I first came across the subject of orthogonal polynomials when I was a student at Cairo University in 1964. It was part of a senior-level course on special functions taught by the late Professor Foad M. Ragab. The instructor used his own notes, which were very similar in spirit to the way Rainville treated the subject. I enjoyed Ragab's lectures and, when I started graduate school in 1968 at the Univerity of Alberta, I was fortunate to work with Waleed Al-Salam on special functions and q-series. Jerry Fields taught me asymptotics and was very generous with his time and ideas. In the late 1960s, courses in special functions were a rarity at North American universities and have been replaced by Bourbaki-type mathematics courses. In the early 1970s, Richard Askey emerged as the leader in the area of special functions and orthogonal polynomials, and the reader of this book will see the enormous impact he made on the subject of orthogonal polynomials. At the same time, George Andrews was promoting q-series and their applications to number theory and combinatorics. So when Andrews and Askey joined forces in the mid-1970s, their combined expertise advanced the subject in leaps and bounds. I was very fortunate to have been part of this group and to participate in these developments. My generation of special functions / orthogonal polynomials people owes Andrews and Askey a great deal for their ideas which fueled the subject for a while, for the leadership role they played, and for taking great care of young people.

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 (zn) = g(zn) for a sequence {zn} 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.