Exceptional Krall polynomials

TL;DR

This paper introduces a new class of exceptional Hermite-type orthogonal polynomials that are eigenfunctions of a fourth-order differential operator, with a missing degree sequence and higher-order recurrence relations.

Contribution

It presents the first known exceptional Hermite-type polynomials that satisfy a fourth-order differential equation and have a missing degree sequence, expanding the theory of exceptional orthogonal polynomials.

Findings

Polynomials are orthogonal with respect to a Hermite-type weight.

They are eigenfunctions of a fourth-order differential operator.

The polynomial family has a missing degree zero and satisfies a 5th order recurrence relation.

Abstract

In this paper we exhibit and study a novel class of exceptional Krall orthogonal polynomials of Hermite type. This means that the polynomials in question are (i) orthogonal with respect to a Hermite-type weight; (ii) are the eigenfunctions of a higher-order differential operator; (iii) the degree sequence of the polynomial family in question is missing a finite number of degrees. Regarding the second point, unlike the known class of exceptional Hermite polynomials that satisfy a second-order eigenvalue equation, the polynomials we introduce here are not eigenfunctions of any 2nd order differential operator, but are for one of 4th order. Regarding the third point, our family does not include a polynomial of degree zero and consequently satisfies a 5th order recurrence relation instead of the classical 3-term relation.