Exceptional Hermite Polynomials and Calogero-Moser Pairs

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.

• The paper introduces a novel formula linking Exceptional Hermite Polynomials to Calogero-Moser pairs using generating functions.
• The paper employs integrable dynamics and matrix theory to derive explicit representations that blend classical and exceptional orthogonal polynomials.
• The paper’s findings open new avenues for computational techniques and further research on orthogonal polynomials and integrable systems.

Exceptional Hermite Polynomials and Calogero-Moser Pairs (2507.21935)

Introduction

The paper explores the interplay between Exceptional Hermite Polynomials (XHPs) and Calogero-Moser (CM) Pairs, two objects of mathematical interest with prior independent developments. While CM Pairs are traditionally associated with integrable dynamics of particle systems, XHPs are a generalization of classical orthogonal polynomials, initially defined in 2009 with specific consideration in 2014. The paper builds on a 2020 discovery that identified a connection between the generating functions of XHPs and Wilson's adelic Grassmannian $Gr^{ad}$, which also characterizes certain bispectral wave functions of the KP Hierarchy in terms of CM Pairs.

Background on Key Mathematical Constructs

Calogero-Moser Pairs (CM Pairs)

CM Pairs are pairs of $N \times N$ matrices $(X, Z)$ whose commutator $[X, Z]$ deviates from the identity matrix by a rank one matrix. These pairs, first attracting interest in the 1970s, are crucial in understanding the integrable dynamics of the Calogero-Moser particle system. The significance of CM Pairs extends to the field of integrable systems, notably influencing the theory behind the KP Hierarchy.

Exceptional Hermite Polynomials (XHPs)

Exceptional Hermite Polynomials represent a class of orthogonal polynomials, which, unlike classical polynomials, can lack members of certain degrees. The paper focuses on these polynomials defined by an innovative generating function that lies in $Gr^{ad}$, utilizing the CM Pairs as a tool for deriving new expressions and properties.

Implementation of Concepts

Calogero-Moser Pair for XHP Generating Functions

The research addresses the relationship between XHPs and CM Pairs by providing a concrete formula that represents XHPs in terms of CM Pairs. The paper establishes that each family of XHPs has a generating function expressible in terms of CM Pairs. This lays the groundwork for further exploration and applications in the analysis and computation of XHPs.

Novel Formula for XHPs

A significant contribution of this work is a new formula that presents XHPs as a linear combination of classical Hermite polynomials, with coefficients dependent on the associated CM Pair. This connection offers a different toolset for analyzing XHPs, particularly in computational contexts.

import numpy as np from sympy import symbols, S def calculate_xhp(n, X, Z, a, b, tau, weighted_degree, K): # Placeholder for the calculation of XHPs utilizing CM Pairs det_X = np.linalg.det(X - weighted_degree * Z) res = det_X * sum( S('n!') / (np.math.factorial(weighted_degree - k)) * np.polynomial.Hermite(hermweight=...)(...) * np.dot(a.T, np.linalg.inv(X)) for k in range(len(K)) ) return res * weighted_degree X = np.array([[0, 0], [-1, 0]]) Z = np.array([[0, 1], [0, 0]]) K = [1, 2, 3, ...] x, y = symbols('x y') weighted_degree = x * x + 4 * ((-1)**2) a = np.array([[-1, -1]]) weighted_degree *= 2

This pseudocode demonstrates a high-level computation of XHPs using the CM Pair representation described in the paper.

Implications and Future Directions

This research delivers a novel intersection between the theory of special functions and integrable systems, providing alternative methodologies for deriving properties of XHPs through algebraic structures of CM Pairs. Although the paper establishes initial connections, the potential applications span a broader mathematical landscape. Investigating these relationships further may reveal deeper insights into orthogonal polynomials, integrable systems, and their roles in both theoretical and computational mathematics. Opportunities for future research exist in streamlining proofs of XHP orthogonality through CM Pair frameworks and generalizing results to arbitrary CM Pairs. The integration of flow dynamics in matrix transformations opens up avenues for alternative algorithmic implementations and potentially offers novel computational efficiencies.