Q n ( cos θ ) = 2 2 n + 1 ( n ! ) 2 ( 2 n + 1 ) ! ∑ k = 0 ∞ 1 ⋅ 3 ⋯ ( 2 k − 1 ) k ! ( n + 1 ) ( n + 2 ) ⋯ ( n + k ) ( 2 n + 3 ) ( 2 n + 5 ) ⋯ ( 2 n + 2 k + 1 ) cos ( ( n + 2 k + 1 ) θ ) \mathsf{Q}_{n}\left(\cos\theta\right)=\frac{2^{2n+1}(n!)^{2}}{(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)} \cos\left((n+2k+1)\theta\right)