Exceptional Gegenbauer polynomials via isospectral deformation

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 exceptional polynomials of the second kind. We illustrate this construction by describing the class of exceptional Gegenbauer polynomials of the second kind.