\[\frac{P^{(\alpha,\beta)}_{n}\left(\cos\theta\right)}{P^{(\alpha,\beta)}_{n}%
\left(1\right)}=\frac{2\Gamma\left(\alpha+1\right)}{{\pi}^{\frac{1}{2}}\Gamma%
\left(\alpha-\beta\right)\Gamma\left(\beta+\tfrac{1}{2}\right)}\*\int_{0}^{1}%
\int_{0}^{\pi}\left((\cos\tfrac{1}{2}\theta)^{2}-r^{2}(\sin\tfrac{1}{2}\theta)%
^{2}+ir\sin\theta\cos\phi\right)^{n}(1-r^{2})^{\alpha-\beta-1}r^{2\beta+1}(%
\sin\phi)^{2\beta}\,\mathrm{d}\phi\,\mathrm{d}r,\]