Hermite and Laguerre Symmetric Functions Associated with Operators of Calogero-Moser-Sutherland Type

TL;DR

This paper develops generalized Hermite and Laguerre symmetric functions as eigenfunctions of infinite-dimensional Calogero-Moser-Sutherland operators, exploring their properties, limits, and associated deformed operators with polynomial eigenfunctions.

Contribution

It introduces new symmetric functions related to CMS operators, establishes their properties, and constructs deformed operators with polynomial eigenfunctions, expanding the theory of integrable systems.

Findings

Derived generating functions, duality relations, and Pieri formulas.

Identified ideals spanned by Hermite and Laguerre symmetric functions.

Constructed deformed CMS operators with polynomial eigenfunctions.

Abstract

We introduce and study natural generalisations of the Hermite and Laguerre polynomials in the ring of symmetric functions as eigenfunctions of infinite-dimensional analogues of partial differential operators of Calogero-Moser-Sutherland (CMS) type. In particular, we obtain generating functions, duality relations, limit transitions from Jacobi symmetric functions, and Pieri formulae, as well as the integrability of the corresponding operators. We also determine all ideals in the ring of symmetric functions that are spanned by either Hermite or Laguerre symmetric functions, and by restriction of the corresponding infinite-dimensional CMS operators onto quotient rings given by such ideals we obtain so-called deformed CMS operators. As a consequence of this restriction procedure, we deduce, in particular, infinite sets of polynomial eigenfunctions, which we shall refer to as super Hermite and super Laguerre polynomials, as well as the integrability, of these deformed CMS operators. We also introduce and study series of a generalised hypergeometric type, in the context of both symmetric functions and 'super' polynomials.