Integration of product of Associated Legendre Polynomial I am interested in the following integral $$I=\int_{-1}^{1} P_s^t(x)P_u^v(x)\mathrm{d}x~.$$ Does any one know if a closed form exist for a general $s, t, u, v$, and for $t\neq v$ and $s\neq u$?
 A: I think your answer is in the paper, "The overlap integral of three associated Legendre polynomials" by Shi-Hai Dong and R. Lemus (2002) (https://www.sciencedirect.com/science/article/pii/S0893965902800040). The relevant expression is $I(s,t;u,v)$, given by equations (7-10), and yields
$$
\int_{-1}^1 P^t_s(x) P^v_u(x)dx=(-1)^\delta|v-t|2^{|v-t|-2}\sqrt{\frac{(s+t)!(u+v)!}{(s-t)!(u-v)!}}\times\sum_\alpha (2\alpha+1)\left(1+(-1)^{\alpha+|v-t|}\right)\sqrt{\frac{(\alpha-|v-t|)!}{(\alpha+|v-t|)!}}\frac{\Gamma\left(\frac{\alpha}{2}\right)\Gamma\left(\frac{\alpha+|v-t|+1}{2}\right)}{\left(\frac{\alpha-|v-t|}{2}\right)!\Gamma\left(\frac{\alpha+3}{2}\right)},
$$
with the phase 
$$
\delta=\left\{\begin{matrix} m_1 & \mbox{ if } & m_2\geq m_1, \\ m_2 & \mbox{ if } &  m_2<m_1,\end{matrix}\right.
$$
and this integral will only be non-zero if 
$$
(|s-u|\leq\alpha\leq s+u)\wedge(\alpha\geq|v-t|)\wedge(\alpha + s + u\mbox{ is even}).
$$ 
Note that there is nothing specifying that $t\neq v$ or $s\neq u$, so this formula may simplify a little under those assumption, I haven't checked.
