Exceptional Charlier and Hermite orthogonal polynomials

Abstract

Using Casorati determinants of Charlier polynomials, we construct for each finite set $F$ of positive integers a sequence of polynomials $r_n^F$, $n\in {\sigma }_F$, which are eigenfunction of a second order difference operator, where ${\sigma }_F$ is an infinite set of nonnegative integers, $σ_F \varsubsetneq \NN$. For certain finite sets $F$ (we call them admissible sets), we prove that the polynomials $r_n^F$, $n\in {\sigma }_F$, 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.