Exceptional Charlier and Hermite orthogonal polynomials

Using Casorati determinants of Charlier polynomials (C-n(a))(n), we construct for each finite set F of positive integers a sequence of polynomials C-n,(a) n is an element of F-sigma, which are eigenfunctions of a second order difference operator, where F-sigma is certain infinite set of nonnegative integers, sigma F not subset of N. For suitable finite sets F (we call them admissible sets), we prove that the polynomials C-n,(a) n is an element of F-sigma, 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. (C) 2014 Elsevier Inc. All rights reserved.