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 $(P_n)_{n\geq 0}$ 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.