Exceptional Hermite Polynomials and Calogero-Moser Pairs

TL;DR

This paper explores the connection between Exceptional Hermite Polynomials and Calogero-Moser Pairs, providing a new formula and explicit distributions that annihilate these polynomials, advancing the understanding of their algebraic structure.

Contribution

It introduces a novel formula linking Exceptional Hermite Polynomials with Calogero-Moser Pairs and demonstrates how to produce explicit distributions that annihilate these polynomials.

Findings

New formula for Exceptional Hermite Polynomials in terms of Calogero-Moser Pairs

Explicit finitely-supported distributions that annihilate Exceptional Hermite Polynomials

Enhanced understanding of the algebraic structure of Exceptional Hermite Polynomials

Abstract

There are two equivalent descriptions of George Wilson's adelic Grassmannian $Gr^{ad}$, one in terms of differential ``conditions'' and another in terms of Calogero-Moser Pairs. The former approach was used in the 2020 paper by Kasman-Milson which found that each family of Exceptional Hermite Polynomials has a generating function which lives in $Gr^{ad}$. This suggests that Calogero-Moser Pairs should also be useful in the study of Exceptional Hermite Polynomials, but no researchers have pursued that line of inquiry prior to the first author's thesis. The purpose of this note is to summarize highlights from that thesis, including a novel formula for Exceptional Hermite Polynomials in terms of Calogero-Moser Pairs and a theorem utilizing this correspondence to produce explicit finitely-supported distributions which annihilate them.