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 \emph{exceptional polynomials of the second kind}. We illustrate this construction by describing the class of exceptional Gegenbauer polynomials of the second kind.