A difference operator of infinite order with Sobolev-type Charlier polynomials as eigenfunctions

Polynomials are considered which are orthogonal with respect to the inner product [f,g] = (x=0)<SIGMA>(infinity) f(x)g(x) e(-a)a(x)\x! + lambda f(0)g(0) + mu Delta f(0), a > 0, lambda greater than or equal to 0, mu greater than or equal to 0. A representation for these polynomials is presented. It is shown that in the case lambda = 0 and mu > 0 these polynomials are eigenfunctions of a difference operator, which is shown to be of infinite order for all values of a > 0. The difference operator and the eigenvalues are linear perturbations of those in the Charlier case (lambda = 0, mu = 0). A formula for the eigenvalues and a representation for the coefficients in the differential operator are presented.