A characterization of continuous $q$-Jacobi, Chebyshev of the first kind and Al-Salam Chihara polynomials

TL;DR

This paper characterizes specific orthogonal polynomial sequences, including continuous $q$-Jacobi, Chebyshev of the first kind, and Al-Salam Chihara polynomials, based on their behavior under the Askey-Wilson operator.

Contribution

It provides a new characterization of these polynomial families through a differential-type relation involving the Askey-Wilson operator.

Findings

Characterization of continuous $q$-Jacobi, Chebyshev, and Al-Salam Chihara polynomials.

Identification of the polynomial relations involving the Askey-Wilson operator.

Extension of classical orthogonal polynomial theory to the $q$-analogue setting.

Abstract

The purpose of this note is to characterize those orthogonal polynomials sequences $(P_n)_{n\geq0}$ for which $$ \pi(x)\mathcal{D}_q P_n(x)=(a_n x+b_n)P_n(x)+c_n P_{n-1}(x),\quad n=0,1,2,\dots, $$ where $\mathcal{D}_q$ is the Askey-Wilson operator, $\pi$ is a polynomial of degree at most 2, and $(a_n)_{n\geq0}$, $(b_n)_{n\geq0}$ and $(c_n)_{n\geq0}$ are sequences of complex numbers such that $c_n\neq0$ for $n=1,2,\dots$.