Exceptional Legendre Polynomials and Confluent Darboux Transformations

TL;DR

This paper introduces a new method for constructing multi-parameter exceptional Legendre polynomials using confluent Darboux transformations, expanding the family of orthogonal polynomials with explicit formulas and arbitrary parameters.

Contribution

It presents a novel construction of multi-parameter exceptional Legendre polynomials via isospectral deformation and confluent Darboux transformations, with explicit operator descriptions.

Findings

Explicit formulas for multi-parameter exceptional Legendre polynomials

Construction method applicable to arbitrary number of parameters

Extension of classical orthogonal polynomial families

Abstract

Exceptional orthogonal polynomials are families of orthogonal polynomials that arise as solutions of Sturm-Liouville eigenvalue problems. They generalize the classical families of Hermite, Laguerre, and Jacobi polynomials by allowing for polynomial sequences that miss a finite number of "exceptional" degrees. In this paper we introduce a new construction of multi-parameter exceptional Legendre polynomials by considering the isospectral deformation of the classical Legendre operator. Using confluent Darboux transformations and a technique from inverse scattering theory, we obtain a fully explicit description of the operators and polynomials in question. The main novelty of the paper is the novel construction that allows for exceptional polynomial families with an arbitrary number of real parameters.