Exceptional Krall polynomials
Alex Kasman, Robert Milson
TL;DR
The paper addresses the Krall problem for Hermite-type weights by constructing a novel exceptional Krall Hermite polynomial family that are orthogonal with a Hermite-type weight and are eigenfunctions of a fourth-order differential operator. The authors implement a bispectral Darboux transformation of the classical Hermite operator via an intertwiner $\hat{A}$, producing the fourth-order eigenoperator $\hat{T}_4$ and a two-parameter family depending on $b$, with a degree sequence that omits degree $0$ and a five-term recurrence. Key results include an explicit exponential generating function, a 4th-order eigenvalue equation, contour-orthogonality with modified weights, and a 5th-order recurrence, with degeneracies at characteristic values $b^2=2j$ leading to linear dependencies and Jordan blocks. Placed within the bispectral framework, the work shows how classical Hermite relations lift to higher-order operators and recurrences through rank-2 Darboux techniques, enriching both Krall-type and exceptional polynomial theory with a parameter-dependent family and its dual algebras.
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.