Askey R. (ed.) Theory and Application of Special Functions

New York, San Francisco, London: Academic Press Inc., 1975. — XI, 560 p. — Proceedings of an Advanced Seminar Sponsored by the Mathematics Research Center The University of Wisconsin-Madison March 31-April 2, 1975. — Advanced Seminar on Special Functions, Madison, Wis., 1975.Theory and Application of Special Functions contains the proceedings of the Advanced Seminar on Special Functions sponsored by the Mathematics Research Center of the University of Wisconsin-Madison and held from March 31 to April 2, 1975. The seminar tackled the theory and application of special functions and covered topics ranging from the asymptotic estimation of special functions to association schemes and coding theory. Some interesting results, conjectures, and problems are given. Comprised of 13 chapters, this book begins with a survey of computational methods in special functions, followed by a discussion on unsolved problems in the asymptotic estimation of special functions. The reader is then introduced to periodic Bernoulli numbers, summation formulas, and applications; problems and prospects for basic hypergeometric functions; and linear growth models with many types and multidimensional Hahn polynomials. Subsequent chapters explore two-variable analogues of the classical orthogonal polynomials; special functions of matrix and single argument in statistics; and some properties of the determinants of orthogonal polynomials. This monograph is intended primarily for students and practitioners of mathematics.Twenty years ago it was widely believed that the existence of large, fast, computing machines spelled the end of the study and use of special functions. Differential equations would be solved numerically, integrals would be evaluated numerically, and special functions would become a fossil. Many mathematicians acted on this belief, and special functions disappeared from their traditional place in the curriculum as part of a course in complex variables. They were often replaced in the mathematical physics course by a section on Hilbert space and functional analysis.To paraphrase Mark Twain, reports of the death of special functions were premature. A sociologist of science would accept the following two facts as proof. Over 150,000 copies of the Handbook of Mathematical Functions have been sold. Two of the five most widely cited mathematics books (as measured by the citations in Science Citation Index) are this book and Higher Transcendental Functions by A. Erdélyi et al. Most of us are more concerned with quality than quantity, and so would not be convinced by these two facts. A more compelling proof is given in this book. Some interesting results, conjectures, and problems are given. In addition there are surprising connections with other fields.Foreword
PrefaceComputational Methods in Special Functions - A Survey
Unsolved Problems in the Asymptotic Estimation of Special Functions
Periodic Bernoulli Numbers, Summation Formulas and Applications
Problems and Prospects for Basic Hypergeometric Functions
An Introduction to Association Schemes and Coding Theory
Linear Growth Models with Many Types and Multidimensional Hahn Polynomials
Orthogonal Polynomials Revisited
Symmetry, Separation of Variables, and Special Functions
Nicholson-Type Integrals for Products of Gegenbauer Functions and Related Topics
Positivity and Special Functions
Two-Variable Analogues of the Classical Orthogonal Polynomials
Special Functions of Matrix and Single Argument in Statistics
Some Properties of Determinants of Orthogonal PolynomialsIndex