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Connection formulas for Askey--Wilson polynomials and related expansions

Howard S. Cohl, Wolter Groenevelt

TL;DR

This work develops explicit connection formulas and expansive representations for Askey--Wilson polynomials and their $q$-analogue families. By combining nonstandard generating functions, Poisson kernels for $q^{-1}$-Al-Salam--Chihara polynomials, and bilinear/double-sum expansions, the authors obtain ${}_4\phi_3$-coefficient connection formulas and new biorthogonal structures tied to Gupta--Masson rational functions. The results unify and extend known expansions by providing multiple limiting regimes to continuous dual $q$-Hahn, Al-Salam--Chihara, and continuous big $q$-Hermite polynomials, as well as their $q^{-1}$-counterparts, with implications for representation theory of quantum algebras. Overall, the paper advances the analytic toolkit for AW polynomials, enabling explicit expansions and biorthogonal relations that connect orthogonality, generating functions, and operator representations in a coherent framework.

Abstract

We derive and study expansions of and over the Askey--Wilson polynomials. We study these expansions and examine some limits to the continuous dual $q$-Hahn, Al-Salam--Chihara, continuous big $q$-Hermite and continuous $q$-Hermite polynomials and their $q^{-1}$-analogues. The Poisson kernel for the infinite discrete orthogonality relation for the $q^{-1}$-Al-Salam--Chihara polynomials is derived which in a special case reduces to the Gupta--Masson biorthogonal rational ${}_4φ_3$-functions. This Poisson kernel implies new infinite series connection relations for the Askey--Wilson polynomials involving these rational ${}_4φ_3$-functions. We also consider various interesting limits.

Connection formulas for Askey--Wilson polynomials and related expansions

TL;DR

This work develops explicit connection formulas and expansive representations for Askey--Wilson polynomials and their -analogue families. By combining nonstandard generating functions, Poisson kernels for -Al-Salam--Chihara polynomials, and bilinear/double-sum expansions, the authors obtain -coefficient connection formulas and new biorthogonal structures tied to Gupta--Masson rational functions. The results unify and extend known expansions by providing multiple limiting regimes to continuous dual -Hahn, Al-Salam--Chihara, and continuous big -Hermite polynomials, as well as their -counterparts, with implications for representation theory of quantum algebras. Overall, the paper advances the analytic toolkit for AW polynomials, enabling explicit expansions and biorthogonal relations that connect orthogonality, generating functions, and operator representations in a coherent framework.

Abstract

We derive and study expansions of and over the Askey--Wilson polynomials. We study these expansions and examine some limits to the continuous dual -Hahn, Al-Salam--Chihara, continuous big -Hermite and continuous -Hermite polynomials and their -analogues. The Poisson kernel for the infinite discrete orthogonality relation for the -Al-Salam--Chihara polynomials is derived which in a special case reduces to the Gupta--Masson biorthogonal rational -functions. This Poisson kernel implies new infinite series connection relations for the Askey--Wilson polynomials involving these rational -functions. We also consider various interesting limits.
Paper Structure (9 sections, 28 theorems, 73 equations)

This paper contains 9 sections, 28 theorems, 73 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The Askey--Wilson polynomials and their symmetric subfamilies
  4. Connection coefficients for Askey--Wilson polynomials
  5. Nonstandard generating function for Askey--Wilson polynomials
  6. Poisson kernel for $q^{-1}$-Al-Salam--Chihara polynomials
  7. A bilinear sum for Askey--Wilson polynomials
  8. A general expansion for Askey--Wilson polynomials
  9. The Gupta--Masson rational ${\ThisStyle{\raisebox{-2\LMpt}{$\SavedStyle_$}}}4\phi{\ThisStyle{\raisebox{-2\LMpt}{$\SavedStyle_$}}}3$-functions

Key Result

Theorem 2.2

Let $q\in{{\mathbb C}^\dag}$, $a,b,c,d,e\in{{\mathbb C}^\ast}$. Then

Theorems & Definitions (34)

  • Definition 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Theorem 3.1
  • Corollary 3.2
  • Corollary 3.3
  • Corollary 3.4
  • Corollary 3.5
  • Corollary 3.6
  • Corollary 3.7
  • ...and 24 more