New examples of Krall-Meixner and Krall-Hahn polynomials, with applications to the construction of exceptional Meixner and Laguerre polynomials

TL;DR

This paper introduces new Krall discrete orthogonal polynomials derived from Meixner and Hahn measures by removing finite mass points, leading to novel exceptional Meixner and Laguerre polynomial families.

Contribution

The paper constructs new Krall orthogonal polynomials from classical measures with finite mass point removals and develops associated exceptional polynomial families.

Findings

New Krall-Meixner polynomials constructed

New Krall-Hahn polynomials constructed

Exceptional Meixner and Laguerre polynomials derived

Abstract

We construct new examples of Krall discrete orthogonal polynomials, i.e., orthogonal polynomials with respect to a measure which are also eigenfunctions of a higher order difference operator. The new examples include the orthogonal polynomials with respect to the measures obtained from the Meixner measure $\rho_{a,c}$ and Hahn measure $\rho_{a,b,N}$ by removing a finite number of their mass points when the parameter $c$ of the Meixner measure and $a$ or $b$ of the Hahn measure are positive integers. From the new Krall-Meixner families we construct new families of exceptional Meixner and Laguerre polynomials.