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

Abstract

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