Plot
Meixner-Pollaczek Polynomials
Definition
The Meixner-Pollaczek polynomials are defined as
\[
\begin{align}
P_n^\lambda(x;\phi) &= \frac{(2 \lambda)_n}{n!} e^{i n \phi} \sum_{k=0}^n \frac{(-n)_k (\lambda + i x)_k}{(2\lambda)_k k!} \left(1-e^{-2i\phi}\right)^k\\
&= \frac{(2 a)_n}{n!} e^{i n \phi} {}_2F_1 \left(\left. {-n, \lambda+i x \atop 2\lambda} \; \right| 1-e^{-2i \phi} \right)
\end{align}
\]
