A solid reference on the subject. Material on generalized hypergeometric functions (starting with Gauss' hypergeometric function) is presented followed by the q analogy's. The material is advanced and is well written with a tight and readable typeface. The introduction to q series will satisfy the beginner. The list of about 500 references covering the entire subject is worth the price alone. Lorenz H. Menke, Jr.

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Gasper, George

Greenwich Medical Media Ltd

Encyclopedia of mathematics and its applications, Second edition, Cambridge, UK, 2004

Cambridge University Press, Cambridge, UK, 2004

United Kingdom and Ireland, United Kingdom

2, PS, 2004

2, 2009

0

lg273136

producers:
CVISION Technologies

{"edition":"2","isbns":["0511526253","0521833574","9780511526251","9780521833578"],"last_page":455,"publisher":"Cambridge University Press","series":"Encyclopedia of Mathematics and its Applications"}

Includes bibliographical references and indexes.

Cover......Page 1
ENCYCLOPEDIA OF MATHEMATICS AND ITS APPLICATIONS 96......Page 2
BASIC HYPERGEOMETRIC SERIES - Second edition......Page 4
Copyright - ISBN: 0521833574......Page 5
Contents......Page 8
Foreword......Page 14
Preface......Page 22
Preface to the second edition......Page 26
1.2 Hypergeometric and basic hypergeometric series......Page 28
1.3 The q-binomial theorem......Page 35
1.4 Heine’s transformation formulas for _2φ_1 series......Page 40
1.5 Heine’s q-analogue of Gauss’ summation formula......Page 41
1.6 Jacobi’s triple product identity, theta functions, and elliptic numbers......Page 42
1.7 A q-analogue of Saalschütz’s summation formula......Page 44
1.9 q-analogues of the Karlsson–Minton summation formulas......Page 45
1.10 The q-gamma and q-beta functions......Page 47
1.11 The q-integral......Page 50
Exercises......Page 51
Notes......Page 61
2.1 Well-poised, nearly-poised, and very-well-poised hypergeometric and basic hypergeometric series......Page 65
2.2 A general expansion formula......Page 67
2.3 A summation formula for a terminating very-well-poised _4φ_3 series......Page 68
2.5 Watson’s transformation formula for a terminating very-well-poised _8φ_7 series......Page 69
2.6 Jackson’s sum of a terminating very-well-poised balanced _8φ_7 series......Page 70
2.7 Some special and limiting cases of Jackson’s and Watson’s formulas: the Rogers–Ramanujan identities......Page 71
2.8 Bailey’s transformation formulas for terminating _5φ_4 and _7φ_6 series......Page 72
2.9 Bailey’s transformation formula for a terminating _{10}φ_9 series......Page 74
2.10 Limiting cases of Bailey’s _{10}φ_9 transformation formula......Page 75
2.11 Bailey’s three-term transformation formula for VWP-balanced _8φ_7 series......Page 80
2.12 Bailey’s four-term transformation formula for balanced _{10}φ_9 series......Page 82
Exercises......Page 85
Notes......Page 94
3.1 Introduction......Page 96
3.2 Two-term transformation formulas for _3φ_2 series......Page 97
3.3 Three-term transformation formulas for _3φ_2 series......Page 100
3.4 Transformation formulas for well-poised _3φ_2 and very-well-poised _5φ_4 series with arbitrary arguments......Page 101
3.5 Transformations of series with base q^2 to series with base q......Page 104
3.6 Bibasic summation formulas......Page 107
3.7 Bibasic expansion formulas......Page 111
3.8 Quadratic, cubic, and quartic summation and transformation formulas......Page 115
3.9 Multibasic hypergeometric series......Page 122
3.10 Transformations of series with base q to series with base q^2......Page 123
Exercises......Page 127
Notes......Page 138
4.1 Introduction......Page 140
4.2 Watson’s contour integral representation for _2φ_1(a, b; c; q, z) series......Page 142
4.3 Analytic continuation of _2φ_1(a, b; c; q, z)......Page 144
4.4 q-analogues of Barnes’ first and second lemmas......Page 146
4.5 Analytic continuation of _{r+1}φ_r series......Page 147
4.6 Contour integrals representing well-poised series......Page 148
4.7 A contour integral analogue of Bailey’s summation formula......Page 150
4.8 Extensions to complex q inside the unit disc......Page 151
4.9 Other types of basic contour integrals......Page 152
4.10 General basic contour integral formulas......Page 153
4.11 Some additional extensions of the beta integral......Page 156
4.12 Sears’ transformations of well-poised series......Page 157
Exercises......Page 159
Notes......Page 162
5.1 Notations and definitions......Page 164
5.2 Ramanujan’s sum for _1ψ_1(a; b; q, z)......Page 165
5.3 Bailey’s sum of a very-well-poised _6ψ_6 series......Page 167
5.4 A general transformation formula for an _rψ_r series......Page 168
5.5 A general transformation formula for a very-well-poised _{2r}ψ_{2r} series......Page 170
5.6 Transformation formulas for very-well-poised _8ψ_8 and _{10}ψ_{10} series......Page 172
Exercises......Page 173
Notes......Page 179
6.1 The Askey–Wilson q-extension of the beta integral......Page 181
6.2 Proof of formula (6.1.1)......Page 183
6.3 Integral representations for very-well-poised _8φ_7 series......Page 184
6.4 Integral representations for very-well-poised _{10}φ_9 series......Page 186
6.5 A quadratic transformation formula for very-well-poised balanced _{10}φ_9 series......Page 189
6.6 The Askey–Wilson integral when max (|a|, |b|, |c|, |d|) >= 1......Page 190
Exercises......Page 195
Notes......Page 200
7.1 Orthogonality......Page 202
7.2 The finite discrete case: the q-Racah polynomials and some special cases......Page 204
7.3 The infinite discrete case: the little and big q-Jacobi polynomials......Page 208
7.4 An absolutely continuous measure: the continuous q-ultraspherical polynomials......Page 211
7.5 The Askey–Wilson polynomials......Page 215
7.6 Connection coefficients......Page 222
7.7 A difference equation and a Rodrigues-type formula for the Askey–Wilson polynomials......Page 224
Exercises......Page 226
Notes......Page 240
8.1 Introduction......Page 244
8.2 A product formula for balanced _4φ_3 polynomials......Page 245
8.3 Product formulas for q-Racah and Askey–Wilson polynomials......Page 248
8.4 A product formula in integral form for the continuous q-ultraspherical polynomials......Page 250
8.5 Rogers’ linearization formula for the continuous q-ultraspherical polynomials......Page 253
8.6 The Poisson kernel for C_n(x; β|q)......Page 254
8.7 Poisson kernels for the q-Racah polynomials......Page 256
8.8 q-analogues of Clausen’s formula......Page 259
8.9 Nonnegative basic hypergeometric series......Page 263
8.10 Applications in the theory of partitions of positive integers......Page 266
8.11 Representations of positive integers as sums of squares......Page 269
Exercises......Page 272
Notes......Page 284
9.1 Introduction......Page 286
9.2 The little q-Jacobi polynomials......Page 287
9.3 A generating function for Askey–Wilson polynomials......Page 289
9.4 A bilinear sum for the Askey–Wilson polynomials I......Page 292
9.5 A bilinear sum for the Askey–Wilson polynomials II......Page 296
9.6 A bilinear sum for the Askey–Wilson polynomials III......Page 297
Exercises......Page 299
Notes......Page 308
10.2 q-Appell and other basic double hypergeometric series......Page 309
10.3 An integral representation for Φ^{(1)}(q^a; q^b, q^{b'} ; q^c; q; x, y)......Page 311
10.4 Formulas for Φ^{(2)}(q^a; q^b, q^{b'} ; q^c, q^{c'} ; q; x, y)......Page 313
10.5 Formulas for Φ^{(3)}(q^a, q^{a'}; q^b, q^{b'} ; q^c; q; x, y)......Page 315
10.6 Formulas for a q-analogue of F_4......Page 317
10.7 An Askey–Wilson-type integral representation for a q-analogue of F_1......Page 321
Exercises......Page 323
Notes......Page 328
11.1 Introduction......Page 329
11.2 Elliptic and theta hypergeometric series......Page 330
11.3 Additive notations and modular series......Page 339
11.4 Elliptic analogue of Jackson’s _8φ_7 summation formula......Page 348
11.5 Elliptic analogue of Bailey’s transformation formula for a terminating _{10}φ_9 series......Page 350
11.6 Multibasic summation and transformation formulas for theta hypergeometric series......Page 352
11.7 Rosengren’s elliptic extension of Milne’s fundamental theorem......Page 358
Exercises......Page 363
Notes......Page 376
Appendix I Identities involving q-shifted factorials, q-gamma functions and q-binomial coefficients......Page 378
Appendix II Selected summation formulas......Page 381
Appendix III Selected transformation formulas......Page 386
References......Page 394
Symbol index......Page 442
Author index......Page 445
Subject index......Page 450

Cover 1
ENCYCLOPEDIA OF MATHEMATICS AND ITS APPLICATIONS 96 2
BASIC HYPERGEOMETRIC SERIES - Second edition 4
Copyright - ISBN: 0521833574 5
Contents 8
Foreword 14
Preface 22
Preface to the second edition 26
1 Basic hypergeometric series 28
1.1 Introduction 28
1.2 Hypergeometric and basic hypergeometric series 28
1.3 The q-binomial theorem 35
1.4 Heine’s transformation formulas for _2φ_1 series 40
1.5 Heine’s q-analogue of Gauss’ summation formula 41
1.6 Jacobi’s triple product identity, theta functions, and elliptic numbers 42
1.7 A q-analogue of Saalschütz’s summation formula 44
1.8 The Bailey–Daum summation formula 45
1.9 q-analogues of the Karlsson–Minton summation formulas 45
1.10 The q-gamma and q-beta functions 47
1.11 The q-integral 50
Exercises 51
Notes 61
2 Summation, transformation, and expansion formulas 65
2.1 Well-poised, nearly-poised, and very-well-poised hypergeometric and basic hypergeometric series 65
2.2 A general expansion formula 67
2.3 A summation formula for a terminating very-well-poised _4φ_3 series 68
2.4 A summation formula for a terminating very-well-poised _6φ_5 series 69
2.5 Watson’s transformation formula for a terminating very-well-poised _8φ_7 series 69
2.6 Jackson’s sum of a terminating very-well-poised balanced _8φ_7 series 70
2.7 Some special and limiting cases of Jackson’s and Watson’s formulas: the Rogers–Ramanujan identities 71
2.8 Bailey’s transformation formulas for terminating _5φ_4 and _7φ_6 series 72
2.9 Bailey’s transformation formula for a terminating _{10}φ_9 series 74
2.10 Limiting cases of Bailey’s _{10}φ_9 transformation formula 75
2.11 Bailey’s three-term transformation formula for VWP-balanced _8φ_7 series 80
2.12 Bailey’s four-term transformation formula for balanced _{10}φ_9 series 82
Exercises 85
Notes 94
3 Additional summation, transformation, and expansion formulas 96
3.1 Introduction 96
3.2 Two-term transformation formulas for _3φ_2 series 97
3.3 Three-term transformation formulas for _3φ_2 series 100
3.4 Transformation formulas for well-poised _3φ_2 and very-well-poised _5φ_4 series with arbitrary arguments 101
3.5 Transformations of series with base q^2 to series with base q 104
3.6 Bibasic summation formulas 107
3.7 Bibasic expansion formulas 111
3.8 Quadratic, cubic, and quartic summation and transformation formulas 115
3.9 Multibasic hypergeometric series 122
3.10 Transformations of series with base q to series with base q^2 123
Exercises 127
Notes 138
4 Basic contour integrals 140
4.1 Introduction 140
4.2 Watson’s contour integral representation for _2φ_1(a, b; c; q, z) series 142
4.3 Analytic continuation of _2φ_1(a, b; c; q, z) 144
4.4 q-analogues of Barnes’ first and second lemmas 146
4.5 Analytic continuation of _{r+1}φ_r series 147
4.6 Contour integrals representing well-poised series 148
4.7 A contour integral analogue of Bailey’s summation formula 150
4.8 Extensions to complex q inside the unit disc 151
4.9 Other types of basic contour integrals 152
4.10 General basic contour integral formulas 153
4.11 Some additional extensions of the beta integral 156
4.12 Sears’ transformations of well-poised series 157
Exercises 159
Notes 162
5 Bilateral basic hypergeometric series 164
5.1 Notations and definitions 164
5.2 Ramanujan’s sum for _1ψ_1(a; b; q, z) 165
5.3 Bailey’s sum of a very-well-poised _6ψ_6 series 167
5.4 A general transformation formula for an _rψ_r series 168
5.5 A general transformation formula for a very-well-poised _{2r}ψ_{2r} series 170
5.6 Transformation formulas for very-well-poised _8ψ_8 and _{10}ψ_{10} series 172
Exercises 173
Notes 179
6 The Askey–Wilson q-beta integral and some associated formulas 181
6.1 The Askey–Wilson q-extension of the beta integral 181
6.2 Proof of formula (6.1.1) 183
6.3 Integral representations for very-well-poised _8φ_7 series 184
6.4 Integral representations for very-well-poised _{10}φ_9 series 186
6.5 A quadratic transformation formula for very-well-poised balanced _{10}φ_9 series 189
6.6 The Askey–Wilson integral when max (|a|, |b|, |c|, |d|) >= 1 190
Exercises 195
Notes 200
7 Applications to orthogonal polynomials 202
7.1 Orthogonality 202
7.2 The finite discrete case: the q-Racah polynomials and some special cases 204
7.3 The infinite discrete case: the little and big q-Jacobi polynomials 208
7.4 An absolutely continuous measure: the continuous q-ultraspherical polynomials 211
7.5 The Askey–Wilson polynomials 215
7.6 Connection coefficients 222
7.7 A difference equation and a Rodrigues-type formula for the Askey–Wilson polynomials 224
Exercises 226
Notes 240
8 Further applications 244
8.1 Introduction 244
8.2 A product formula for balanced _4φ_3 polynomials 245
8.3 Product formulas for q-Racah and Askey–Wilson polynomials 248
8.4 A product formula in integral form for the continuous q-ultraspherical polynomials 250
8.5 Rogers’ linearization formula for the continuous q-ultraspherical polynomials 253
8.6 The Poisson kernel for C_n(x; β|q) 254
8.7 Poisson kernels for the q-Racah polynomials 256
8.8 q-analogues of Clausen’s formula 259
8.9 Nonnegative basic hypergeometric series 263
8.10 Applications in the theory of partitions of positive integers 266
8.11 Representations of positive integers as sums of squares 269
Exercises 272
Notes 284
9 Linear and bilinear generating functions for basic orthogonal polynomials 286
9.1 Introduction 286
9.2 The little q-Jacobi polynomials 287
9.3 A generating function for Askey–Wilson polynomials 289
9.4 A bilinear sum for the Askey–Wilson polynomials I 292
9.5 A bilinear sum for the Askey–Wilson polynomials II 296
9.6 A bilinear sum for the Askey–Wilson polynomials III 297
Exercises 299
Notes 308
10 q-series in two or more variables 309
10.1 Introduction 309
10.2 q-Appell and other basic double hypergeometric series 309
10.3 An integral representation for Φ^{(1)}(q^a; q^b, q^{b'} ; q^c; q; x, y) 311
10.4 Formulas for Φ^{(2)}(q^a; q^b, q^{b'} ; q^c, q^{c'} ; q; x, y) 313
10.5 Formulas for Φ^{(3)}(q^a, q^{a'}; q^b, q^{b'} ; q^c; q; x, y) 315
10.6 Formulas for a q-analogue of F_4 317
10.7 An Askey–Wilson-type integral representation for a q-analogue of F_1 321
Exercises 323
Notes 328
11 Elliptic, modular, and theta hypergeometric series 329
11.1 Introduction 329
11.2 Elliptic and theta hypergeometric series 330
11.3 Additive notations and modular series 339
11.4 Elliptic analogue of Jackson’s _8φ_7 summation formula 348
11.5 Elliptic analogue of Bailey’s transformation formula for a terminating _{10}φ_9 series 350
11.6 Multibasic summation and transformation formulas for theta hypergeometric series 352
11.7 Rosengren’s elliptic extension of Milne’s fundamental theorem 358
Exercises 363
Notes 376
Appendix I Identities involving q-shifted factorials, q-gamma functions and q-binomial coefficients 378
Appendix II Selected summation formulas 381
Appendix III Selected transformation formulas 386
References 394
Symbol index 442
Author index 445
Subject index 450

This revised and expanded new edition will continue to meet the needs for an authoritative, up-to-date, self contained, and comprehensive account of the rapidly growing field of basic hypergeometric series, or q-series. 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, orthogonal polynomials in several variables, and generating functions. Chapters 9-11 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. Some sections and exercises have been added to reflect recent developments, and the Bibliography has been revised to maintain its comprehensiveness.

This updated edition will continue to meet the needs for an authoritative comprehensive analysis of the rapidly growing field of basic hypergeometric series, or q-series. It includes deductive proofs, exercises, and useful appendices. Three new chapters have been added to this edition covering q-series in two and more variables: linear- and bilinear-generating functions for basic orthogonal polynomials; and summation and transformation formulas for elliptic hypergeometric series. In addition, the text and bibliography have been expanded to reflect recent developments. First Edition Hb (1990): 0-521-35049-2

2010-05-31