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

Krall-Charlier polynomials (cna;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 DaF formed by all difference operators with respect to which the family of Krall-Charlier polynomials (cna;F)n are eigenfunctions. Each operator D∈DaF is characterized by the so called eigenvalue polynomial λD: λD is the polynomial satisfying D(cna;F)=λD(n)cna;F. We characterize the algebra of difference operators DaF by means of the algebra of polynomials D̃aF={λ∈C[x]:λ(x)=λD(x),D∈DaF}. We associate to the family (cna;F)n a polynomial ΩFa and prove that, except for degenerate cases, the algebra D̃aF is formed by all polynomials λ(x) such that ΩFa divides λ(x)−λ(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 (cna;F)n are orthogonal with respect to a positive measure.