Characterization of Orthogonal Polynomials on lattices

Abstract

We consider two sequences of orthogonal polynomials (Pn) n≥0 and (Qn) n≥0 such thatwith k, m, M, N ∈ N, a j,n and b j,n are sequences of complex numbers,), 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 (Pn) n≥0 and (Qn) n≥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. This work generalizes and extends a previous result [arXiv:2202.10167] concerning a characterization of Askey-Wilson polynomials.