Characterization of Orthogonal Polynomials on lattices

Abstract: We consider two sequences of orthogonal polynomials $(P_n){n\geq 0}$ and $(Q_n){n\geq 0}$ 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, $$2\mathrm{S}xf(x(s))=(\triangle +2\,\mathrm{I})f(z),~~ \mathrm{D}_xf(x(s))=\frac{\triangle}{\triangle x(s-1/2)}f(z),$$ $z=x(s-1/2)$, $\mathrm{I}$ is the identity operator, $x$ defines a lattice, and $\triangle f(s)=f(s+1)-f(s)$. We show that under some natural conditions, both involved orthogonal polynomials sequences $(P_n){n\geq 0}$ and $(Q_n)_{n\geq 0}$ are semiclassical whenever $k=m$. Some particular cases are studied closely where we characterize the continuous dual Hahn and Wilson polynomials for quadratic lattices.