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