11 - q-Series Preliminaries

Summary

Introduction

Most of the second half of this monograph is a brief introduction to the theory of q-orthogonal polynomials. We have used a novel approach to the development of those parts needed from the theory of basic hypergeometric functions. This chapter contains preliminary analytic results needed in the later chapters. One important difference between our approach to basic hypergeometric functions and other approaches, for example those of Andrews, Askey and Roy (Andrews et al., 1999), Gasper and Rahman (Gasper and Rahman, 1990), or of Bailey (Bailey, 1935) and Slater (Slater, 1964) is our use of the divided difference operators of Askey and Wilson, the q-difference operator, and the identity theorem for analytic functions.

The identity theorem for analytic functions can be stated as follows.

Theorem 11.1.1Let f(z) and g(z) be analytic in a domain Ω and assume that f (z_n) = g(z_n) for a sequence {z_n} converging to an interior point of Ω. Then f(z) = g(z) at all points of Ω.

A proof of Theorem 11.1.1 is in most elementary books on complex analysis, see for example, (Hille, 1959, p. 199), (Knopp, 1945).

In Chapter 12 we develop those parts of the theory of basic hypergeometric functions that we shall use in later chapters. Sometimes studying orthogonal polynomials leads to other results in special functions. For example the Askey–Wilson polynomials of Chapter 15 lead directly to the Sears transformation, so the Sears transformation (12.4.3) is stated and proved in Chapter 12 but another proof is given in Chapter 15.