The expression of \(P_n(\cos2\theta)\) in terms of \(P_n(\cos\theta)\). (Q1512656)

Ableitung der Formel: \[ \begin{multlined} P_n(\cos2\theta) = \sum_0^n (-1)^r 2^{n-r} \frac{4n-4r+1}{4n-2r+1} \frac{A(r)A(n-r)}{A(2n-r)} P_{2n-2r}(\cos\theta)\\ = (-1)^n\sum_0^n (-1)^p2^p \frac{4n+1}{2n+2p+1} \frac{A(n-p)A(p)}{A(n+p)} P_{2p}(\cos\theta),\end{multlined} \] wo \(A(m) = 1.3.5\dots(2m-1)/m!\) ist.