Exceptional Charlier and Hermite orthogonal polynomials

TL;DR

This paper constructs new classes of exceptional Charlier and Hermite polynomials using Casorati and Wronskian determinants, proving their orthogonality, completeness, and eigenfunction properties for specific finite sets.

Contribution

It introduces a novel method to generate exceptional orthogonal polynomials via Casorati and Wronskian determinants, expanding the family of known exceptional polynomials.

Findings

Exceptional Charlier polynomials are orthogonal and complete for certain sets.

Limit transition from Charlier to Hermite yields exceptional Hermite polynomials.

New eigenfunction properties for these polynomials are established.

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, $\sigma_F \varsubsetneq \NN$. 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.