Recurrence Relations for Exceptional Hermite Polynomials

TL;DR

This paper develops explicit formulas for difference operators with Hermite exceptional orthogonal polynomials as eigenfunctions, using the bispectral anti-isomorphism and stabilizer ring elements.

Contribution

It introduces a method to derive all difference operators associated with Hermite exceptional polynomials via bispectral anti-isomorphism.

Findings

Explicit formulas for difference operators with Hermite exceptional polynomials as eigenfunctions.

Connection between stabilizer ring elements and eigenvalue polynomials.

Framework applicable to other exceptional orthogonal polynomials.

Abstract

The bispectral anti-isomorphism is applied to differential operators involving elements of the stabilizer ring to produce explicit formulas for all difference operators having any of the Hermite exceptional orthogonal polynomials as eigenfunctions with eigenvalues that are polynomials in $x$.