On difference operators for symmetric Krall-Hahn polynomials

TL;DR

This paper explores how to reduce the order of difference operators for symmetric Krall-Hahn polynomials by imposing symmetries on parameters and associated sets, advancing the understanding of their algebraic structure.

Contribution

It demonstrates that symmetry assumptions on parameters and sets lead to lower-order difference operators for Krall-Hahn polynomials.

Findings

Order reduction of difference operators achieved through symmetry assumptions

Explicit construction of symmetric Krall-Hahn polynomials

Enhanced understanding of algebraic properties of these polynomials

Abstract

The problem of finding measures whose orthogonal polynomials are also eigenfunctions of higher-order difference operators have been recently solved by multiplying the classical discrete measures by suitable polynomials. This problem was raised by Richard Askey in 1991. The case of the Hahn family is specially rich due to the number of parameters involved. These suitable polynomials are in the Hahn case generated from a quartet $F_1,F_2,F_3$ and $F_4$ of finite sets of positive integers. In this paper, we show that the order of the corresponding difference operators can be reduced by assuming certain symmetries on both, the parameters of the Krall-Hahn family and the associated quartet $F_1,F_2,F_3$ and $F_4$.