Complex exceptional orthogonal polynomials and quasi-invariance

TL;DR

This paper generalizes the theory of exceptional orthogonal polynomials derived from Hermite polynomial Wronskians to all partitions, introducing complex contours and non-positive inner products, and explores their orthogonality, density, and quasi-invariance properties.

Contribution

It extends previous results to all partitions by considering complex integration contours and non-positive inner products, establishing orthogonality and density in specific subspaces, and introduces a Laurent version linked to Schrödinger operators.

Findings

Polynomials are orthogonal and dense in a finite-codimensional subspace.

Extension of exceptional orthogonal polynomials to all partitions.

Introduction of a Laurent version related to monodromy-free trigonometric Schrödinger operators.

Abstract

Consider the Wronskians of the classical Hermite polynomials $$H_{\lambda, l}(x):=\mathrm{Wr}(H_l(x),H_{k_1}(x),\ldots, H_{k_n}(x)), \quad l \in \mathbb Z_{\geq 0},$$ where $k_i=\lambda_i+n-i, \,\, i=1,\dots, n$ and $\lambda=(\lambda_1, \dots, \lambda_n)$ is a partition. G\'omez-Ullate et al showed that for a special class of partitions the corresponding polynomials are orthogonal and dense among all polynomials with certain inner product, but in contrast to the usual case have some degrees missing (so called exceptional orthogonal polynomials). We generalise their results to all partitions by considering complex contours of integration and non-positive Hermitian products. The corresponding polynomials are orthogonal and dense in a finite-codimensional subspace of $\mathbb C[x]$ satisfying certain quasi-invariance conditions. A Laurent version of exceptional orthogonal polynomials, related to monodromy-free trigonometric Schr\"odinger operators, is also presented.