Basic Hypergeometric Series: (Series Number 96 Encyclopedia of Mathematics and its Applications)

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. Elsewhere, errata have been corrected, some sections and exercises have been added to reflect recent developments, and the Bibliography has been revised to maintain its comprehensiveness.

Table of Contents:
Foreword; Preface; 1. Basic hypergeometric series; 2. Summation, transformation, and expansion formulas; 3. Additional summation, transformation, and expansion formulas; 4. Basic contour integrals; 5. Bilateral basic hypergeometric series; 6. The Askey-Wilson q-beta integral and some associated formulas; 7. Applications to orthogonal polynomials; 8. Further applications; 9. Linear and bilinear generating functions for basic orthogonal polynomials; 10. q-series in two or more variables; 11. Elliptic, modular, and theta hypergeometric series; Appendices; References; Author index; Subject index; Symbol index.

Review :
'I love this book! It is great! This really is a book you can learn the subject from. The plentiful exercises vary from elementary to challenging with lots of each. Congratulations and thanks are due the authors.' George Andrews, American Math. Monthly 'The book is remarkable in many ways. It is comprehensive, at least, comprehensive to date. As is typical of most works on the subject, it is clearly and carefully written. While no book can conceivably incorporate all the important results, particularly those obtained in the last decade, many of them are included as exercises. And this is the feature all other books on the subject lack: a set of exercises. Each chapter is topped off by a challenging series of problems which lead the reader to recreate recent discoveries. Anyone who works even a small percentage of them will soon be an expert. A generous series of historical notes concludes each chapter. The book is user friendly in every respect. The book has two excellent Appendices which summarize the identities and summation formulas derived in the text, an exhaustive 25 page list of references, and a nontrivial index. Now anyone working in combinatorics, group representation theory, coding theory, and related fields will want to own it. Many physicists will find it bears directly on matters of interest to them. Computer scientists may find the book increasingly timely. Those who have refrained from entering the field because of the tortuous notation can now have untroubled access to its mysteries. I say, come in, the water's fine.' Jet Wimp, SIAM Review 'This is an excellent and very informative book on the subject. After a gentle introduction to basic series and some special cases (such as the 'q'-binomial theorem) the authors bring the reader up to the latest results on the general theory and its extensions, many such results are due to them. The exercises are utilized to include results that found no room in the detailed treatment. In addition to these exercises, notes at the end of each chapter point the reader to related topics. This alone makes the book an invaluable reference to those who are interested in basic series.' Waleed A. Al-Salam, Math. Reviews 'Thus the present book, devoted to 'q'-hypergeometric series, appears at a very timely moment. The result is excellent. The first chapter presents a clear and elementary introduction to the subject. At the end of the book there are excellent indices and compendia of formulas.' Tom H. Koornwinder, Bulletin of London Mathematical Society '... a very modern, self-contained, comprehensive and successful monograph, interesting and useful, for physicists as well as for mathematicians from various branches, who wish to learn about the subject.' European Mathematical Society Newsletter