\[P^{(\lambda)}_{n}\left(x;\phi\right)=\frac{{\left(2\lambda\right)_{n}}}{n!}{%
\mathrm{e}}^{n\mathrm{i}\phi}F\left({-n,\lambda+\mathrm{i}x\atop 2\lambda};1-{%
\mathrm{e}}^{-2\mathrm{i}\phi}\right).\]