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A handbook of generalized special functions for statistical and physical sciences. (English) Zbl 0770.33001
Oxford Science Publications. Oxford: Clarendon Press. xi, 235 p. £35.00/hbk (1993).

This book is devoted to generalized special functions and their applications in statistics and physical sciences. The following are the main features of the book: There are four chapters. Chapter 1 deals with gamma, beta, psi and zeta functions, their important properties including a simplified technique in calculating the residues of gamma functions, when the product of several gammas are involved. Chapter 2 and 3 are devoted mainly in developing the theory of $G$-functions and its various properties. The various paths of integration of $G$-function as given in the monograph by A. Erdélyi, W. Magnus, F. Oberhettinger and F. G. Tricomi [Higher transcendental functions. Vol. I (1953; Zbl 0051.303)] are shown clearly by drawing the contours. Application of the $G$-functions in the various fields of statistics and physical sciences are illustrated by suitable examples. In Chapter 4, the theory of generalized special functions of scalar variables is extended to the corresponding matrix variables in the case of real symmetric positive definite matrices.

Each chapter contains a list of exercises for solving, most of them extracted from research papers. The printing is clear and uncrowded. The author discusses in detail nearly all the special functions of scalar and matrix arguments and indicates some of their applications in statistical distributions.

This is a fast growing area with a vast potential of its applications in various statistical problems. This book can be used as a test or a reference book for research workers in the field of statistical distributions as well as generalized hypergeometric functions. As a conclusion remark, it goes without saying that anyone who is working on hypergeometric functions will find it essential to have ready access to a copy of this book. It makes a good companion to the earlier volumes on the $G$- and $H$-functions by A. M. Mathai and the reviewer [Generalized hypergeometric functions with applications in statistics and physical sciences (1973; Zbl 0272.33001), and the $H$-function with applications in statistics and other disciplines (1978; Zbl 0382.33001)].

##### MSC:
 33-02 Research monographs (special functions) 33C60 Hypergeometric integrals and functions defined by them 33C20 Generalized hypergeometric series, ${}_{p}{F}_{q}$