Dual forms of the orthogonality relations of some classical q-orthogonal polynomials

TL;DR

This paper introduces new matrix operations and inverse relations to establish dual orthogonality relations for various classical q-orthogonal polynomials, revealing deep connections within the Askey-scheme.

Contribution

It presents novel dual forms of orthogonality relations for key q-polynomials, including the Askey-Wilson family, using innovative matrix and inverse relation techniques.

Findings

Dual orthogonality relations for q-polynomials established

Askey-Wilson q-beta integral shown as a dual form

New matrix operations facilitate the derivation of dual relations

Abstract

In this paper, by introducing new matrix operations and using a specific inverse relation, we establish the dual forms of the orthogonality relations for some well-known discrete and continuous $q$-orthogonal polynomials from the Askey-scheme such as the little and big $q$-Jacobi, $q$-Racah, (generalized) $q$-Laguerre, as well as the Askey-Wilson polynomials. As one of the most interesting results, we show that the Askey-Wilson $q$-beta integral represented in terms of the VWP-balanced $\,_8\phi_7$ series is just a dual form of the orthogonality relation of the Askey-Wilson polynomials.