##
**Basic hypergeometric series.**
*(English)*
Zbl 0695.33001

Encyclopedia of Mathematics and Its Applications, 35. Cambridge etc.: Cambridge University Press. xx, 287 p. £35.00; $ 59.50 (1990).

This book gives a systematic treatment of the theory and applications of the basic hypergeometric series. The importance of this function lies in the fact that during the last two decades, it has found many applications in areas of combinatorics, orthogonal polynomials, number theory and modular forms, Lie algebras etc. In this direction, the reader is referred to the work of Richard Askey, G. E. Andrews and their collaborators.

All the chapters are based on original research papers. Each chapter contains exercises as well as notes, which further enhances the utility of the book.

Chapter 1 introduces the hypergeometric and basic hypergeometric series and gives various theorems such as the \(q\)-binomial theorem, Heine’s transformation formulas for \({}_ 2\Phi_ 1\) series etc. The \(q\)-gamma and \(q\)-beta functions and the \(q\)-integral are also discussed.

Chapter 2 is devoted to summation and transformation formulas for very- well-poised basic hypergeometric series.

Chapter 3 deals with recent results on bibasic summation formulas, and on quadratic and quartic summation and transformation formulas.

In Chapter 4, Watson’s \(q\)-analogue of the Barnes contour integral representation for the hypergeometric function is presented and it is further used in deriving an analytic continuation formula for the \({}_ 2\Phi_ 1\) series. \(q\)-analogues of the well-known Barnes first and second lemmas are also given.

It is interesting to observe that the reviewer, G. C. Modi and S. L. Kalla have already extended the Watson’s integral on the \(q\)-analogue of the hypergeometric function to a \(q\)-analogue of the H-function [Rev. Tec. Fac. Ing., Univ. Zulia 6, 139–143 (1983; Zbl 0562.33004)] and to a \(q\)-analogue of the H-function of several variables [ibid. 10, No. 2, 35–39 (1987; Zbl 0617.33002)], which can provide elegant generalization and unification of some of the results of this chapter. An account of the H- function with applications is available from the monograph written by A. M. Mathai and the reviewer [The H-function with applications in statistics and other disciplines. New Delhi etc.: Wiley Eastern (1978; Zbl 0382.33001)].

Chapter 5 is devoted to the description of the basic properties of the bilateral basic hypergeometric series, which, besides other results, deals with Ramanujan’s sum for \({}_ 1\Psi_ 1(a;b,q,z).\)

In the last three chapters, recent work on the extension of Euler’s beta integral and, applications to orthogonal polynomials and other related areas are presented.

The book ends with three appendices and references. The book under review can be recommended as a text to research scholars. This can also serve as a reference book for workers in special functions and related fields.

All the chapters are based on original research papers. Each chapter contains exercises as well as notes, which further enhances the utility of the book.

Chapter 1 introduces the hypergeometric and basic hypergeometric series and gives various theorems such as the \(q\)-binomial theorem, Heine’s transformation formulas for \({}_ 2\Phi_ 1\) series etc. The \(q\)-gamma and \(q\)-beta functions and the \(q\)-integral are also discussed.

Chapter 2 is devoted to summation and transformation formulas for very- well-poised basic hypergeometric series.

Chapter 3 deals with recent results on bibasic summation formulas, and on quadratic and quartic summation and transformation formulas.

In Chapter 4, Watson’s \(q\)-analogue of the Barnes contour integral representation for the hypergeometric function is presented and it is further used in deriving an analytic continuation formula for the \({}_ 2\Phi_ 1\) series. \(q\)-analogues of the well-known Barnes first and second lemmas are also given.

It is interesting to observe that the reviewer, G. C. Modi and S. L. Kalla have already extended the Watson’s integral on the \(q\)-analogue of the hypergeometric function to a \(q\)-analogue of the H-function [Rev. Tec. Fac. Ing., Univ. Zulia 6, 139–143 (1983; Zbl 0562.33004)] and to a \(q\)-analogue of the H-function of several variables [ibid. 10, No. 2, 35–39 (1987; Zbl 0617.33002)], which can provide elegant generalization and unification of some of the results of this chapter. An account of the H- function with applications is available from the monograph written by A. M. Mathai and the reviewer [The H-function with applications in statistics and other disciplines. New Delhi etc.: Wiley Eastern (1978; Zbl 0382.33001)].

Chapter 5 is devoted to the description of the basic properties of the bilateral basic hypergeometric series, which, besides other results, deals with Ramanujan’s sum for \({}_ 1\Psi_ 1(a;b,q,z).\)

In the last three chapters, recent work on the extension of Euler’s beta integral and, applications to orthogonal polynomials and other related areas are presented.

The book ends with three appendices and references. The book under review can be recommended as a text to research scholars. This can also serve as a reference book for workers in special functions and related fields.

Reviewer: Ram Kishore Saxena (Jodhpur, India)

### MSC:

33Dxx | Basic hypergeometric functions |

33-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to special functions |

33-02 | Research exposition (monographs, survey articles) pertaining to special functions |

05A30 | \(q\)-calculus and related topics |

05E35 | Orthogonal polynomials (combinatorics) (MSC2000) |