Remarks on Askey-Wilson polynomials and Meixner polynomials of the second kind

TL;DR

This paper characterizes orthogonal polynomial sequences satisfying a specific difference equation involving Askey-Wilson and Meixner polynomials of the second kind, and introduces a method to analyze such problems.

Contribution

It provides a complete characterization of polynomial sequences satisfying a particular difference relation and offers a practical approach to study these types of orthogonal polynomials.

Findings

Identifies all orthogonal polynomial sequences satisfying the given difference equation.

Provides a systematic method for analyzing similar orthogonal polynomial problems.

Connects the characterization to Askey-Wilson and Meixner polynomials of the second kind.

Abstract

The purpose of this note is twofold: firstly to characterize all the sequences of orthogonal polynomials $(P_n)_{n\geq 0}$ such that $$ \frac{\triangle}{{\bf \triangle} x(s-1/2)}P_{n+1}(x(s-1/2))=c_n(\triangle +2\,\mathrm{I})P_n(x(s-1/2)), $$ where $\mathrm{I}$ is the identity operator, $x$ defines a class of lattices with, generally, nonuniform step-size, and $\triangle f(s)=f(s+1)-f(s)$; and secondly to present, in a friendly way, a method to deal with these kind of problems.