Introduction

The main objective of this chapter is to consider q-analogues of Appell's four well-known functions F1, F2, F3 and F4. We start out with Jackson's [1942] φ(1), φ(2), φ(3) and φ(4) functions, defined in terms of double hypergeometric series, which are q-analogues of the Appell functions. It turns out that not all of Jackson's q-Appell functions have the properties that enable them to have transformation and reduction formulas analogous to those for the Appell functions. Also, starting with a q-analogue of the function on one side of a hypergeometric transformation of a reduction formula may lead to a different q-analogue of the formula than starting with a q-analogue of the function on the other side of the formula. We find, further, that the alternative approach of using the q-integral representations of these q-Appell functions can be very fruitful. For example, it immediately leads to the fact that a general φ(1) series is indeed equal to a multiple of a 3ø2 series (see (10.3.4) below). The q-integral approach can be used to derive q-analogues of the Appell functions that are quite different from the ones given by Jackson. In the last section we give a completely different q-analogue of F1, based on the so-called q-quadratic lattice, which has a representation in terms of an Askey-Wilson type integral. We do not attempt to consider Askey-Wilson type q-analogues of F2 and F3 because these are probably the least interesting of the four Appell functions and nothing seems to be known about these analogues.

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.