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

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 $L,L^{\ast }$ denote the corresponding Leonard pair. Assume that $r≠0$ and $r+s=0$. We show that $L,(L^{\ast }+\frac{r-d}{2}{)}^2$ 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.