Characterization of Orthogonal Polynomials on lattices

Abstract

We consider two sequences of orthogonal polynomials $(P_n{)}_{n\ge 0}$ and $(Q_n{)}_{n\ge 0}$ such that $$\sum _{j=1}^M{a}_{j,n}{\mathrm{S}}_x{\mathrm{D}}_x^k{P}_{k+n-j}(z)=\sum _{j=1}^N{b}_{j,n}{\mathrm{D}}_x^m{Q}_{m+n-j}(z),$$ with $k,m,M,N\in \mathbb{N}$, $a_{j,n}$ and $b_{j,n}$ are sequences of complex numbers, $$2{\mathrm{S}}_{x}f(x(s))=(△+2\mathrm{I})f(z),{\mathrm{D}}_{x}f(x(s))=\frac{△}{△x(s-1/2)}f(z),$$ $z=x(s-1/2)$, $\mathrm{I}$ is the identity operator, $x$ defines a lattice, and $△f(s)=f(s+1)-f(s)$. We show that under some natural conditions, both involved orthogonal polynomials sequences $(P_n{)}_{n\ge 0}$ and $(Q_n{)}_{n\ge 0}$ are semiclassical whenever $k=m$. Some particular cases are studied closely where we characterize the continuous dual Hahn and Wilson polynomials for quadratic lattices.