On another characterization of Askey-Wilson polynomials

TL;DR

This paper characterizes Askey-Wilson polynomials as the unique orthogonal polynomial sequences satisfying a specific q-difference relation involving the Askey-Wilson operator and averaging operator.

Contribution

It provides a new characterization of Askey-Wilson polynomials through a differential relation involving well-chosen polynomial coefficients.

Findings

Askey-Wilson polynomials are uniquely characterized by the given operator relation.

The polynomials satisfying the relation are either Askey-Wilson polynomials or their special/limiting cases.

The result extends the understanding of the structural properties of Askey-Wilson polynomials.

Abstract

In this paper we show that the only sequences of orthogonal polynomials $(P_n)_{n\geq 0}$ satisfying \begin{align*} \phi(x)\mathcal{D}_q P_{n}(x)=a_n\mathcal{S}_q P_{n+1}(x) +b_n\mathcal{S}_q P_n(x) +c_n\mathcal{S}_q P_{n-1}(x), \end{align*} ($c_n\neq 0$) where $\phi$ is a well chosen polynomial of degree at most two, $\mathcal{D}_q$ is the Askey-Wilson operator and $\mathcal{S}_q$ the averaging operator, are the multiple of Askey-Wilson polynomials, or specific or limiting cases of them.