10 - q-series in two or more variables

Summary

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.