$q$-Difference Operators, Orthogonal Polynomials, and Symmetric Expansions
- ISBN-10
- 082182774X
- ISBN-13
- 9780821827741
- Category
- Difference operators
- Pages
- 73
- Language
- English
- Published
- 2002
- Publisher
- American Mathematical Soc.
- Author
- Douglas Bowman
Description
The author explores 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 this approach. This leads naturally to the evaluation of the Askey-Wilson integral and generalizations. Expansions of certain generalized basic hypergeometric functions in terms of the symmetric polynomials are also found. This provides a quick route to understanding the group structure generated by iterating the two-term transformations of these functions. Some infrastructure is also laid for more general investigations in the future
