2025/10/20 by Alex Kasman, Kasman, Alex, Robert Milson +1
Mathematics · Physics and Astronomy · #33C45 #33C47 #Advanced Mathematical Identities #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Mathematical functions and polynomials #Quantum Mechanics and Non-Hermitian Physics
paper · pdf · doi:10.48550/arxiv.2510.17571
openalex publication_date 2025/10/20 · openalex created_date 2025/10/22 · openalex updated_date 2026/07/28
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.