\[\left(\sin\tfrac{1}{2}\theta\right)^{\alpha+\frac{1}{2}}\left(\cos\tfrac{1}{2}%
\theta\right)^{\beta+\frac{1}{2}}P^{(\alpha,\beta)}_{n}\left(\cos\theta\right)%
={\pi}^{-1}2^{2n+\alpha+\beta+1}\mathrm{B}\left(n+\alpha+1,n+\beta+1\right)\*%
\left(\sum_{m=0}^{M-1}\frac{f_{m}(\theta)}{2^{m}{\left(2n+\alpha+\beta+2\right%
)_{m}}}+O\left(n^{-M}\right)\right),\]