A Handbook of Generalized Special Functions for Statistical and Physical Sciences

This book is a blend of theory and applications of generalized special functions, especially Meijer's G-functions. Properties of these functions are illustrated with worked examples and thus these complicated functions should be accessible to the potential user. Applications of these functions into mathematical statistics and astrophysics are integrated into the text. Thus this book should be a valuable reference source and manual for those working in any branch of mathematical statistics, mathematical physics and several branches of mathematics. After developing the scalar variable case vector and matrix variable analogues are also developed in this book. A comprehensive development of special functions of matrix argument is given. Introductory development of various topics in this area are in this chapter. These give rise to a large variety of research problems for people working on one or more scaler variable cases of generalized special functions.

Table of Contents:
Part 1 Mathematical preliminaries: the gamma function; Bernoulli polynomials; asymptotic expansions of gamma functions; the psi functions; the generalized zeta functions; the beta function; calculation of residues for gamma functions; the Mellin transform; density functions; methods of deriving distributions; exercises. Part 2 The G-function: some basic properties of the G-function; the Mellin transform of a G-function; properties connected with the derivatives of a G-function; series representation for a G-function; G-function as multiple integrals or as solutions of integral equations; differential equation for a G-function; asymptotic expansions for a G-function; exercises. Part 3 Elementary special functions and the G-function: gamma and related functions - notations and definitions; hypergeometric functions - notations and special cases; confluent hypergeometric function and related function; exponential integral and related function; Bessel functions and associated functions; other special functions; orthogonal polynomials; elementary special functions expressed in terms of elementary special functions; some integrals involving G-functions; the H-function; computational aspects of G- and H-functions; orders of the special functions for small and large values of the argument; exercises. Part 4 Generalizations to matrix variables: scalar functions of a symmetric positive definite matrix; scalar functions of matrix arguments; Laplace transform; hypergeometric functions of matrix arguments; generalized matrix transform of M-transform; zonal polynomial; matrix variate Dirichlet distribution; hypergeometric functions of many scalar variables; hypergeometric functions of many matrix arguments; G- and H-functions of two variables; exercises.