\[P^{(\lambda)}_{n}\left(\cos\theta;a,b\right)=\frac{{\left(\lambda-\mathrm{i}%
\tau_{a,b}(\theta)\right)_{n}}}{n!}\,{\mathrm{e}}^{\mathrm{i}n\theta}\*F\left(%
{-n,\lambda+\mathrm{i}\tau_{a,b}(\theta)\atop-n-\lambda+1+\mathrm{i}\tau_{a,b}%
(\theta)};{\mathrm{e}}^{-2\mathrm{i}\theta}\right)=\sum_{\ell=0}^{n}\frac{{%
\left(\lambda+\mathrm{i}\tau_{a,b}(\theta)\right)_{\ell}}}{\ell!}\,\frac{{%
\left(\lambda-\mathrm{i}\tau_{a,b}(\theta)\right)_{n-\ell}}}{(n-\ell)!}\,{%
\mathrm{e}}^{\mathrm{i}(n-2\ell)\theta},\]