A conjecture on Exceptional Orthogonal Polynomials
Mathematical Physics
2013-09-04 v1 math.MP
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.
Cite
@article{arxiv.1203.6857,
title = {A conjecture on Exceptional Orthogonal Polynomials},
author = {David Gomez-Ullate and Niky Kamran and Robert Milson},
journal= {arXiv preprint arXiv:1203.6857},
year = {2013}
}