Exceptional Charlier and Hermite orthogonal polynomials

Abstract

Using Casorati determinants of Charlier polynomials (cna)n, we construct for each finite set F of positive integers a sequence of polynomials cnF, n∈σF, which are eigenfunctions of a second order difference operator, where σF is certain infinite set of nonnegative integers, σF⊊N. For suitable finite sets F (we call them admissible sets), we prove that the polynomials cnF, n∈σF, are actually exceptional Charlier polynomials; that is, in addition, they are orthogonal and complete with respect to a positive measure. By passing to the limit, we transform the Casorati determinant of Charlier polynomials into a Wronskian determinant of Hermite polynomials. For admissible sets, these Wronskian determinants turn out to be exceptional Hermite polynomials.