The algebra of difference operators associated to a family of orthogonal polynomials

Abstract

Given a weight matrix W of arbitrary size N×N on the real line and a sequence of matrix valued orthogonal polynomials (Pn)n with respect to W, we study the algebra D(W) of difference operators D with matrix polynomial coefficients such that D(Pn)=ΓnPn, with Γn∈CN×N. As a consequence, we deduce that scalar polynomials orthogonal with respect to a nondegenerate positive measure can satisfy only difference equations of even order, and prove that the algebra of difference operators associated to any of the four discrete classical families of Charlier, Meixner, Krawtchouk and Hahn is generated from the second order difference operator (it is unique up to constants). We also introduce three illustrative matrix examples showing that the situation in the matrix valued case is much more interesting. These matrix families are the first non-trivial examples of weight matrices appearing in the literature whose orthogonal polynomials satisfy second order difference equations.