Legendre Functions, Spherical Rotations, and Multiple Elliptic Integrals

A closed-form formula is derived for the generalized Clebsch-Gordan integral $ \int_{-1}^1 {[}P_ν(x){]}^2P_ν(-x)\D x$, with $P_ν$ being the Legendre function of arbitrary complex degree $ν\in\mathbb C$. The finite Hilbert transform of $P_ν(x)P_ν(-x),-1<x<1$ is evaluated. An analytic proof is provided for a recently conjectured identity $\int_0^1[\mathbf K(\sqrt{1-k^2})]^{3}\D k=6\int_0^1[\mathbf K(k)]^2\mathbf K(\sqrt{1-k^2})k\D k=[Γ(1/4)]^{8}/(128π^2) $ involving complete elliptic integrals of the first kind $\mathbf K(k)$ and Euler's gamma function $Γ(z)$.

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