj,aq/ej; q)= (aq/e,aq/d,aq/e,aq/j,be/a,bd/a,be/a,bj/a;q)= '
where qa 2 = bedej.
1
=
(11.25)
Appendix II
357
q-Analogues of the Karlsson-Minton sums,
a , b, b1 qTn1 , ... , br qTnr . a-I 1-(Tn1 +·+Tn r )] b b b ,q, q q, 1,···, r = (q, bq/ a; q)oo (bdb; q)Tn1 ... (br/b; q)Tnr bTn1 +.+Tn r (bq, q/a; q)00(b 1; q)Tn1 ... (b r ; q)Tnr
A, r+2'1-'r+l [
and A, r+ 1 'l-'r [
a , b1 qTn 1, ... , br qTnr . -1 -( Tn1+···+Tnr) ] - 0 b b ' q, a q - ,
(11.26)
(II.27)
1, ... , r
where ml, ... ,mr are arbitrary nonnegative integers.
Sums of bilateral basic series: Jacobi's triple product, 00
L
qk 2 zk = (q2, -qz, -q/z; q2)00.
(11.28)
k=-oo Ramanujan's sum, nl,
1 '1-'1
(.b. ) _ (q,b/a,az,q/az;q)oo a, ,q, z - ( b,q/ a,z,b / az;q ) 00 .
(11.29)
The sum of a well-poised 2't/J2 series,
2't/J2(b, e; aq/b, aq/e; q, -aq/be) (aq/be; q)00(aq2/b 2, aq2/e2, q2, aq, q/a; q2)00 (aq/b, aq/e, q/b, q/e, -aq/be; q)oo
(II.30)
Bailey's sum of a well-poised 3't/J3,
3't/J3 [
b, e, d
q ]
;q,q/b, q/e, q/d bed (q,q/be, q/bd,q/ed; q)oo (q/b, q/e,q/d, q/bed;q)oo·
(11.31 )
A basic bilateral analogue of Dixon's sum, nl,
[
4'1-'4
-qd , b, 1 -a 2 , aq/b,
e,
aq/e,
d. qa ~ ,q'bd aq/d e
1
(aq, aq/be, aq/bd, aq/ed, qa! /b, qa! /e, qa! /d, q, q/a; q)oo (aq/b, aq/e, aq/d, q/b, q/e, q/d, qa!, qa-!, qa~ /bed; q)oo .
(11.32)
The sum of a very-well-poised 6't/J6 series, nl,
qa !2 , -qa !2 , b, e, d, e. qa 2 ] 1 1 ,q,a2 , -a 2 , aq/b, aq/e, aq/d, aq/e bede (aq, aq/be, aq/bd, aq/be, aq/ed, aq/ee, aq/de, q, q/a;q)oo (aq/b,aq/e,aq/d,aq/e,q/b,q/e,q/d, q/e,qa 2/bede; q)oo .
6'1-'6 [
(11.33)
Appendix II
358 Bibasic sums: Gosper's indefinite bibasic sum, ~ 1- apkqk ~
k=O
(a;ph(c; q)k k (q; q)k(ap/c;ph
- - - --:----,----,-----'--:----,----,------'----:--q =
1- a
(ap;P)n(cq; q)n -n c (q; q)n (ap/c; P)n .
(11.34)
An extension of (11.34), ~ (1 - apkqk) (1 - bpkq-k)
6
(1 - a)(l - b)
(a, b; ph (c, a/bc; q)k qk (q, aq/b; q)k(ap/c, bCp;p)k
(ap,bP;P)n(cq,aq/bc;q)n (q,aq/b;q)n(ap/c,bcP;P)n
(11.35)
and, more generally, ~ (1 - adpkqk)(l - bpk /dqk)
~
k=-m
(1 - ad)(l - bid)
(a, b;P)k(C, ad 2/bc; q)k k (dq, adq/b; qh(adp/c, bcp/d;p)k q
(1 - a)(l - b)(l- c)(l- ad2/bc) { (ap, bP;P)n(cq, ad 2q/bc; q)n - d(l - ad)(l - b/d)(l - c/d)(l - ad/bc) (dq, adq/b; q)n(adp/c, bcp/d; P)n _ (c/ad, d/bc;P)m+1(1/d, b/ad; q)m+l} (l/c, bc/ad2; q)m+l (l/a, l/b;P)m+l '
(11.36)
where m is an integer or +00. An extension of the formula for the n-th q-difference of (apk; q)n-l,
t
(apk, bp- k ; q)n-l (1 - ap2k /b) (_l)kp(~) = 8n o. ( 1 _ ~) (1 _ ~) q q k=O (p;p)k(P;P)n_k(ap k /b;P)n+l ' (11.37)
Append ix III SELECTED TRANSFORMATION FORMULAS
Heine's transformations of 2¢1 (a,
2¢1
series:
(b, az; q)oo b; C; q, z ) = ( ) c, Z; q
( /
2¢1 C
= (c/b, ( bz; )q)oo c, Z; q
=
2¢1
( abz/ c, b; bz; q, c / b)
(IlL2)
00
2¢1, 2¢2
and
3¢2
(IlL3)
series:
) b; C; q, z ) = (az; ( q)oo ) 2¢2 ( a, / c b; c, az; q, bz z;q 00 _ (abz/c;q)oo A. [a,c/b,o. ] - (bz/c;q)oo 3'1-'2 c,cq/bz,q,q
2¢1 (a,
(a,bz,c/b;q)oo
+ (c,z,c /b) z;q Transformations of terminating
-n
(IlLl )
(abz/c; q)oo () 2¢1(c/a, c/b; C; q, abz/c). z;q 00
Jackson's transformations of
2¢1 (q
) b, Z; az; q, b
00
,b;c;q,z
)
00
A.
3'1-'2
(IlI.4)
[z,abz/c,o. ] bz, bzq / c ,q, q .
(IlL5)
series:
2¢1
(c/b;q)n (bZ)n (.) c,q n q X 3¢2(q-n, q/z, c- 1q1-n; bc- 1q1-n, 0; q, q) (IlL6) _ (c/b; q)n [q-n, b, bzq-n /c. ] (III. 7) (c,.q) n 3¢2 bq 1-n/ C, ,q, q
=
°
=
(c/b;q)nbn ( c,.q) n
A.
3'1-'1
[q-n,b,q/z. ~] bq1-n/ C ' q, C ,
(IlL8)
where, as elsewhere in this appendix, n denotes a non-negative integer. Transformations of 3¢2 series: A.
3'1-'2
[a,d,b,e c.,q, abc de ]
2:]
_ (e/ a, de/bc; q)oo ¢ [a, d/b, d/ c . - (e, de/abc; q)oo 3 2 d, de/bc ,q, a _ (b, del ab, de/bc; q)oo (d ,e, d e / abC; q) 00
3¢2
359
[d/b, e/b, de/abc. ] de / ab, de / bc ,q, b ,
(IlL9) (IlI.10)
Appendix III
360
q-n, b, e ] d, e ; q, q
[
3¢2
= (de/be;q)n (be)n d
( e; ) q n _ (e/e;q)n n - (e; q ) n e
A-.
3'1'2
[q-n,e,d/b. bq ] d, eq 1 -n /e' q, e ,
A-.
3'1'2
[q-n, b, e. de qn ] _ (e/e; q)n d , e ,q, be - (e,.q) n
A-.
3'1'2
(III. 11)
[q-n, d/b, die. ] d, de / be ' q, q
A-.
3'1'2
( ) III. 12 [q-n, e, d/b. ] (111.13) d ,eq I-n/' e q, q .
The Sears-Carlitz transformation of a terminating well-poised [
3¢2
b, aq / b,
a,
_ (az;q)oo -
(
provided that a
z;q
=
5'1'4 00
[d,-a!,(aq)!,-(aq)!,aq/be. ] ,q, q aq/b,aq/e,az,q/z
[
(III.14)
q-n. See (III.35) for a nonterminating case.
Sears' transformations of terminating balanced 4¢3
series,
e aqz] aq / e ;q'-be
A-.
)
3¢2
4¢3
series:
q-n, a, b, e ] d, e, f ; q, q
_ (e/a, f fa; q)n n ( e, f;) q n a
-
A-.
[
4'1'3
_ (a,ef/ab,ef/ae;q)n (e, f, ef /abe; q)n
4
q-n, a, d/b, d/e . ] 1 / 1 / ,q, q d, aq -n e, aq -n f
(III.15)
¢ [q-n,e/a,f/a,ef/abe. ] 3 ef lab, eflae, ql-n /a' q, q ,
(III.16)
where def = abeql-n. Watson's transformation formulas: [
8¢7
=
a,
qa!, 1
a"2,
-qa!, 1
-a"2,
b, aq/b,
d,
e,
aq/e,
(aq, aq/de, aq/df, aq/ef; q)oo (aq/ d, aq/ e, aq/ f, aq/ def; q)oo
aq/d,
¢ [ 4 3
e,
aq/e,
a 2q2 ]
f ;
q, b d
aq/ f e e
f
aq/be, d, e, f ] aq/b, aq/ e, def / a; q, q
(III.17) whenever the f = q-n, 8¢7
=
[
a,
8¢7
1
qa"2, 1
a"2,
series converges and the 1
-qa"2, 1
-a"2,
(aq, aq/de; q)n (aq/d, aq/e; q)n
b,
e,
d,
aq/b,
aq/e,
aq/d,
[aq/be, 4¢3
4¢3
d, aq/b,
e, aq/e,
series terminates, and, when e, aq/e,
q-n a 2qn+2] aqn+l; q, bede
de~=: / a ; q, q] (III.18)
Appendix III
361
or, equivalently, 4
¢ [q-n, a, b, c. ] _ (d/b, d/c; q)n 3 d,e,j ,q,q - (d,d/bc;q)n
x
S¢7
[
1
0",
1
q0"2,
-q0"2,
1
j la,
e/a,
b,
c,
e,
j,
ej lab,
ej lac,
1
0"2,
-0"2,
where dej = abcql-n and 0" = ej /aq. Another transformation of a terminating balanced poised S¢7 series,
4¢3
series to a very-well-
¢ [q-n, a, b, c . 4 3
] _ (abq/ j, acq/ j, bcq/ j, q/ j; q)oo ,q, q - (aq/ j, bq/ j, cq/ j, abcq/ j; q)oo
d, e, j
b, p,q/b,
c, JLq/c,
eqn. de] d ,q, abc '
(III.20) where dej = abcql-n and JL = abc/ f.
Singh's quadratic transformation: a2,b2,c,d ] [ b 1 1 d; q, q a q2, -a bq2,-C a 2 , b2, c 2, d 2 4¢3 [ 2b2 d d; q ,q a q, -c ,-c q
4¢3
=
2 2] ,
(II1.21 )
provided the series terminate.
A q-analogue of Clausen's formula:
]}2
a,b,abz,ab/z . [ b 1 1 b,q,q a q2, -abq2,-a _ [ a 2 , b2 , ab, abz, ab/z . ] 5¢4 b 1 b 1 b 2b2 ' q, q a q 2 ,-a q 2 ,-a ,a
{
4¢3
(III.22)
provided both series terminate. A non-terminating q-analogue of Clausen's formula is given in (8.8.17).
Transformations of very-well-poised S¢7
[
a,
1
qa 2 ,
1
-qa 2 ,
1
b,
1
a2, -a 2 , aq/b, (aq,aq/ej,Aq/e, Aq/j;q)oo (aq/e,aq/j,Aq, Aq/ej;q)oo
x
S¢7
[
A,
qA!, \ 1
/\2,
c, aq/c,
S¢7
d, aq/d,
series: e, aq/e,
j
aq/ j; q, bcdej
-qA!,
Ab/a,
Ac/a,
Ad/a,
e,
\! /\,
aq/b,
aq/c,
aq/d,
Aq/e,
_
a 2 q2 ]
j
aq ]
Aq/j;q, ej
(III.23)
362
Appendix III
(aq, b, bcf..t/a, bdf..t/a, bef..t/a, bff..t/a; q)oo (aq/c,aq/d,aq/e,aq/f,f..tq,bf..t/a;q)oo
x
8¢7
1
f..t,
[
1
qf..t2,
-qf..t2,
1
1
f..t2,
-f..t2,
1
aq/bc,
aq/bd,
aq/be,
bcf..t/a,
bdf..t/a,
bef..t/a , bff..t/a,
aq/bf,
bf..t/a aq/b; q, b ,
(III.24)
Transformations of a nearly-poised
[
5¢4
b,
c,
d,
aq/b,
aq/c,
aq/d,
a,
1
_ (Aq/a,A 2q/a;q)n (Aq, A2q/a 2; q)n
rI-. 12'i"11
-
1
a2
[q-n,
5 r1-. 4
-qA2,
bA/ a,
CA/ a,
dA/ a,
\ .1
-A! ,
aq/b,
aq/c,
aq/d,
1\ 2 ,
(aq)!,
-(aq)!,
A2qn+1/a,
-Aq/d,
A(q/a)!,
-A(q/a)!,
aq-n /A,
b,
d,
c,
q1-n/b,
'i"
1
qA2,
-a!,
,
Aq/d,
[A,
series:
5¢4
e
q1-n/ c , q1-n/d,
]
eq-2n/f..t2;q,q
(f..t2 qn+1, J.tq/ e; q)n (f..t 2qn+1/e,f..tq;q)n
x
12¢11
[
f..t'
1
qf..t2, .1 f..t2,
1
-qf..t2, 1 -f..t2,
f..tbqn, q1-n /b,
f..tcqn, q1-n /c,
f..tdqn, q1-n /d,
] q-n/2, _ q-n/2, q(1-n)/2, _q(1-n)/2, e, f..t2 qn+1/ e f..tq(n+2)/2, _f..tq(n+2)/2, f..tq(n+1)/2, _f..tq(n+1)/2, f..tq/e, eq-n / f..t; q, q ,
(III.26) where A = qa 2/bcd and f..t = q1-2n /bcd. Transformation of a nearly-poised rI-. 7'i"6
[a,
1
qa 2 , 1 a2,
1
-qa 2 , b, 1 -a2, aq/b,
c,
aq/c,
(Ajaq, A2/aq; q)n (1 - A2q2n-1/a) (1-A2/aq)
= (Aq,A 2/a 2q;q)n
7¢6
series:
d, aq/d,
q-n ] 2 ; q, q a q -n /A 2 2
[A,
qA!, A!,
12¢11
-qA!, -A!,
bAja,
cA/a,
aq/b,
aq/c,
1
1 1qa2, 1 1A 2 1 la, q-n dA/a, (aq)2, -(aq)2, -qa2, qn1
1
1
1
aq/d, A(q/a)2, -A(q/a)2, Aja2, -A/a2, aq2-n /A, Aqn+1
;
q, q ,
(III.27) where A = qa 2/bcd.
363
Appendix III
Bailey's
1O¢9
1O¢9 [
transformation formula:
1
! \ n+l/ e, j q -n a, qa! 2 , -qa 2, b,C, d ,e, j ,/\aq 1
1
a2,-a2,aq/b,aq/c,aq/d,aq/e,aq/j,ejq-n/A,aqn+l
_ (aq,aq/ej,Aq/e,Aq/j;q)n - (aq/e, aq/ j, Aq/ej, Aq; q)n Ad/a, aq/d,
e, Aq/e,
¢ [A, 10 9
j, Aq/ j,
qA!, A!,
Aaqn+l/ej, ejq-n la,
-qA!, -A!,
;q,q
Ab/a, aq/b,
Ac/a, aq/c,
q-n, ] Aqn+l ; q, q ,
(III.28)
where A = qa 2/bcd.
Transformations of r+2¢r+1 series: do
r+2'!-'r+l =
[
a ' b,1 b qml , ... , br qmr . a-I m+l-(ml +···+m r ) ] b l+m b ' q, q q ,1,···, b r
(q, bq/a; q)oo (bq; q)m(bI/b; q)ml ... (br/b; q)mr bml+.+mr-m (bq, q/a; q)oo (q; q)m(b 1 ; q)ml ... (b r ; q)mr X
r+2¢r+l
[
q-m, b, bq/b 1 , ... , bq/br ] bq / a, bq I-m1/b 1,···, bq I-mr /b r ; q, q
(III.29)
and do r+2'!-'r+ 1 [
a ' b,1 b qml , ... , br qmr . a-I 1-(ml+···+mr ) ] b b b ,q, q cq, 1,···, r
(III.30) where m, ml, ... ,mr are arbitrary nonnegative integers.
Three-term transformation formulas: 2¢1 (a,
q/c; q)oo b; c; q, z ) = (abz/c, ( / / ) az c, q a; q 00
(b, q/c, cia, az/q, q2 /az; q)oo - (/ b / c q, q c, q / a, az / c, cq / az; q) 00
2¢1 (a,
b; c; q, z )
=
(b,c/a,az,q/az;q)oo (b/ /) c, a, z, q z; q 00
bz, q/bz; q)oo + (a,(c,c/b, /b / ) a , z, q z; q 00
2¢1
(
( / a, cq / abz; cq / az; q, bq / c )
2¢1 C
2¢1
(/ / 2/ ) aq c,bq c;q c;q,z.
2¢1
(
(
III.31
)
/ / /) a, aq c; aq b; q, cq abz
/ / /) b, bq c; bq a; q, cq abz .
(III.32)
364
Appendix III
3¢2
a, b, e de] [ d,e ;q, abc
_ (e/b, e/ e, eq/ a, q/ d; q)oo ¢ [e, d/ a, eq/ e . bq ] - (e, eq/d, q/a, e/be; q)oo 3 2 eq/a, beq/e' q, d (q/ d, eq/ d, b, e, d/ a, de/beq, beq2 / de; q)oo ¢ [a q/ d, bq/ d, eq/ d . de ] q2/d, eq/d ,q, abc . - (d/q, e, bq/d, eq/d, q/a, e/be, beq/e; q)oo 3 2 (III.33) _ (e/b, e/e; q)oo [ a, b, e.,q, ~] b d, e a c e(, e /b e; q) 00
3¢2
(d/ a, b, e, de/be; q)oo
+ (d, e, be/e, de/abc; q)oo ¢ [a, 3
2
b, aq/b,
_ (ax;q)oo (x; q)oo
-
r/,
3¢2
3¢2
[d/a, b, e. ] / ,q, q d, beq e
[e/b, e/ e, del abc ] de/be, eq/be ; q, q .
(III. 34)
e aqx] aq/e; q, bc
[a!,
5'1'4
-a!, aq / b,
(aq)!, aq / e,
-(aq)!, ax,
aq/bc. ] q/x ,q,q
(a,aq/be,aqx/b,aqx/e;q)oo
+ (aq/b, aq/e, aqx/be, X-I; q)oo r/,
x ¢ [a, 8 7
5'1'4
qa!, a!,
[xa!,-xd,x(aq)!,-x(aq)!,aqx/be ] ; q, q . 2 aqx/b,aqx/e,xq,ax -qa!, -a!,
b, aq/b,
e, aq/e,
d, aq/d,
(III.35)
f. a 2q2 ] aq/ f' q, bedef
e, aq/e,
_ (aq, aq/de, aq/df, aq/ef; q)oo ¢ [ aq/be, d, e, f . ] - (aq/ d, aq/ e, aq/ f, aq/ def; q)oo 4 3 aq/b, aq/ e, def / a' q, q (aq,aq/be,d,e,f,a2q2/bdef,a2q2/edef;q)00 (aq/b, aq/e, aq/d, aq/e,aq/f, a 2q2/bedef,def/aq; q)oo
+~~~~--~~~~-7~~--~~~--
x
[ 8¢7
4¢3
a,
[
aq/de,aq/df,aq/ef,a2q2/bedef ] 2 2 2 2 2 ; q, q . a q /bdef,a q /edef,aq /def
qa!, a!,
-qa!, -a!,
b, aq/b,
e,
aq/e,
d, aq/d,
e,
aq/e,
(III.36)
f a 2q2 ] aq/f;q, bedef
(aq,aq/de,aq/df,aq/ef,eq/e, fq/e, b/a, bef/a;q)oo (aq/d, aq/e, aq/f,aq/def,q/e, efq/e, be/a, bf/a;q)oo x 8¢7 [
ef/e,q(ef/e)! ,-q(ef/e)! ,aq/be,aq/ed,ef/a,e,J 1
(ef /ep
1
,- (ef /ep
,bef la, def la, aq/e, fq/e, eq/ e
;
bd] q, --;;:
Appendix III
365
b (aq, bq/a, bq/c, bq/d, bq/e, bq/ j, d, e, j, aq/bc; q)oo a (aq/b,aq/c,aq/d,aq/e,aq/j,bd/a,be/a;q)oo
+-~~~~~~~~~~~~~~~
(bdej /a 2, a 2q/bdej; q)oo (bj/a,dej/a,aq/dej,q/c,b 2q/a;q)oo
x-:-::--:--;--.:..,,-::-'-:-'-----'---:-:-=-'-::---;:..-.c-~=:------:--
x
8
¢ [b 2/a, bqa-!, -bqa-!, b, bc/a, bd/a, be/a, bj /a. a 2q2 ] 7 ba-!,-ba-!,bq/a,bq/c,bq/d,bq/e,bq/j ,q, bcdej .
(III.37) Transformation of an 8'¢8 series: (aq/b, aq/c, aq/d, aq/e,q/ab, q/ac, q/ad, q/ae; q)oo (ja,ga, j/a,g/a,qa 2, q/a 2;q)oo
x
[
8'¢8
qa,
-qa
ba,
ca,
da,
ea,
ja,
a,
-a,
aq/b,
aq/c,
aq/d,
aq/e,
aq/ j,
(q,q/bj,q/cj,q/dj,q/ej,qj/b,qj/c,qj/d,qj/e;q)oo (ja,q/ja,aq/j,j/a,g/j,jg,qj2;q)oo
x
[P,
8¢7
+
qj,
-qj,
jb,
jc,
jd,
je,
j,
- j,
jq/b,
jq/c,
jq/d,
jq/e,
idem (1; g). 1O¢9
1
+
[
transformation: 1
a,qa'l,-qa'l,b,c,d,e,j,g,h 1
1
(III.38)
Bailey's four-term 1O¢9
jg q2 jq/g; q, bcdejg
1
a'l,-a'l,aq/b,aq/c,aq/d,aq/e,aq/j,aq/g,aq/h
;q,q
]
(aq,b/a,c,d,e,j,g,h,bq/c;q)oo 2 (b q/a,a/b,aq/c,aq/d,aq/e,aq/j,aq/g,aq/h,bc/a;q)oo (bq/ d, bq/ e, bq/ j, bq/ g, bq/ h; q)oo (bd/a, be/a, bj la, bg/a, bh/a; q)oo
x-c--.:..,-------'-,---:'--c--'-7------'---:---':--
x 1O¢9 [
1
1
;
ba-'l, -ba-'l, bq/a, bq/c, bq/d, bq/e, bq/ j, bq/g, bq/h
(aq,b/a, Aq/j,Aq/g, Aq/h,bj/A,bg/A,bh/A; q)oo (Aq,b/A, aq/j,aq/g, aq/h, bj/a, bg/a, bh/a; q)oo X 1O¢9
[
1
b2/a, qba-!, -qba-!, b, bc/a, bd/a, be/a, bj la, bg/a, bh/a
A, qA!, -qA!, b, Ac/a, Ad/a, Ae/a, j, g, h 1
1
;
A'l, -A'l, Aq/b, aq/c, aq/d, aq/e, Aq/ j, Aq/g, Aq/h
q, q
1
q, q
(aq,b/a,j,g,h,bq/j,bq/g,bq/h,Ac/a,Ad/a,Ae/a,abq/Ac;q)oo
+ (b 2q/A,A/b,aq/c,aq/d,aq/e,aq/j,aq/g,aq/h,bc/a,bd/a,be/a,bj/a;q)oo (abq/Ad, abq/Ae; q)oo x (bg/a,bh/a;q)oo
1O¢9
[ b2/A, qbA-!, -qbA-!, b, bc/a, bd/a, bA-!,-bA-!,bq/A,abq/cA,abq/dA,
be/a, bj /A, bg/A, bh/A ] .q q abq/eA,bq/j,bq/g,bq/h" ,
(III.39)
366
Appendix III
where a 3q2
=
bcdefgh and A = qa 2/cde.
Transformation of a
1O'l)i1O
series:
(aq/b,aq/c,aq/d,aq/e,aq/f,q/ab,q/ac,q/ad,q/ae,q/af;q)oo (ag,ah,ak,g/a,h/a,k/a,qa 2,q/a2;q)00 [ X 1O'l)i1O
qa, -qa, ba, ca, da, ea, fa, ga, ha, ka
q3
]
a, -a, aq/b, aq/ c, aq/ d, aq/ e, aq/ f, aq/ g, aq/ h, aq/ k ; q, bcdef ghk
(q,q/bg,q/cg,q/dg,q/eg,q/fg,qg/b,qg/c,qg/d,qg/e,qg/f;q)00 (gh,gk,h/g,k/g,ag,q/ag,g/a,aq/g, qg2;q)00
x
[ 1O
g2, qg, _qg, gb, ge, gd, ge, gf, gh, gk
q3]
g, _g, qg/b, qg/e, qg/d, qg/e, qg/ f, qg/h, qg/k; q, bedefghk
+ idem (g; h, k).
(III.40)
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Symbol index
(a)n (a; q)n (a; q)l/ (al' a2, ... ,am; q)n (a; q)oo (al' a2, ... ,am; q)oo (a; q,P)n, (al' a2,···, am; q,P)n (a;p, q)r,s ,al, laJ [a]q [n]q! [a]q,n [al' a2,·· . ,am]q,n [a; 0"] [n; O"]! [a;O"]n [a;O",T] [n;O",T]! [a; 0", T]n, [al' a2,··· ,am; 0", T]n [al' a2,·· . ,am; 0", T]n an(a, ;3lq) A(z) bk,j b(k, n;;3) bn (a,;3; q) B(z) B((}) B(x,y) Bq(x, y) B~(x; k) BE(t; q,p) cosq(x) Ck,n
2, 5 3, 6 208 6 6 6 304 112 205 7 7 7 7 7 8 8 17 312 312 312 192 185 195 185 192 189 265 23 23 205 344 28 195
Cj (;3lq) cn(x; a; q) c~(x; k)
C
CN (Contour) Cj,k,m,n (Contour) C(¢) Cosq(x) Cq(x;w) Cn(x; ;3lq) C~(x; ;3lq) C~(x)
C.T.
'Dq Dq Dn(x; ;3lq) r+ler res eq(z) eXPq(z) Eq(z) E± q r+1Er rEs Eq(x; a) Eq(x; a, b) Eq(x, y; a) F(a, b; c; z) rFs Fl ,F2,F3 F4 F(x, t) F(x, ylq) Fl (x, ylq), F2(x, ylq) 415
187 202 204 312 126 345 265 28 212 31 255 2 171 27 197 212 312 316, 319 11 12 11 197 304 309 12 12 111
2 3 283 219, 283 259 265 266
416 FA:B;C D:E;F
r9s rGr rGs 9o(a,b,e,d,f) G(x, t) Gl(x, ylq) G2(x, ylq) Gt(x; a, b, e, dlq) Gt(x; a, elq) h(x; a), h(x; a; q) h(x; aI, a2,··· ,am) h(x; aI, a2,··· ,am; q) hn(q),hn(a,b,e,N;q) ht(a) hn(a, b, e; q) hn(a, b, e, dlq) hn(a, b, e, d, 1) hn ((3lq) hn(x; q) Hn(x) Hn(xlq) Hn(x;q) Ht(x), Ht(x; a, b, e, dlq) H(x,y, t) idem (b; e) idem (al;a2, ... ,ar+1) I(a,b,e,d) 1m Ja(x) JP) (x; q), JS 2)(X; q), JS 3)(x; q) J(a,b,e,d,j,9) kn ]( (Contour)
]((x, y, z) ]((x, y, Z; (3lq) ](n(x; a, N; q) ](n(x; a, Nlq) ](t(x, y) ](t(x, y; (3lq) ](:ff (x; a, N; q) Lm(a; b, e, d) Lt(x, y; (3lq)
Symbol index
283 316,319 309 310 159 260 269 270 262 264 154 154 154 180 261 182 190 256 186 209 4 31 209 263 259 69 121 156 125 4 30 157 265 125 225 223 201 201 227, 261 227 202 164 227
231 4 L~(x) 210 L~(x; q) Mn(x; a, e; q) 202 M(x, y), M(x, y; a, b, e, e' ; q) 291 p(n) 239 240 PN (n), Pe (n), Pdist (n), Po( n) 241 Peven (n), Podd (n) 175 Pn(x) 195 Pn(x; a, (3) 32, 181 Pn(x; a, b; q) 194 Pn(eos(B + ¢); a, blq) 59, 189 Pn(x; a, b, e, dlq) 253 Pv(qX; a, b; q) P(z) 125 2 Pn(x) Pn(x; a, b, e; q) 182 255 Pn(xlq) 255 Pn(x; a, b, e, d, a2, a3, . .. , aslq)
Lt(x, y; a, b, e, a", M, N; q)
p~a,(3)(x)
p~a,(3) (x;
2
q)
191
p~a,(3) (xlq)
191 229 229 180 273 189 243 265 254 333 181 257 345 216 28 28 315 294 61 214 257
Pz(x, y) Pz(x,y;a,b,e,a",](,M,N;q) Qn(x), Qn(x; a, b, N; q) Qn(x; a, blq) Qn(z; a, b, e, dlq) r2k(n) 224, rn(x;a,b,e,dlq),rn(x) r~(x), r~(x; a, b, e, dlq) RN Rn(~(x)), Rn(~(x); b, e, N; q) Rm(x), Rm(x; a, b, e, d, 1) Rm,j(z) sn(x) sinq(x) Sinq(x) SL(2, 2':) S(a,b,e,d,j,9;A,~) S(A,~,V,p)
Sn(x;p, q) Sn(x),Sn(x;a,b,e,d,j)
Symbol index
Sq(x; w) rts ']['
,][,n Tn (X) Tn,k(Z) Un (X) U~a)(x; q) Vn r+lVr v(x; a, b, c, d, f) V(e iO ) r+1 v;.
Wj w(x) w(x; a, b, c, d) w(x; a, b, c, dlq) w( 0; q) wk(a, b, c, dlq) W(O) W(e iO ), W(eiO,a,b,c,dlq) W,a(xlq) Wn(x; b,q) Wn(x;q), Wn(x;a,b,c,fV;q) r+lWr x+ z,zqk Z 2Z Z+rZ J J(t) dqt J(n) "-' g(n)
L:~=m ak rr~=m ak
[~L' [~L [kl' ~ ,kmL
212 7 344 347 2 345 2 209 176 312 159 198 306 175, 179 175 157 190 192 191 194 198 185 214 59, 180 39 114 331 312 320 312 23 185 81 325 24 25
0
[~Lp' [~Lp
311
[~Lp
417
312
175 21 20, 29 rq(X) r(z; q,p) 311,338 345 r(Zl'oo 0 ,zn;q,p) t(z; q,p) 338 42 Om,n 11, 197 Oq 11 /).J(z), '\l J(z) /).Uk 80 32 /).bJ(Z) /).(Z) , /).(zqk) 331 /).(z;p), /).n(z;p) 333 /).E(Z; t; q,p) 344 /).E(Z; t) 345 303 O(x;p), O(Xl' 0 0 0, xm;P) 16 171 (x, q), 172 (X, q), 173 (X, q), 174 (x, q) 190 K(a, b, c, dlq) 180 Aj 156 An(q) 156 An(a, b, c, d, a2, a3, 0 0 0, aslq) 181 JL(x) 269 Vn 270 1rn p(x;q),p(x;a,b,c,fV;q) 180 194 Pn(a, blq) 156 p(xlq) 156 p(x; a, b, c, d, a2, a3, 0 0 0, aslq) ¢(a,b;c;q,z) 3 2¢1 (a, b; c; q, z) 3 4 r¢s [0 0 oj 95 (l) , (2) ,(3), (4) 283 294 1(a; {3, (3'; 'I; q; x, y) D(a; bl , 0 0 0, br ; c; q; Xl, 0 0 0, x r ) 300 A:B;C 283 D:E;F 137 r1/Js 315 r-l W r-2 a(x) r(x)
Author index
Adams, C. R., 36 Adiga, C., 35, 67, 152 Agarwal, A. K., 111, 257 Agarwal, N., 112 Agarwal, R. P., 34-36, 68, 111, 112, 119, 122, 124, 136, 301 Aigner, M., 35 Alder, H. L., 257 Alexanderson, G. L., 35 Alladi, K., 35, 67 Allaway, Wm. R., 204, 205, 214, 216 AI-Salam, W. A., 35, 36, 52, 83, 86, 204, 205, 209, 214-216, 273 Andrews, G. E., 9, 15, 21, 26, 29, 33-37, 52, 53, 59, 65, 67, 68, 111, 112, 138, 141, 147, 149, 152, 174, 181, 182, 184, 213, 215, 216, 241-243, 245, 257, 296, 297, 300, 341, 342 Aomoto, K., 174 Appell, P., xv, 282 Artin, E., 36 Askey, R., 2, 9, 17, 21, 26, 29, 31, 33-35, 52, 53, 59, 67, 84, 99, 113, 125, 129, 136, 138, 141, 149, 152, 154, 165, 170-172, 174, 177, 180-182, 185, 188, 191, 193, 195, 197, 198, 200, 204-206, 214, 215, 232, 236, 242, 257, 273, 274, 281 Atakishiyev, M. N., 215, 257 Atakishiyev, N. M., 12, 35, 195, 213-215, 257 Atkin, A. O. L., 152 Atkinson, F. V., 176 Bailey, W. N., xxiii, 5, 9, 18, 41, 46, 47, 50, 53, 54, 57, 58, 60, 61, 64, 67, 73, 96100, 112, 140, 148-150, 152, 219, 236, 261, 287-290, 293, 298, 299 Baker, M., 35 Baker, T. H., 68 Bannai, E., 213 Barnes, E. W., 113, 117
418
Bateman, H., 221, 222 Baxter, R. J., 35, 67, 302, 307 Beckmann, P., 257 Beerends, R. J., 301 Bellman, R., 35 Bender, E. A., 34 Berg, C., 214 Berkovich, A., 35, 67, 341, 342 Berman, G., 35 Berndt, B. C., 35, 67, 152, 257 Bhargava, S., 35, 67, 152 Bhatnagar, G., 257, 258, 325, 349, 350 Biedenharn, L. C., 213 Bohr, H., 29 Baing, H., 34 Borwein, J. M., 67 Borwein, P. B., 67 Bowman, D., 36 de Branges, L., xv, 84, 232, 237, 257 Bressoud, D. M., 35, 36, 67, 68, 83, 111, 112, 213, 226, 257, 301 Bromwich, T. J. I'A., 5 Brown, B. M., 214, 216 Burchnall, J. L., 111, 112, 291 Burge, W. H., 67 Bustoz, J., 36, 213, 273, 280, 281 Carlitz, L., 28, 31, 35, 36, 58, 64, 111, 152, 209, 210, 215, 216, 275, 281 Carlson, B. C., 34 Carmichael, R. D., 36 Carnovale, G., 35 Cauchy, A.-L., 9, 112, 132 Cayley, A., 112 Charris, J., 215 Chaundy, T. W., 111, 291 Cheema, M. S., 35 Chen, Y., 216 Cherednik, 1., 68 Chihara, L., 213-215
Author index
Chihara, T. S., 34, 176, 209, 213-216, 259, 273 Chu Shih-Chieh, 2 Chu, VV., 111, 151, 152, 257, 325, 349, 350 Chudnovsky, D. V., xv, 232 Chudnovsky, G. V., xv, 232 Chung, VV. S., 258 Ciccoli, N., 215 Cigler, J., 33, 35, 112, 216 Clausen, T., xiv, 103, 232 Cohen, H., 258 Comtet, L., 258 Coon, D., 35 Cooper, S., 68, 258 Crippa, D., 34 Date, E., 302, 307 Daum, J. A., 18 Dehesa, J. S., 216 Delsarte, P., 181, 202, 213, 215 Denis, R Y., 301, 350 Desarmenien, J., 112, 216 Dickson, 1. E., 243 van Diejen, J.F., 258, 302, 317, 347, 349 Di Vizio, L., 36 Dixon, A. C., 38 Dobbie, J. M., 67 Dougall, J., 38 Dowling, T. A., 35 Dunkl, C. F., 35, 181, 202, 259 Duren, P. L., 257 Dyson, F. J., 67, 68, 242, 257 Edwards, D., 112 Eichler, M., 307 Erdelyi, A., 34, 66, 111, 174, 236, 249, 265, 284, 287, 302 Euler, 1., 3, 62, 239, 241 Evans, R J., 68, 174 Evans, VV. D., 214 Ewell, J. A., 35 Exton, H., 36, 283 Faddeev, L. D., 36 Fairlie, D. B., 34 Favard, J., 176 Feinsilver, P., 33 Felder, G., 312, 338, 339, 350 Feldheim, E., 281 Fields, J. L., 84, 86 Fine, N. J., 67, 112, 242 Floreanini, R, 215, 301
419
Floris, P. G. A., 258 Foata, D., 36, 112 Foda, 0., 67 Forrester, P. J., 67, 68, 257 Fox, C., 19 Frenkel, I. B., 7, 17, 258, 302, 307, 315 Freud, G., 176 Fryer, K. D., 35 Fiirlinger, J., 112 Gangolli, R., 195 Garoufalidis, S., 34 Garrett, K., 67, 148 Garsia, A. M., 67, 112 Gartley, M. G., 12 Garvan, F. G., 34, 35, 67, 174 Gasper, G., 19, 32-34, 36, 58, 61, 65-67, 74,78,80,81,83-85,88,91,93,94,105107, 109-111, 130, 135, 136, 172, 174, 186, 195, 207, 208, 213, 215, 217, 223, 226, 227, 229, 232, 233, 235, 236, 238, 239, 248, 250, 251, 256-258, 279, 300, 302, 326, 327, 329, 343, 344, 349, 350 Gauss, C. F., xiv, 1 Gegenbauer, 1., 225, 249 Geronimo, J. S., 258 Gessel, I., 83, 93, 112, 258, 349, 350 Goethals, J. M., 181, 215 Goldman, J., 35, 258 Gonnet, G. H., 34, 174 Gordon, B., 152 Gosper, R. VVm., 21, 34, 36, 81, 93, 111, 148 Goulden, I. P., 37, 68, 257 Greiner, P. C., 19 Grosswald, E., 243 Gupta, D. P., 215 Gustafson, R A., 35, 152, 173, 258, 349, 350 Habsieger, L., 174 Hahn, VV., 28, 30, 36, 138, 180, 181 Hall, N. A., 72 Handa, B. R., 35 Hardy, G. H., 44, 45, 138, 240-242 Heine, E., xviii, 3, 9, 13, 14, 21, 26-28 Henrici, P., 32, 34 Hickerson, D., 242, 257 Hilbert, D., 243 Hirschhorn, M. D., 152, 245, 258 Hodges, J., 215 Hofbauer, J., 112
420
Author index
Hou, Q.-H., 216 Hua, L. K., 243 Ihrig, E., 35 Ismail, M. E. H., 12, 22, 30, 31, 35-37, 61, 65, 67, 68, 86, 111, 138, 141, 148, 152, 155, 172-174, 177, 184, 185, 204, 209, 212-216, 251, 254, 257-259, 261, 272277, 280, 281, 301 Ito, M., 174,349 Ito, T., 213 Jackson, D. M., 37 Jackson, F. H., 14, 18, 21-23 28 30 32 35, 36, 43, 58, 87, 103, 11~, 1'12, '136: 138, 232, 282-284, 311, 338 Jackson, M., 144, 146, 152 Jacobi, C. G. J., 15, 34 Jain, V. K., 36, 60, 61, 63, 78, 97, 98, 101, 111, 112, 152, 232, 242, 301 Jimbo, M., 35, 302, 307 Joichi, J. T., 34, 35 Joshi, C. M., 68
Lakin, A., 141 Lam, H. Y., 258 Lapointe, L., 301 Lascoux, A., 216 Lassalle, M., 174, 201 Le, T. T., 34 Leininger, V., 258, 349 Leonard, D. A., 213 Lepowsky, J., 35, 68 LeTourneux, J., 215 Lewis, R. P., 35 Li, X., 215 Libis, C. A., 215 Lilly, G. M., 258, 349 Littlewood, J. E., 118 Liu, Zhi-Guo, 258 Lorch, L., 22 Louck, J. D., 213 Lubinsky, D. S., 216 Luke, Y. L., 34, 88
Kac, V. G., 35, 152 Kadell, K W. J., 36, 68, 152, 174, 258 Kairies, H.-H., 36 Kajihara, Y., 302, 346 Kajiwara, K, 302 Kalnins, E. G., 135, 214, 257, 258 Kampe de Feriet, J., xv, 282 van Kampen, N. G., 35 Kaneko, J., 68, 174 Karlsson, Per W., 18 Kashaev, R. M., 36 Kendall, M. G., 35 Kirillov, A. N., 258 Klein, F., xv Klimyk, A. U., 215, 257 Knopfmacher, A., 34 Knuth, D., 35, 36 Koekoek, R., 259, 273 Koelink, H. T., 213-215, 252, 256, 258, 274, 279, 280, 302 Koepf, W., 34 Koornwinder, T. H., 21, 33-35, 101, 172, 202, 210, 213-216, 257, 258, 349 Krattenthaler, C., 34, 83, 111, 112, 258, 301, 346, 349 Kummer, E. E., xv Kuniba, A., 302, 307
Macdonald, 1. G., 35, 68, 174 MacMahon, P. A., 35 Masson, D. R., 36, 173, 213, 215, 275, 276, 280, 281 Masuda, T., 37, 302 McCoy, B. M., 67, 68 Meijer, H. G., 36 Menon, P. K., 35 Merkes, E., 36 Milin, 1. M., 257 Miller, W., 34, 36, 135, 214, 258 Milne, S. C., 35, 67, 152, 174, 213, 257, 258, 303, 325, 331, 333, 335, 349, 350 Mimachi, K., 36 Minton, B. M., 18, 19 Misra, K. C., 68 Miwa, T., 302, 307 Moak, D. S., 21, 29, 211, 216 Mohanty, S. G., 35 Molk, J., 151, 331 Mollerop, J., 29 Monsalve, S., 215 Mordell, L. J., 258 Morris, W., 68, 174 Mu, Y.-P., 216 Muldoon, M. E., 22, 36 Mulla, F. S., 257 Mullin, R., 258 Muttalib, K A., 216
Laine, T. P., 216
Nakagami, Y., 37
Author index
Narukawa, A., 350 Nassrallah, B., 96, 104, 105, 112, 158, 163, 232, 295, 301, 341 Needham, J., 2 Nelson, C. A., 12 Nevai, P. G., 186, 190, 215, 259 Newcomb, J. W., 258, 350 Nikiforov, A. F., 34, 135, 136, 213 Nishizawa, M., 350 van Norden, Y., 302 Noumi, M., 258, 302, 346 O'Hara, K. M., 35 Ohta, Y., 302 Okado, M., 302, 307 Onofri, E., 257 Opdam, E. M., 68 Orr, W. McF., 112 Pastro, P. I., 216 Paule, P., 34, 67, 68, 112 Pearce, P. A., 35, 67 Perline, R, 37 Perron, 0., 176 Petkovsek, M., 34 Pfaff, J. F., xiv, 17 Phragmen, E., 302 Pitman, J., 301 Pitre, S. N., 301 P6lya, G., 35 Potter, H. S. A., 33 Qazi, Tariq M., 214, 254, 255, 257 Quano, Y.-H., 67 Rademacher, H., 241 Rahman, M., 34, 36, 58, 61, 65, 74, 78, 81, 91-94, 96, 104, 108-112, 136, 152, 153, 155, 158, 159, 161, 163, 169, 171-174, 185, 191, 207, 208, 211-217, 223, 225229, 232, 235, 248, 249, 251, 253-258, 269, 274-279, 281, 295, 341, 342, 350 Rains, E., 174, 302, 347, 349 Rainville, E. D., 34 Rakha, M. A., 258, 349 Ramanujan, S., 68, 138, 152, 172, 232, 236, 240 Ramanathan, K. G., 67 Rankin, R A., 67 Remmel, J., 112 Reshetikhin, N. Yu, 258 Riese, A., 34
421
Rogers, L. J., xviii, 31, 68, 184, 226 Rogov, V.-B. K., 35 Rosengren, H., 258, 281, 301-303, 331, 332, 335, 337, 346-350 Rota, G.-C., 35, 258 Rothe, H. A., 9 Roy, R, 9, 113, 125, 129, 172 Ruedemann, R. W., 213 Ruijsenaars, S. N. M., 311, 338 Saalschiitz, L., 17, 113 Saff, E. B., 216 Sahai, V., 301 Sauloy, J., 36 Schempp, W., 257 Schilling, A., 67, 68 Schlosser, M., 152, 174, 257, 258, 302, 326, 327, 329, 343, 344, 347, 349, 350 Schur, I. J., 68 Schiitzenberger, M.-P., 33 Schwartz, A. L., 257 Schweins, Ferd. F., 9 Sears, D. B., 15, 49, 51, 61, 64, 70, 71, 74, 130, 131, 136, 152 Selberg, A., 174 Shohat, J., 214 Sills, A. V., 34, 68 Simon, K., 34 Singh, V. N., 99, 105, 232 Slater, L. J., 5, 9, 32, 36, 68, 125, 126, 128, 130, 141-144, 151, 152, 242, 331 Spiridonov, V. P., 215, 302-306, 309, 310, 313, 316, 317, 320, 338, 341, 345, 347, 349 Srinivasa Rao, K., 301 Srivastava, B., 301 Srivastava, H. M., 84, 107, 232, 283 Stanton, D., 34-36, 65, 67, 68, 83, 93, 111, 112, 148, 155, 174, 181, 201, 213, 215, 216, 258, 276, 277, 280, 349, 350 Starcher, G. W., 36 Stembridge, J. R., 68 Stevens, L., 350 Stokman, J. V., 174, 215, 253, 258, 279 Stone, M. H., 176 Stuart, A., 35 Styer, D., 36 Subbarao, M. V., 152, 349 Sudler, C., 35 Suslov, S. K., 12, 34-36, 111, 135, 136, 152, 153, 195, 213, 214, 216, 257, 258, 274-276, 280, 281
422
Author index
Swarttouw, R. F., 36, 258, 259, 273 Swinnerton-Dyer, P., 152 Sylvester, J. J., 35 Szego, G., 31, 34, 35, 176, 177, 213, 216, 249, 259, 260, 265 Takacs, L., 2 Tamarkin, T., 214 Tannery, J., 151, 331 Tarasov, V., 174 Temme, N., 259 Terwilliger, P., 213 Thomae, J., xviii, 21, 23, 24, 69, 70 Toeplitz, 0., 35 Touhami, N., 258 Tratnik, M. V., 213, 215, 258 Trebels, W., 257 Trjitzinsky, W. J., 36 Trutt, D., 257 Turaev, V. G., 7, 17, 258, 302, 307, 315 Ueno, K., 37 Upadhyay, M., 301 Uvarov, V. B., 34, 136, 213 Valent, G., 213 Vandermonde, A. T., 2 Van Assche, W., 258 Van der Jeugt, J., 252, 258, 280, 301 Varchenko, A., 174, 312, 339, 350 Verma, A., 36, 52, 61, 63, 65, 68, 78, 83, 86, 97, 98, 111, 112, 185, 207, 214, 216, 225, 228, 242, 249, 257, 342, 349 Vidyasagar, M., 152 Viennot, G., 67, 216
Vilenkin, N. Ja., 34 Vinet, L., 215, 301 Volkov, A. Yu., 36 Wallisser, R., 36 Warnaar, S. 0., 34, 67, 68, 302-304, 306, 323, 325, 329, 339-342, 346, 347, 349 Watson, G. N., xv, 16, 17, 21, 34, 35, 42, 67, 68, 112, 114, 115, 117, 119, 124, 151, 152, 220, 317 Whipple, F. J. W., 49, 68, 112 Whittaker, E. T., 16, 17, 21, 34, 151 Wilf, H. S., 34 Wilson, J. A., 59, 99, 132, 152, 154, 165, 177, 180, 188, 191, 193, 195, 197, 198, 200, 206, 214, 215, 259, 261, 272, 274, 317 Wilson, R. L., 68 Wimp, J., 37, 84 Wintner, A., 176 Wright, E. M., 35, 241, 242 Wu, M.-Y., 34 Xu, Y., 259 Yamada, Y., 302 Yang, K.-W., 33 Yoon, G. J., 213 Zagier, D., 307 Zaslavsky, T., 35 Zeilberger, D., 34, 35, 68, 174 Zhang, R., 12, 204, 251 Zhedanov, A. S., 215, 302 Zimmermann, B., 34
Subject index
Addition formula for continuous q-ultraspherical polynomials, 249 Eq functions, 111 little q-Jacobi polynomials, 210 little q- Legendre functions, 254 Affine q-Krawtchouk polynomials, 202 Al-Salam-Chihara polynomials, 273 Almost-poised series, 111 Analytic continuation of 21 series, 117 r+" cPr series, 120 Andrews' q-Dyson conjecture, 68 Appell functions and series, 282, 283 Askey-Gasper inequality, 232 q-analogues of, 236-238 Askey-Wilson polynomials, 59, 188 multivariable extension of, 255 Askey-Wilson q-beta integral, 154, 163-168 Associated Askey-Wilson polynomials, 254 Associated q-ultraspherical polynomials, 255 Bailey-Daum summation formula, 18, 354 Bailey's four-term transformation formulas for balanced very-well-poised lOcP9 series, 55-58, 64, 365 Bailey's identity, 61 Bailey's lemma, 41 Bailey's product formulas, 219, 236 Bailey's sum of a very-well-poised 6'I/J6 series, 140, 357 Bailey's summation formula, 54, 124, 356 Bailey's three-term transformation formula for a very-well-poised 8cP7 series, 53, 364 Bailey's transformation formulas for 2'I/J2 series, 150 terminating 5cP4 and 7cP6 series, 45-47, 363 terminating lOcP9 series, 47, 263 limiting cases of, 48-53, 360-362
423
Bailey's 3'I/J3 and 6'I/J6 summation formulas, 149, 150, 357 Balanced series (and k-balanced), 5 Barnes' beta integral, 114 Barnes' contour integral, 113 Barnes' first and second lemmas, 113 q-analogues of, 119 Base modular parameter, 304 Basic contour integrals, 113 Basic hypergeometric functions, 5 Basic hypergeometric series, 1-4 Basic integrals, 23 Basic number, 4 Bateman's product formula, 221, 222 Bessel function, 4 Beta function, 22 Beta function integral, 23 Bibasic expansion formulas, 84-87 series, 80-87 summation formulas, 80-83, 328, 358 transformation formulas, 105-107 Big q-Jacobi polynomials, 181, 182 Bilateral basic hypergeometric series, 137 Bilateral bibasic series, 82 Bilateral q-integral, 23 Bilateral theta hypergeometric series, 309, 316 Bilinear generating functions, 227, 259, 281 Binomial theorem, 8 q-analogue of, 8, 354 Biorthogonal rational functions, 35, 173, 213, 257, 258, 345 Cauchy's beta integral, 132 Christoffel-Darboux formula, 200 Chu-Vandermonde formula, 3 q-analogue of, 14, 354 Clausen's formula, 103, 232 q-extensions of, 232, 235, 236, 251, 261 Connection coefficients, 33, 195-197
424
Subject index
Continuous q-Hahn polynomials, 193 in base q-l, 279 Continuous q-Hermite polynomials, 31 Continuous q-Jacobi polynomials, 191 Continuous q-ultraspherical polynomials, 31, 184 Contour integral representations of 2Fl series, 113 2<1>1 series, 115 very-well-poised series, 121-124 Convergence of basic hypergeometric series, 5 bilateral basic series, 137 hypergeometric series, 5 Cubic summation and transformation formulas, 93, 108-110
Darboux's method, 259 Discrete q-Hermite polynomials, 209 Dixon's formula, 38 q-analogue of, 44, 355 Double product theta function, 303 Dougall's summation formulas, 38, 39 q-analogues of, 43, 44, 356 Dual orthogonality, 176 Dual q-Hahn polynomials, 181 Dyson's conjecture, 68
Elliptic analogue of Bailey's transformation formula for a terminating 1O9 series, 307, 323 Elliptic analogue of Jackson's 87 summation formula, 307, 321 Elliptic balancing conditions (E-balanced), 305, 309, 313, 314 Elliptically balanced series, see Elliptic balancing conditions Elliptic beta function, 344 Elliptic Jackson sum, 347 Elliptic Dn Jackson sum, 348 Elliptic gamma function, 311, 338 Elliptic hypergeometric series, 302, 305 Elliptic integrals, 344, 345, 347 Elliptic numbers, 17 Elliptic shifted factorials, 304, 312 Erdelyi's formula, 174 Euler's identity, 62 Euler's integral representation, 24 Euler's partition identity, 240 Euler's transformation formulas, 13, 86
en
Expansion formulas (also see Addition formula, Connection coefficients, Linearization formula, Nonnegativity, Product formulas, and Transformation formulas), basic series, 19, 40, 41, 62--67, 84-87, 107,143 elliptic and theta hypergeometric series, 329-331, 337 multi basic series, 95-97 Fields-Wimp expansion, 84 q-extensions of, 84-87 q,p-extensions of, 329-331 Gamma function, 2, 21, 22 q-analogues of, 20, 29, 353 Gauss' hypergeometric series, 1, 2 Gauss' multiplication formula, 22 Gauss' summation formula, 3 Gegenbauer's addition formula, 249 q-extension of, 249 Gegenbauer polynomials, 2 Gegenbauer's product formula, 225 General basic contour integral formulas, 126 General transformations for r'l/Jr series, 138-145 Generalized hypergeometric series, 3 Generalized Stieltjes-Wigert polynomials, 214 Generating functions for orthogonal polynomials, 31, 184, 203, 259-281 partitions, 239, 240 q- Bessel functions, 31 Gosper's sums, 81, 93, 358 extensions, 81, 82, 93, 358 Hall's formula, 72 Heine's series, 3 Heine's summation formula, 14 Heine's transformation formulas, 13, 14, 359 Hermite polynomials, 4 q-analogues of, 31 Hypergeometric functions, 5 Identities involving q-shifted factorials, 6, 24, 25, 351-353 Indefinite summation formulas, 80-83, 322, 324-329, 342-344 Infinite products, 6, 20-23, 352, 353 sums of, 61, 150-152, 304-310, 312-349
Subject index Integral representations of 2Fl series, 24 21 series, 24 associated Askey-Wilson polynomials, 255 very-well-poised 81>7 series, 157, 158 very-well-poised 101>9 series, 159-161 Integral representations for q-Appell series, 284, 286, 288, 289, 294 Inversion in the base of a basic series, 25 theta hypergeometric series, 337 Jackson's product formula, 93 Jackson's summation formulas, 43, 355, 356 Jacobi polynomials, 2 Jacobi's triple product identity, 15, 357 Jain's transformation formula, 60 Jain and Verma's transformation formulas, 63, 78 Jump function, 175 Kampe de Feriet series, 284 Karlsson-Minton summation formulas, 18, 19 q-extensions of, 19, 20 Kummer's formula, 18 q-analogue of, 18, 354 Laguerre polynomials, 4 Legendre polynomials, 2 Level basic series, 111 Linearization formula for Cn(x;,Blq), 226 inverse of, 249 Linearization formula for continuous q-Jacobi polynomials, 253 associated q-ultraspherical polynomials, 255 Little q-Jacobi polynomials, 32, 181, 182, 245 Little q-Legendre function, 253 Milne's fundamental theorem, 331 Rosengren's elliptic extension of, 331, 332 Modular balancing conditions (M-balanced), 315-317 Modular group, 315 Modular hypergeometric function, 315 Modular invariant, 316 Modular parameters, 304, 312 Modular series, 316 Modular symmetry relations, 317 Moments, 175
425
Multibasic hypergeometric series, 95-97, 105-107, 112 Multibasic summation and transformation formulas for theta hypergeometric series, 325-331 Multinomial coefficients, 68 q-analogue of, 25, 68 Multivariable sums, 331-336, 345-350
Nearly poised series, 38, 39 of the first and second kinds, 38, 39 Nome modular parameter, 304 N onnegativity of 3F2 series, 232, 237 4F3 series, 239 51>4 series, 237 61>5 series, 238, 239 71>6 series, 238 connection coefficients, 197, 215 linearization coefficients, 226, 227, 257 Poisson and other kernels, 229-232, 257
Orthogonality relations for affine q-Krawtchouk polynomials, 201 Al-Salam-Carlitz polynomials, 209 Askey-Wilson polynomials, 190, 191, 206-208 big q-Jacobi polynomials, 182 continuous q-Hermite polynomials, 31, 204 continuous q-Jacobi polynomials, 191, 192 continuous q-ultraspherical polynomials, 31, 184-187 discrete q-Hermite polynomials, 209 dual q-Hahn polynomials, 181 little q-Jacobi polynomials, 182 q-Charlier polynomials, 202 q-Hahn polynomials, 180 q-Krawtchouk polynomials, 201, 203 q-Laguerre polynomials, 210 q-Meixner polynomials, 202 q-Racah polynomials, 180 sieved orthogonal polynomials, 204, 205 Orthogonal system of polynomials, 175 in several variables, 213, 255, 258
Partial sums, 58, 63, 80-83, 106, 358 Partitions, 239-242 Pentagonal numbers, 241 Pfaff-Saalschiitz summation formula, 17 q-analogue of, 17, 355
426
Subject index
Poisson kernels for continuous q-ultraspherical polynomials, 227-229 q-Racah polynomials, 229-232 Product formulas for 2Fl series, 103, 219-221, 232, 236 Askey-Wilson polynomials, 222 balanced 43 polynomials, 218-223 basic hypergeometric series, 103-105, 232-236, 251, 361 big q-Jacobi polynomials, 245 continuous q-Jacobi polynomials, 253 continuous q-ultraspherical polynomials, 223, 227, 236, 249 Gegenbauer polynomials, 225 hypergeometric series, 103,219-221,232, 236 Jacobi polynomials, 220, 222 little q-Jacobi polynomials, 210, 245 q-Hahn polynomials, 221, 246 q-Racah polynomials, 221, 246, 247 Projection formulas, 195
q-analogue of Fl (a;b,b' ;c;x,y), 283-285, 294-296 q-analogue of F 4 (a,b;c,c';X(1-y),Y(1-X)), 219, 220, 283, 290-294 q-Appell functions, 282, 283 q-Bessel functions, 30, 104 q-beta function, 22, 23 q- beta integrals of Andrews and Askey, 53 Askey, 170, 171 Askey and Roy, 129 Askey and Wilson, 154, 155 Gasper, 130 Nassrallah and Rahman, 158,295 Ramanujan, 172 Wilson, 132 q-binomial theorem, 8, 354 q-binomial coefficients, 24, 353 q-Cayley-Orr type formulas, 105, 112 q-Charlier polynomials, 202 q-Clausen formulas, 232-235, 361 q-contiguous relations for 43 polynomials, 200 Heine's series, 27 q-Cosine functions, 28 q-deformation, 7 q-derivative operators, 27, 197, 208, 251 q-difference equations, 32, 199
q-difference equations for Askey-Wilson polynomials, 199 q-difference operators, 32 forward and backward, 11 symmetric, 11 q-differential equations, 27 q-Dixon sums, 44, 58, 355 q-Dougall sum, 44, 356 q-exponential functions, 11, 12,33,34, 111, 354 q-extension, 7 q-gamma functions, 20-22, 29, 353 q-Gauss sum, 14, 354 q-generalization, 7 q-Hahn polynomials, 180, 181 q-integral representations of 2(1)1 series, 24 Askey-Wilson polynomials, 207 continuous q-ultraspherical polynomials, 185 very-well-poised 87 series, 52 q-integrals, 23, 24, 52-55, 57, 58, 65, 76, 149, 156-158, 162, 163, 166, 171, 172, 183, 185, 203, 207 q-Kampe de Feriet series, 284 q-Karlsson-Minton sums, 19, 20, 357 q-Krawtchouk polynomials, 201, 202 q-Kummer sum, 18,354 q-Lagrange inversion theorem, 107 q-Laguerre polynomials, 210 q-Lauricella function, 300, 301 q-Leibniz formula, 27 q-Mehler's formula, 275 q-Meixner polynomials, 202 q-multinomial coefficients, 25, 68 q-multinomial theorem, 25 q-number,7 q-number factorial, 7 q-number shifted factorial, 7 q-quadratic lattice, 12, 294 q-Racah polynomials, 59, 197-180 q-Saalschiitz sum, 17, 355 q-series, 4, 8, 282 q-shifted factorial, 3, 351 q-sine functions, 28 q-trigonometric functions, 28, 212 q-ultraspherical function of the second kind, 211 q-Vandermonde sum, 14,354 q-Watson sum, 61, 355 q-Whipple sum, 61, 355 q,p-binomial coefficient, 311
Subject index
q,p -shifted factorial, 304 Quadratic elliptic transformation formula, 341 Quadratic summation and transformation formulas, 69-90, 91,92, 96-100, 162, 163, 341, 361, 362 Quartic summation and transformation formulas, 94, 111, 109 Quasi-periodicity relations, 317 Quintuple product identity, 147
Ramanujan's identities, 171, 172 Ramanujan's summation formula, 52, 138, 357 Recurrence relations for Askey-Wilson polynomials, 188 big q-Jacobi polynomials, 202 continuous q-ultraspherical polynomials, 203, 204 discrete q-Hermite polynomials, 209 little q-Jacobi polynomials, 204, 205 orthogonal polynomials, 175, 200 q-Racah polynomials, 177 sieved orthogonal polynomials, 201, 202 Reduction formulas for q-Appell series, 282, 284-289, 299 Reversal of terminating basic series, 25 theta hypergeometric series, 338 Rodrigues-type formula for the Askey-Wilson polynomials, 199 Rogers' linearization formula, 226 inverse of, 249 Rogers-Ramanujan identities, 44, 241 Rogers-Szego polynomials, 210 Saalschiitzian series, 5 Saalschiitz's formula, 17 Sears' 43 transformation formula, 49, 360 Sears' nonterminating extension of the q-Saalschiitz sum, 51, 356 Sears' transformations of well-poised series, 130, 131 Sears-Carlitz transformation formulas, 64, 75, 360 Selberg's integral, 174 Shifted factorial, 2 Sieved ultraspherical polynomials of the first kind, 204 second kind, 205 Singh's quadratic transformation, 99, 361 6j-coefficients, 180, 302, 307
427
Split-poised series, 106 Squares of 2 F1 series, 232 2
428
Subject index
Transformation formulas (selected): r+1
very-well-poised 2r'
