Topics and Methods in q-Series

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.

Contents:

• Foreword

• Introduction

• Basic Notation

• The q-Binomial Theorem

• Heine's Transformation

• Other Important Basic Hypergeometric Transformations

• The Jacobi Triple Product Identity

• Ramanujan's 1ψ1 Summation Formula

• Bailey's 6ψ6 Summation

• The Rogers–Fine Identity

• Bailey Pairs

• Bailey Chains

• WP-Bailey Pairs and Chains

• Further Results on Bailey/WP-Bailey Pairs and Chains

• Gaussian Polynomials

• Bijective Proofs of Basic Hypergeometric Identities

• q-Continued Fractions

• Lambert Series

• Mock Theta Functions

• Appendices:

  • Frequently Used Theorems
  • WP-Bailey Chains
  • WP-Bailey Pairs
  • Bailey Chains
  • Bailey Pairs
  • Mock Theta Functions
  • Selected Summation Formulae

• Bibliography

• Author Index

• Subject Index

Readership: Undergraduate and graduate students.
Key Features:

• The book is essentially self-contained, all the pre-requisites are essentially covered in earlier chapters of the book
• There is a comprehensive selection of guided exercises at the end of each chapter, allowing the reader to go deeper into the subject, and permitting the book to be used both as a graduate text book, and an undergraduate text book for a topics course
• No other book that treats the same or nearby subjects is as accessible to a reader as this one, nor does any such book cover the same selection of topics