On orthogonal polynomials spanning a non-standard flag

TL;DR

This paper reviews recent advances in orthogonal polynomials, highlighting the discovery of exceptional families that extend classical polynomials by featuring non-standard polynomial flags and analyzing their properties.

Contribution

It provides a classification of codimension 1 exceptional orthogonal polynomials and introduces a new proof of the fundamental classification theorem.

Findings

Existence of orthogonal polynomials outside classical families

Classification of codimension 1 exceptional polynomials

Analysis of parameter ranges for non-singular weights

Abstract

We survey some recent developments in the theory of orthogonal polynomials defined by differential equations. The key finding is that there exist orthogonal polynomials defined by 2nd order differential equations that fall outside the classical families of Jacobi, Laguerre, and Hermite polynomials. Unlike the classical families, these new examples, called exceptional orthogonal polynomials, feature non-standard polynomial flags; the lowest degree polynomial has degree $m>0$. In this paper we review the classification of codimension $m=1$ exceptional polynomials, and give a novel, compact proof of the fundamental classification theorem for codimension 1 polynomial flags. As well, we describe the mechanism or rational factorizations of 2nd order operators as the analogue of the Darboux transformation in this context. We finish with the example of higher codimension generalization of Jacobi polynomials and perform the complete analysis of parameter values for which these families have non-singular weights.