The algebras of difference operators associated to Krall-Charlier orthogonal polynomials

Krall-Charlier polynomials (c(n)(a; F))(n) are orthogonal polynomials which are also eigenfunctions of a higher order difference operator. They are defined from a parameter a (associated to the Charlier polynomials) and a finite set F of positive integers. We study the algebra D-a(F) formed by all difference operators with respect to which the family of Krall-Charlier polynomials (c(n)(a; F))(n) are eigenfunctions. Each operator D is an element of D-a(F) is characterized by the so called eigenvalue polynomial lambda(D): lambda(D) is the polynomial satisfying D(c(n)(a; F)) = lambda(D)(n)c(n)(a; F). We characterize the algebra of difference operators D-a(F) by means of the algebra of polynomials D-a(F) = {lambda is an element of C[x] : lambda(x) = lambda(D)(x), D is an element of D-a(F)}. We associate to the family (c(n)(a; F))(n) a polynomial Omega(a)(F) and prove that, except for degenerate cases, the algebra D-a(F) is formed by all polynomials lambda(x) such that Omega(a)(F) divides lambda(x) - lambda(x - 1). We prove that this is always the case for a segment F (i.e., the elements of F are consecutive positive integers), and conjecture that it is also the case when the Krall-Charlier polynomials (c(n)(a; F))(n) are orthogonal with respect to a positive measure. (C) 2018 Elsevier Inc. All rights reserved.