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$q$-Difference Operators, Orthogonal Polynomials, and Symmetric Expansions
 
Douglas Bowman University of Illinois, Urbana, IL
q-Difference Operators, Orthogonal Polynomials, and Symmetric Expansions
eBook ISBN:  978-1-4704-0350-8
Product Code:  MEMO/159/757.E
List Price: $51.00
MAA Member Price: $45.90
AMS Member Price: $30.60
q-Difference Operators, Orthogonal Polynomials, and Symmetric Expansions
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$q$-Difference Operators, Orthogonal Polynomials, and Symmetric Expansions
Douglas Bowman University of Illinois, Urbana, IL
eBook ISBN:  978-1-4704-0350-8
Product Code:  MEMO/159/757.E
List Price: $51.00
MAA Member Price: $45.90
AMS Member Price: $30.60
  • Book Details
     
     
    Memoirs of the American Mathematical Society
    Volume: 1592002; 56 pp
    MSC: Primary 33; 05;

    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
     
     
    • Chapters
    • 1. Introduction and preliminaries
    • 2. New results and connections with current research
    • 3. Vector operator identities and simple applications
  • Requests
     
     
    Review Copy – for publishers of book reviews
    Permission – for use of book, eBook, or Journal content
    Accessibility – to request an alternate format of an AMS title
Volume: 1592002; 56 pp
MSC: Primary 33; 05;

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.

  • Chapters
  • 1. Introduction and preliminaries
  • 2. New results and connections with current research
  • 3. Vector operator identities and simple applications
Review Copy – for publishers of book reviews
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
Please select which format for which you are requesting permissions.