\[(\sin\tfrac{1}{2}\theta)^{\alpha+\frac{1}{2}}(\cos\tfrac{1}{2}\theta)^{\beta+%
\frac{1}{2}}P^{(\alpha,\beta)}_{n}\left(\cos\theta\right)=\frac{\Gamma\left(n+%
\alpha+1\right)}{2^{\frac{1}{2}}\rho^{\alpha}n!}\*\left(\theta^{\frac{1}{2}}J_%
{\alpha}\left(\rho\theta\right)\sum_{m=0}^{M}\dfrac{A_{m}(\theta)}{\rho^{2m}}+%
\theta^{\frac{3}{2}}J_{\alpha+1}\left(\rho\theta\right)\sum_{m=0}^{M-1}\dfrac{%
B_{m}(\theta)}{\rho^{2m+1}}+\varepsilon_{M}(\rho,\theta)\right),\]