A Characterization of Askey-Wilson polynomials

TL;DR

This paper characterizes Askey-Wilson polynomials as the unique monic orthogonal polynomials satisfying a specific second-order q-difference relation involving a polynomial of degree at most 4, confirming a conjecture by Ismail.

Contribution

It proves that Askey-Wilson polynomials are uniquely characterized by a particular structure relation involving the Askey-Wilson operator, completing Ismail's conjecture.

Findings

Characterization of Askey-Wilson polynomials through a structure relation.

Derivation of bounds for the zeros of Askey-Wilson polynomials.

Confirmation of a conjecture by Ismail regarding these polynomials.

Abstract

We show that the only monic orthogonal polynomials $\{P_n\}_{n=0}^{\infty}$ that satisfy $$\pi(x)\mathcal{D}_{q}^2P_{n}(x)=\sum_{j=-2}^{2}a_{n,n+j}P_{n+j}(x),\; x=\cos\theta,\;~ a_{n,n-2}\neq 0,~ n=2,3,\dots,$$ where $\pi(x)$ is a polynomial of degree at most $4$ and $\mathcal{D}_{q}$ is the Askey-Wilson operator, are Askey-Wilson polynomials and their special or limiting cases. This completes and proves a conjecture by Ismail concerning a structure relation satisfied by Askey-Wilson polynomials. We use the structure relation to derive upper bounds for the smallest zero and lower bounds for the largest zero of Askey-Wilson polynomials and their special cases.