This book is a monograph on a special branch of mathematics that is losing its special status as it becomes more incorporated in other areas of mathematics and applications. The book is a comprehensive synopsis of hypergeometric series and is written in a friendly way, yielding an easy access to notation, which in this area is rather complicated and thus seems formidable and discouraging. The first five chapters build the general theory, the next five deal with applications (e.g. orthogonal polynomials, the Askey-Wilson integral, q-series) and the last one is devoted to the elements of theta, or elliptic hypergeometric series. Of three very useful appendices, the first two summarize identities and summation formulas and the third concerns transformation formulas. Numerous exercises, varying from elementary problems to those containing new results, that did not find place in the main text complete each chapter and provide a challenge to the reader. The book ends with an extensive list of references. It is 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.

Reviewer:

jd