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Special functions, which include the trigonometric functions, have been used for centuries. Their role in the solution of differential equations was exploited by Newton and Leibniz, and the subject of special functions has been in continuous development ever since. In just the past thirty years several new special functions and applications have been discovered.
This treatise presents an overview of the area of special functions, focusing primarily on the hypergeometric functions and the associated hypergeometric series. It includes both important historical results and recent developments and shows how these arise from several areas of mathematics and mathematical physics. Particular emphasis is placed on formulas that can be used in computation.
The book begins with a thorough treatment of the gamma and beta functions, which are essential to understanding hypergeometric functions. Later chapters discuss Bessel functions, orthogonal polynomials and transformations, the Selberg integral and its applications, spherical harmonics, q-series, partitions, and Bailey chains.
This clear, authoritative work will be a lasting reference for students and researchers in number theory, algebra, combinatorics, differential equations, mathematical computing, and mathematical physics.
This book contains a wealth of fascinating material which is presented in a user-friendly way. If you want to extend your knowledge of special functions, this is a good place to start. Even if your interests are in number theory or combinatorics, there is something for you too. The book can be warmly recommended because the material is written in an excellent manner. This book is recommended warmly as a rich source of information to everybody who is interested in "Special Functions".
The Gamma and Beta Functions
The Hypergeometric Functions
Hypergeometric Transformations and Identities
Bessel Functions and Confluent Hypergeometric Functions
Orthogonal Polynomials
Special Orthogonal Polynomials
Topics in Orthogonal Polynomials
The Selberg Integral and Its Applications
Spherical Harmonics
Introduction to q-Series
Partitions
Bailey Chains
Infinite Products
Summability and Fractional Integration
Asymptotic Expansions
Euler-Maclaurin Summation Formula
Lagrange Inversion Formula
Series Solutions of Differential Equations
Bibliography
Index
Subject Index
Symbol Index