Exceptional Gegenbauer polynomials via isospectral deformation

TL;DR

This paper introduces a method to generate new families of orthogonal polynomials, called exceptional Gegenbauer polynomials of the second kind, using isospectral deformations and confluent Darboux transformations.

Contribution

It presents a novel construction technique for exceptional orthogonal polynomials with multiple parameters, expanding the classical polynomial families.

Findings

Constructed explicit examples of exceptional Gegenbauer polynomials of the second kind.

Demonstrated the use of confluent Darboux transformations in creating isospectral deformations.

Established a framework for polynomial eigenfunctions with arbitrary continuous parameters.

Abstract

We show a method to construct isospectral deformations of classical orthogonal polynomials. The construction is based on confluent Darboux transformations, and it allows to construct Sturm-Liouville problems with polynomial eigenfunctions that have an arbitrary number of continuous parameters. We propose to call these new orthogonal polynomial systems \emph{exceptional polynomials of the second kind}. We illustrate this construction by describing the class of exceptional Gegenbauer polynomials of the second kind.