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Continuous Hahn Polynomials
Definition
The Continuous Hahn polynomials are defined as
\[
\begin{align}
p_n(x;a,b,c,d) &= i^n\frac{(a+c)_n (a+d)_n}{n!} \sum_{k=0}^n \frac{(-n)_k (n+a+b+c+d-1)_k (a+i x)_k}{(a+c)_k (a+d)_k k!}\\
&= i^n\frac{(a+c)_n (a+d)_n}{n!} {}_3F_2 \left(\left. {-n, n+a+b+c+d-1, a+i x \atop a+c, a+d} \; \right| 1 \right)
\end{align}
\]
