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

Published 2024-11-13Version 1

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.