A conjecture on Exceptional Orthogonal Polynomials

TL;DR

This paper explores the structure of exceptional orthogonal polynomial systems, proposing a conjecture relating them to classical systems via Darboux-Crum transformations, and proves this for codimension 2 cases, leading to a classification including new examples.

Contribution

It formulates and proves a conjecture connecting exceptional orthogonal polynomials to classical ones through Darboux-Crum transformations, and classifies all codimension 2 cases.

Findings

Proved the conjecture for codimension 2 exceptional orthogonal polynomials.

Classified all possible X2-OPS, including new examples.

Extended the understanding of the relationship between exceptional and classical orthogonal polynomials.

Abstract

Exceptional orthogonal polynomial systems (X-OPS) arise as eigenfunctions of Sturm-Liouville problems and generalize in this sense the classical families of Hermite, Laguerre and Jacobi. They also generalize the family of CPRS orthogonal polynomials. We formulate the following conjecture: every exceptional orthogonal polynomial system is related to a classical system by a Darboux-Crum transformation. We give a proof of this conjecture for codimension 2 exceptional orthogonal polynomials (X2-OPs). As a by-product of this analysis, we prove a Bochner-type theorem classifying all possible X2-OPS. The classification includes all cases known to date plus some new examples of X2-Laguerre and X2-Jacobi polynomials.