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