\[\mathsf{P}_{n}\left(\cos\theta\right)=\frac{2^{2n+2}(n!)^{2}}{\pi(2n+1)!}\*%
\sum_{k=0}^{\infty}\frac{1\cdot 3\cdots(2k-1)}{k!}\*\frac{(n+1)(n+2)\cdots(n+k%
)}{(2n+3)(2n+5)\cdots(2n+2k+1)}\*\sin\left((n+2k+1)\theta\right),\]