$q$-difference operators, orthogonal polynomials, and symmetric expansions
About this Title
Publication: Memoirs of the American Mathematical Society
Publication Year 2002: Volume 159, Number 757
ISBNs: 978-0-8218-2774-1 (print); 978-1-4704-0350-8 (online)
MathSciNet review: 1921582
MSC (2000): Primary 33D70; Secondary 05A30, 05E05, 33D45, 39A13
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.
Graduate students and research mathematicians interested in special functions and combinatorics.
Table of Contents
- 1. Introduction and preliminaries
- 2. New results and connections with current research
- 3. Vector operator identities and simple applications