TOPICS AND METHODS IN Q-SERIES

by James Mc Laughlin

Overview

The book provides a comprehensive introduction to the many aspects of the subject of basic hypergeometric series. The book essentially assumes no prior knowledge but eventually provides a comprehensive introduction to many important topics. After developing a treatment of historically important topics such as the q-binomial theorem, Heine's transformation, the Jacobi triple product identity, Ramanujan's 1-psi-1 summation formula, Bailey's 6-psi-6 summation formula and the Rogers-Fine identity, the book goes on to delve more deeply into important topics such as Bailey- and WP-Bailey pairs and chains, q-continued fractions, and mock theta functions. There are also chapters on other topics such as Lambert series and combinatorial proofs of basic hypergeometric identities.The book could serve as a textbook for the subject at the graduate level and as a textbook for a topic course at the undergraduate level (earlier chapters). It could also serve as a reference work for researchers in the area.

Product Details

ISBN-13:	9789813223387
Publisher:	World Scientific Publishing Company, Incorporated
Publication date:	09/22/2017
Series:	MONOGRAPHS IN NUMBER THEORY , #8
Sold by:	Barnes & Noble
Format:	eBook
Pages:	400
File size:	24 MB
Note:	This product may take a few minutes to download.

Table of Contents

Foreword v

1 Introduction 1

2 Basic Notation 6

3 The q-Binomial Theorem 12

4 Heine's Transformation 17

4.1 The q-Pfaff-Saalschütz sum 18

4.2 Heine's q-Gauss sum and the q-Kummer sum 20

5 Other Important Basic Hyper geometric Transformations 29

5.1 Gasper's bibasic telescoping sum 29

5.2 A summation formula for a very-well-poised 6φ5 series 30

5.3 Watson's transformation for an 8φ7 series 32

5.4 Bailey's transformation for an 10φ9 series 35

6 The Jacobi Triple Product Identity 44

6.1 Ramanujan's theta functions 47

6.2 Schröter's formula 49

6.2.1 The quintuple product identity 50

7 Ramanujan's 1ψ1 Summation Formula 56

7.1 Constant term results 58

7.2 Vanishing coefficients 59

7.3 Dissections of quotients of infinite products 60

8 Bailey's 6ψ6 Summation 65

9 The Rogers-Fine Identity 71

9.1 False theta series identities 72

9.2 Transformation of series 73

9.3 Other Fine transformations 74

10 Bailey Pairs 80

10.1 The Bailey transform and Bailey pairs 81

10.2 A proof of the Rogers-Ramanujan identities 85

10.3 Example: mod 3 Bailey pairs 86

11 Bailey Chains 95

11.1 The Bailey lattice 99

11.2 Polynomial versions of identities of Rogers-Ramanujan-Slater type 100

11.3 Multiple series Rogers-Ramanujan type identities 101

11.4 The Andrews-Gordon identities 102

12 WP-Bailey Pairs and Chains 110

12.1 A transformation of Sears 111

12.2 WP-Bailey pairs and chains 112

12.2.1 Bailey's WP-Bailey pairs 112

12.2.2 Bressoud's WP-Bailey pairs 113

12.2.3 Singh's WP-Bailey pair 114

12.3 The WP-Bailey chains of Andrews 116

12.4 The WP-Bailey chains of Warnaar 122

12.5 The WP-Bailey chain of Liu and Ma 124

12.6 The WP-Bailey chains of Mc Laughlin and Zimmer 124

13 Further Results on Bailey/WP-Bailey Pairs and Chains 133

13.1 From (α_n(aq^N, q), Β_n(aq^N, q)) to (α_n(a, q), Β_n(α, q)) 133

13.2 A transformation of Warnaar 137

13.3 Bilateral Bailey- and WP-Bailey pairs 139

13.4 Shifted Bailey pairs 148

13.5 Change of base in Bailey pairs 150

13.5.1 Singh's quadratic transformation 151

13.6 The Bailey-Rogers-Ramanujan group 156

13.7 Conjugate Bailey pairs and conjugate WP-Bailey pairs 162

13.7.1 A conjugate Bailey pair of Schilling and Warnaar 165

13.7.2 The conjugate Bailey pair of Rowell 168

13.8 Symmetric- and asymmetric Bailey pairs and transforms 170

13.9 Miscellaneous identities 175

14 Gaussian Polynomials 186

14.1 q-Trinomial Coefficients 189

15 Bijective Proofs of Basic Hypergeometric Identities 193

15.1 Some combinatorial proofs without bijections 196

15.2 Conjugation 198

15.3 Other Bijections 199

15.3.1 Franklin's involution 199

15.3.2 Self-conjugate partitions and partitions into distinct odd parts 202

15.3.3 An example of a more complicated bijection 203

15.4 Bijective proofs of identities involving Gaussian polynomials 204

15.5 From basic hypergeometric identities to partition identities 208

16 q-Continued Fractions 212

16.1 Notation and Some Convergence Criteria 213

16.2 Some Elementary Properties of Continued Fractions 217

16.2.1 Constructing a continued fraction with numerators {A_n} and denominators {B_n} 217

16.2.2 The odd- and even part of a continued fraction 218

16.2.3 Equivalence transformations and equivalent continued fractions 221

16.2.4 From a sequence to a continued fraction 221

16.3 The Bauer-Muir transformation of a continued fraction 224

16.3.1 A class of continued fractions that are easily shown to converge to their Bauer-Muir transforms 228

16.4 General Continued Fraction Identities 229

16.4.1 A first general continued fraction identity 229

16.4.2 Continued fraction expansion of infinite products - I 233

16.4.3 A second general continued fraction identity 235

16.4.4 Continued fraction expansion of infinite products - II 240

16.4.5 A third general continued fraction identity 243

16.4.6 Continued fraction expansion of infinite products - III 245

16.5 Heme's continued fraction and three-term recurrence relations 248

16.6 Regular continued fractions from q-continued fractions 262

17 Lambert Series 271

17.1 Lambert series identities deriving from Bailey's 6ψ6 summation formula 272

17.2 Lambert series identities deriving from Ramanujan's 1ψ1 summation formula 277

17.3 Lambert, series identities arising via differentiation 282

18 Mock Theta Functions 287

18.1 The Third Order Mock Theta Functions 289

18.2 The Fifth Order Mock Theta Functions 297

18.2.1 The Mock Theta Conjectures 303

18.3 The Sixth Order Mock Theta Functions 305

18.4 The Eighth Order Mock Theta Functions 312

18.5 The Second Order Mock Theta Functions 319

18.6 The Mock Theta Functions of Other Orders - Some Brief Comments 322

18.6.1 The Tenth Order Mock Theta Functions 322

18.6.2 The Seventh Order Mock Theta Functions 323

Appendix I Frequently Used Theorems 328

Appendix II WP-Bailey Chains 333

Appendix III WP-Bailey pairs 336

Appendix IV Bailey Chains 340

Appendix V Bailey Pairs 343

Appendix VI Mock Theta Functions 348

Appendix VII Selected Summation Formulae 353

Bibliography 362

Author Index 383

Subject Index 386