DEK-Type orthogonal polynomials and a modification of the Christoffel formula

TL;DR

This paper revisits DEK-type orthogonal polynomials, providing a new characterization via discrete Darboux transformations, and introduces a modified Christoffel formula for a related family of polynomials with missing degrees.

Contribution

It offers a novel characterization of DEK polynomials bypassing differential equations and introduces a modified Christoffel formula for a broader family of orthogonal polynomials.

Findings

Dissection of DEK polynomials using discrete Darboux transformations

Development of a characterization bypassing the differential equation

Derivation of a modified Christoffel formula for polynomials with missing degrees

Abstract

In this note we revisit one of the first known examples of exceptional orthogonal polynomials that was introduced by Dubov, Eleonskii, and Kulagin in relation to nonharmonic oscillators with equidistant spectra. We dissect the DEK polynomials using the discrete Darboux transformations and unravel a characterization bypassing the differential equation that defines the DEK polynomials. This characterization also leads to a family of general orthogonal polynomials with missing degrees and this approach manifests its relation to biorthogonal polynomials introduced by Iserles and N{\o}rsett, which are applicable to a whole range of problems in computational and applied analysis. We also obtain a modification of the Christoffel formula for this family since its classical form cannot be applied in this case.