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Dual addition formulas: the case of continuous $q$-ultraspherical and $q$-Hermite polynomials

Tom H. Koornwinder

Abstract

We settle the dual addition formula for continuous $q$-ultraspherical polynomials as an expansion in terms of special $q$-Racah polynomials for which the constant term is given by the linearization formula for the continuous $q$-ultraspherical polynomials. In a second proof we derive the dual addition formula from the Rahman--Verma addition formula for these polynomials by using the self-duality of the polynomials. We also consider the limit case of continuous $q$-Hermite polynomials.

Dual addition formulas: the case of continuous $q$-ultraspherical and $q$-Hermite polynomials

Abstract

We settle the dual addition formula for continuous -ultraspherical polynomials as an expansion in terms of special -Racah polynomials for which the constant term is given by the linearization formula for the continuous -ultraspherical polynomials. In a second proof we derive the dual addition formula from the Rahman--Verma addition formula for these polynomials by using the self-duality of the polynomials. We also consider the limit case of continuous -Hermite polynomials.

Paper Structure

This paper contains 13 sections, 2 theorems, 77 equations.

Table of Contents

  1. Introduction
  2. Note
  3. The dual addition formula for ultraspherical polynomials
  4. Some $q$-hypergeometric orthogonal polynomials
  5. Askey--Wilson polynomials
  6. Continuous $q$-ultraspherical polynomials
  7. $q$-Racah polynomials
  8. The dual addition formula for continuous $q$-ultraspherical polynomials
  9. The Rahman--Verma addition formula
  10. The dual addition formula
  11. A second proof of the dual addition formula
  12. A limit to continuous $q$-Hermite polynomials
  13. Acknowledgement

Key Result

Theorem 4.1

The sum 15 can be evaluated as Here we use the conventions that $(\pm a;q)_n:=(a;q)_n (-a;q)_n$ and $x=\frac{1}{2}(z+z^{-1})$.

Theorems & Definitions (3)

  • Theorem 4.1
  • Remark 4.2
  • Theorem 4.3: Dual addition formula