We explore ramifications and extensions of a \(q\)-difference operator method first used by L.J. Rogers for deriving relationships between special functions involving certain fundamental \(q\)-symmetric polynomials. In special cases these symmetric polynomials reduce to well-known classes of orthogonal polynomials. A number of basic properties of these polynomials follow from our approach. This leads naturally to the evaluation of the Askey-Wilson integral and generalizations. We also find expansions of certain generalized basic hypergeometric functions in terms of the symmetric polynomials. This provides us with a quick route to understanding the group structure generated by iterating the two-term transformations of these functions. We also lay some infrastructure for more general investigations in the future.

Readership

Graduate students and research mathematicians interested in special functions and combinatorics.

Table of Contents

• Introduction and preliminaries
• New results and connections with current research
• Vector operator identities and simple applications
• Bibliography