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The $q$ and $q^{-1}$-symmetric orthogonal polynomials in the $q$-Askey scheme, their dual polynomials and functions, orthogonality, generating functions and relations and nonterminating $q$-Chaundy double product representations

Howard S. Cohl, Roberto S. Costas-Santos

TL;DR

The paper develops a $q$-analogue of Chaundy diagonal summation to obtain double-product representations for nonterminating basic hypergeometric series, including limiting cases as $q\to 1^{-}$. These $q$-Chaundy representations are then employed to study the $q$- and $q^{-1}$-symmetric orthogonal polynomials of the $q$-Askey scheme—such as Askey–Wilson, continuous dual $q$-Hahn, Al–Salam–Chihara, and the continuous/big/small $q$-Hermite families—focusing on generating functions, duality relations, and orthogonality. The work also derives new terminating and nonterminating basic-hypergeometric representations, generating functions, and connection formulas, and leverages diagonal summation to relate products of series to single terminating expressions; these methods yield new summation and integration formulas across the infinite families, with detailed dualities linking to big and little $q$-Jacobi polynomials and $q$-Bessel/ Stieltjes–Wigert structures. Together, these results enhance analytical tools for basic hypergeometric orthogonal polynomials, enabling streamlined derivations of generating functions and dual relations, and offering new pathways for evaluating products and transforms in the $q$-Askey framework.

Abstract

We derive double-product representations of nonterminating basic hypergeometric series using diagonalization, a method introduced by Theo William Chaundy in 1943. We refer to this result as the $q$-Chaundy theorem and several limiting $q\to 1^{-}$ cases are considered. Using the $q$-Chaundy theorem, we explore properties of the symmetric and $q^{-1}$-symmetric basic hypergeometric orthogonal polynomials in the $q$-Askey scheme. These are the continuous dual $q$ and $q^{-1}$-Hahn polynomials, the $q$ and $q^{-1}$-Al-Salam--Chihara polynomials, the continuous big $q$ and $q^{-1}$-Hermite polynomials and the continuous $q$ and $q^{-1}$-Hermite polynomials. For instance, we show how many known (and unknown) generating functions can be easily derived for these polynomials. We also explore other methods to find generating functions for these polynomials. By applying the $q$-Chaundy theorem to the Ismail--Masson $q$-exponential generating function for continuous $q$ and $q^{-1}$-Hermite polynomials, we are able to derive alternative expansions of these generating functions, and from these, new terminating basic hypergeometric representations for the continuous $q$ and $q^{-1}$-Hermite polynomials. New quadratic transformations for the terminating basic hypergeometric series involved connect these representations. For the $q$ and $q^{-1}$-symmetric subfamilies of the Askey--Wilson polynomials and as well their dual polynomials, which include the big and little $q$-Jacobi polynomials and the $q^{-1}$-Bessel polynomials, we discuss, and show how to exploit special orthogonality relations (integral and infinite series), connection formulas, and duality relations for these infinite families to derive new generating relations, and as well summation and integration formulas.

The $q$ and $q^{-1}$-symmetric orthogonal polynomials in the $q$-Askey scheme, their dual polynomials and functions, orthogonality, generating functions and relations and nonterminating $q$-Chaundy double product representations

TL;DR

The paper develops a -analogue of Chaundy diagonal summation to obtain double-product representations for nonterminating basic hypergeometric series, including limiting cases as . These -Chaundy representations are then employed to study the - and -symmetric orthogonal polynomials of the -Askey scheme—such as Askey–Wilson, continuous dual -Hahn, Al–Salam–Chihara, and the continuous/big/small -Hermite families—focusing on generating functions, duality relations, and orthogonality. The work also derives new terminating and nonterminating basic-hypergeometric representations, generating functions, and connection formulas, and leverages diagonal summation to relate products of series to single terminating expressions; these methods yield new summation and integration formulas across the infinite families, with detailed dualities linking to big and little -Jacobi polynomials and -Bessel/ Stieltjes–Wigert structures. Together, these results enhance analytical tools for basic hypergeometric orthogonal polynomials, enabling streamlined derivations of generating functions and dual relations, and offering new pathways for evaluating products and transforms in the -Askey framework.

Abstract

We derive double-product representations of nonterminating basic hypergeometric series using diagonalization, a method introduced by Theo William Chaundy in 1943. We refer to this result as the -Chaundy theorem and several limiting cases are considered. Using the -Chaundy theorem, we explore properties of the symmetric and -symmetric basic hypergeometric orthogonal polynomials in the -Askey scheme. These are the continuous dual and -Hahn polynomials, the and -Al-Salam--Chihara polynomials, the continuous big and -Hermite polynomials and the continuous and -Hermite polynomials. For instance, we show how many known (and unknown) generating functions can be easily derived for these polynomials. We also explore other methods to find generating functions for these polynomials. By applying the -Chaundy theorem to the Ismail--Masson -exponential generating function for continuous and -Hermite polynomials, we are able to derive alternative expansions of these generating functions, and from these, new terminating basic hypergeometric representations for the continuous and -Hermite polynomials. New quadratic transformations for the terminating basic hypergeometric series involved connect these representations. For the and -symmetric subfamilies of the Askey--Wilson polynomials and as well their dual polynomials, which include the big and little -Jacobi polynomials and the -Bessel polynomials, we discuss, and show how to exploit special orthogonality relations (integral and infinite series), connection formulas, and duality relations for these infinite families to derive new generating relations, and as well summation and integration formulas.
Paper Structure (82 sections, 252 theorems, 405 equations, 1 figure)

This paper contains 82 sections, 252 theorems, 405 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Mathematical preliminaries
  3. Binomial coefficients, shifted and q-shifted factorials
  4. Generalized hypergeometric series
  5. Generalized hypergeometric orthogonal polynomials in the Askey scheme
  6. Basic hypergeometric series
  7. Generalized hypergeometric series: the $q\to1^{-}$ limit of basic hypergeometric series
  8. Important nonterminating summations
  9. Important terminating summations
  10. Important nonterminating transformations
  11. Important terminating transformations
  12. Basic hypergeometric orthogonal po-ly-no-mials and functions
  13. The $q$ and $q^{-1}$-symmetric and dual families
  14. The Askey--Wilson polynomials
  15. The continuous dual q and $q^{-1}$-Hahn polynomials
  16. ...and 67 more sections

Key Result

Proposition 2.2

For any $a\in \mathbb C$, $a\ne 1$, the following identity holds:

Figures (1)

  • Figure 1: This figure depicts the duality relations for the $q$ and $q^{-1}$-symmetric subfamilies of the Askey--Wilson polynomials. Arrows represent limit transitions between the subfamilies of the Askey--Wilson polynomials. Dashed lines represent polynomial duality and double dashed lines represent function duality.

Theorems & Definitions (284)

  • Definition 2.1
  • Proposition 2.2
  • Lemma 2.3
  • Theorem 2.4: Euler
  • Remark 2.5
  • Theorem 2.6
  • Theorem 2.7: Gasper & Rahman's (2004) Inversion Theorem
  • Lemma 2.8
  • Corollary 2.9
  • Proposition 2.10
  • ...and 274 more