Foreword

Summary

My education was not much different from that of most mathematicians of my generation. It included courses on modern algebra, real and complex variables, both point set and algebraic topology, some number theory and projective geometry, and some specialized courses such as one on Riemann surfaces. In none of these courses was a hypergeometric function mentioned, and I am not even sure if the gamma function was mentioned after an advanced calculus course. The only time Bessel functions were mentioned was in an undergraduate course on differential equations, and the only thing done with them was to find a power series solution for the general Bessel equation. It is small wonder that with a similar education almost all mathematicians think of special functions as a dead subject which might have been interesting once. They have no idea why anyone would care about it now.

Fortunately there was one part of my education which was different. As a junior in college I read Widder's book The Laplace Transform and the manuscript of its very important sequel, Hirschman and Widder's The Convolution Transform. Then as a senior, I. I. Hirschman gave me a copy of a preprint of his on a multiplier theorem for Legendre series and suggested I extend it to ultraspherical series. This forced me to become acquainted with two other very important books, Gabor Szego′'s great book Orthogonal Polynomials, and the second volume of Higher Transcendental Functions, the monument to Harry Bateman which was written by Arthur Erdélyi and his co-workers W. Magnus, F. Oberhettinger and F. G. Tricomi.