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Asymptotics, orthogonality relations and duality for the $q$ and $q^{-1}$-symmetric polynomials in the $q$-Askey scheme

Howard S. Cohl, Roberto S. Costas-Santos, Xiang-Sheng Wang

TL;DR

This survey analyzes the orthogonality, asymptotics, and duality structures of the q- and q^{-1}-symmetric subfamilies in the q-Askey scheme, centering on the Askey–Wilson polynomials and their Numerous limits to continuous dual q-Hahn, Al-Salam–Chihara, continuous big q-Hermite, and continuous q-Hermite polynomials, along with their big and little q-Jacobi and q-Bessel counterparts. It highlights that the q^{-1}-symmetric families satisfy an indeterminate moment problem, yielding an infinite spectrum of orthogonality relations, including continuous, discrete, and bilateral forms; a new infinite discrete orthogonality is obtained for the continuous big q^{-1}-Hermite polynomials via symmetric limiting processes. The work develops duality relations among q- and q^{-1}-symmetric families, uses the Darboux method to derive large-degree asymptotics, and introduces a two-parameter generating function for the q^{-1}-Al-Salam–Chihara polynomials to facilitate these analyses. Overall, the paper consolidates orthogonality frameworks and dualities across the q- and q^{-1}-symmetric subfamilies, establishing spectral and asymptotic properties that connect to big and little q-Jacobi families and q-Bessel systems, with implications for degeneration and application within the q-Askey scheme.

Abstract

In this survey we summarize the current state of known orthogonality relations for the $q$ and $q^{-1}$-symmetric and dual subfamilies of the Askey--Wilson polynomials in the $q$-Askey scheme. These polynomials 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 and their dual counterparts which are connected with the big $q$-Jacobi polynomials, the little $q$-Jacobi polynomials and the $q$ and $q^{-1}$-Bessel polynomials. The $q^{-1}$-symmetric polynomials in the $q$-Askey scheme satisfy an indeterminate moment problem, satisfying an infinite number of orthogonality relations for these polynomials. Among the infinite number of orthogonality relations for the $q^{-1}$-symmetric families, we attempt to summarize those currently known. These fall into several classes, including continuous orthogonality relations and infinite discrete (including bilateral) orthogonality relations. Using symmetric limits, we derive a new infinite discrete orthogonality relation for the continuous big $q^{-1}$-Hermite polynomials. Using duality relations, we explore orthogonality relations for and from the dual families associated with the $q$ and $q^{-1}$-symmetric subfamilies of the Askey--Wilson polynomials. In order to give a complete description of the convergence properties for these polynomials, we provide the large degree asymptotics using the Darboux method for these polynomials. In order to apply the Darboux method, we derive a generating function with two free parameters for the $q^{-1}$-Al-Salam--Chihara polynomials which has natural limits to the lower $q^{-1}$-symmetric families.

Asymptotics, orthogonality relations and duality for the $q$ and $q^{-1}$-symmetric polynomials in the $q$-Askey scheme

TL;DR

This survey analyzes the orthogonality, asymptotics, and duality structures of the q- and q^{-1}-symmetric subfamilies in the q-Askey scheme, centering on the Askey–Wilson polynomials and their Numerous limits to continuous dual q-Hahn, Al-Salam–Chihara, continuous big q-Hermite, and continuous q-Hermite polynomials, along with their big and little q-Jacobi and q-Bessel counterparts. It highlights that the q^{-1}-symmetric families satisfy an indeterminate moment problem, yielding an infinite spectrum of orthogonality relations, including continuous, discrete, and bilateral forms; a new infinite discrete orthogonality is obtained for the continuous big q^{-1}-Hermite polynomials via symmetric limiting processes. The work develops duality relations among q- and q^{-1}-symmetric families, uses the Darboux method to derive large-degree asymptotics, and introduces a two-parameter generating function for the q^{-1}-Al-Salam–Chihara polynomials to facilitate these analyses. Overall, the paper consolidates orthogonality frameworks and dualities across the q- and q^{-1}-symmetric subfamilies, establishing spectral and asymptotic properties that connect to big and little q-Jacobi families and q-Bessel systems, with implications for degeneration and application within the q-Askey scheme.

Abstract

In this survey we summarize the current state of known orthogonality relations for the and -symmetric and dual subfamilies of the Askey--Wilson polynomials in the -Askey scheme. These polynomials 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 and their dual counterparts which are connected with the big -Jacobi polynomials, the little -Jacobi polynomials and the and -Bessel polynomials. The -symmetric polynomials in the -Askey scheme satisfy an indeterminate moment problem, satisfying an infinite number of orthogonality relations for these polynomials. Among the infinite number of orthogonality relations for the -symmetric families, we attempt to summarize those currently known. These fall into several classes, including continuous orthogonality relations and infinite discrete (including bilateral) orthogonality relations. Using symmetric limits, we derive a new infinite discrete orthogonality relation for the continuous big -Hermite polynomials. Using duality relations, we explore orthogonality relations for and from the dual families associated with the and -symmetric subfamilies of the Askey--Wilson polynomials. In order to give a complete description of the convergence properties for these polynomials, we provide the large degree asymptotics using the Darboux method for these polynomials. In order to apply the Darboux method, we derive a generating function with two free parameters for the -Al-Salam--Chihara polynomials which has natural limits to the lower -symmetric families.

Paper Structure

This paper contains 27 sections, 69 theorems, 166 equations, 1 figure.

Table of Contents

  1. Mathematical preliminaries
  2. Binomial coefficients and q-shifted factorials
  3. Basic hypergeometric series
  4. Nonterminating and terminating summations
  5. Nonterminating and terminating transformations
  6. The $q$ and $q^{-1}$-symmetric polynomials and their dual families
  7. The Askey--Wilson polynomials
  8. The continuous dual q and $q^{-1}$-Hahn polynomials
  9. The q and $q^{-1}$-Al-Salam--Chihara polynomials
  10. The continuous big q and $q^{-1}$-Hermite polynomials
  11. The continuous q and $q^{-1}$-Hermite polynomials
  12. The big q-Jacobi polynomials and functions
  13. The little q-Jacobi polynomials and functions
  14. The q and $q^{-1}$-Bessel polynomials and functions
  15. Duality relations for the $q$ and $q^{-1}$-symmetric and their dual families
  16. ...and 12 more sections

Key Result

Theorem 1.2

Let $q\in{{\mathbb C}^\dag}$, $z\in{{\mathbb C}}$ such that $|z|<1$. Then

Figures (1)

  • Figure 1: This figure depicts the $q$-Askey scheme including the $q$ and $q^{-1}$-symmetric subfamilies of the Askey--Wilson polynomials and as well their corresponding dual families. 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 (79)

  • Definition 1.1
  • Theorem 1.2: Euler
  • Theorem 1.3: Gasper & Rahman's (2004) Inversion Theorem
  • Lemma 1.4
  • Corollary 1.5
  • Proposition 1.6
  • Remark 2.1
  • Remark 2.2
  • Theorem 2.3
  • Corollary 2.4
  • ...and 69 more