$Q_n\left(\cos \theta \right)=\frac{2^{2n+1}{\left(n!\right)}^2}{\left(2n+1\right)!}{\sum }_{k=0}^{\infty }\frac{1\cdot 3\cdots \left(2k-1\right)}{k!}\frac{\left(n+1\right)\left(n+2\right)\cdots \left(n+k\right)}{\left(2n+3\right)\left(2n+5\right)\cdots \left(2n+2k+1\right)}\cos \left(\left(n+2k+1\right)\theta \right)$