A way to treat dual Hahn polynomials as Racah polynomials via the theory of Leonard pairs

TL;DR

This paper demonstrates that under certain parameter conditions, dual Hahn polynomials can be viewed as Racah polynomials through the lens of Leonard pairs, revealing a new connection between these families of orthogonal polynomials.

Contribution

It establishes a novel relationship showing dual Hahn polynomials are also Racah polynomials via Leonard pairs when specific parameters are satisfied.

Findings

Dual Hahn polynomials can be interpreted as Racah polynomials under certain conditions.

A new Leonard pair involving a shifted and squared operator is constructed.

The polynomials share the same orthogonality with respect to the same inner product.

Abstract

The dual Hahn polynomials $\{u_i(x)\}_{i=0}^d$ are a family of discrete orthogonal polynomials involving two real parameters $r$ and $s$. Let $L,L^*$ denote the corresponding Leonard pair. Assume that $r\not=0$ and $r+s=0$. We show that $L,(L^*+\frac{r-d}{2})^{2}$ is a Leonard pair. According to the theory of Leonard pairs, the polynomials $\{u_i(x)\}_{i=0}^d$ are not only the dual Hahn polynomials but also the Racah polynomials with respect to the same inner product.