Q_{n} (\cos θ) = \frac{2^{2 n + 1} {(n!)}^{2}}{(2 n + 1)!} \sum_{k = 0}^{\infty} \frac{1 \cdot 3 \dots (2 k - 1)}{k!} \frac{(n + 1) (n + 2) \dots (n + k)}{(2 n + 3) (2 n + 5) \dots (2 n + 2 k + 1)} \cos ((n + 2 k + 1) θ)