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Racah Polynomials
Definition
The Racah polynomials are defined as
\[
\begin{align}
R_n(\lambda(x);\alpha,\beta,\gamma,\delta) &= \sum_{k=0}^n \frac{(-n)_k (n+\alpha+\beta+1)_k (-x)_k (x+\gamma+\delta+1)_k}{(\alpha+1)_k (\beta+\delta+1)_k (\delta+1)_k k!}\\
&= {}_4F_3 \left(\left. {-n, n+\alpha+\beta+1, -x, x+\gamma+\delta+1\atop \alpha+1, \beta+\delta+1, \gamma+1} \; \right| 1 \right)
\end{align}
\]
