Integration of Associated Legendre Polynomial I am interested in the following integral $$I=\int_{-1}^1P_\ell^2(x)P_n(x)\mathrm{d}x,$$
where $P_n(x)$ is Legendre Polynomial of $n$th order, and $P_\ell^2$ is Associated Legendre Polynomial. Any one has any idea on how to proceed?

This is not an answer but it is too long for a comment.
Considering $$I_{n,l}=\int_{-1}^{+1} P_l^2(x) P_n(x)\,dx$$ this integral seems to show interesting patterns I give you below (this is just based on numerical evaluation and observation).
For positive values of $n,l$

*

*for $l<n \implies I_{n,l}=0$

*for $l=n+(2k-1)\implies I_{n,l}=0$

*for $l=n+2k\implies I_{n,l}=4$

*for $l=n\implies I_{n,l}=-\frac{2n(n-1)}{2n+1}$
