Unfortunately I have problems to solve the following integral
$\int_{-1}^{1} x^{2} P_{n}(x)P_{m}(x)dx = ?$,
where $P_{n}(x)$, $P_{m}(x)$ denote the associated Legendre Polynomials.
I only know that the result has to be somehow proportional to $\delta_{n,m}$ and that the following relation holds (Bonnet’s recursion formula)
$x^2P_{n}(x)=\frac{(n+1)xP_{n+1}(x)+nxP_{n-1}(x)}{2n+1}$ .
I appreciate any of your ideas.
Ok this was not very clever. Here is the answer:
$\int_{-1}^{1}x^2P_{n}P_{m}(x)dx ~ ~ = \frac{(n+1)*(m+1)}{(2n+1)*(2m+1)}*\frac{2}{2*(n+1)+1}\delta_{n+1,m+1}+\frac{(n+1)*m}{(2n+1)*(2m+1)}*\frac{2}{2(n+1)+1}\delta_{n+1,m-1}+\frac{n*(m+1)}{(2n+1)*(2m+1)}*\frac{2}{2*(n-1)+1}\delta_{n-1,m+1}+\frac{n*m}{(2n+1)*(2m+1)}*\frac{2}{2(n-1)+1}\delta_{n-1,m-1}$