The Adelic Grassmannian and Exceptional Hermite Polynomials

TL;DR

This paper reveals a surprising connection between the adelic Grassmannian and exceptional Hermite polynomials, leading to new algorithms for computing related operators and answering open questions.

Contribution

It establishes a novel link between the adelic Grassmannian and exceptional Hermite polynomials, providing new computational methods and insights.

Findings

Generated functions of exceptional Hermite polynomials from adelic Grassmannian points.

Developed algorithms for differential and difference operator computation.

Resolved open questions about these operators.

Abstract

It is shown that when dependence on the second flow of the KP hierarchy is added, the resulting semi-stationary wave function of certain points in George Wilson's adelic Grassmannian are generating functions of the exceptional Hermite orthogonal polynomials. This surprising correspondence between different mathematical objects that were not previously known to be so closely related is interesting in its own right, but also proves useful in two ways: it leads to new algorithms for effectively computing the associated differential and difference operators and it also answers some open questions about them.