Legendre Functions, Spherical Rotations, and Multiple Elliptic Integrals

A closed-form formula is derived for the generalized Clebsch-Gordan integral ${\int }_{-1}^1[P_{\nu }(x){]}^2{P}_{\nu }(-x)\text{D}x$, with $P_{\nu }$ being the Legendre function of arbitrary complex degree $\nu \in \mathbb{C}$. The finite Hilbert transform of $P_{\nu }(x)P_{\nu }(-x),-1<x<1$ 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=[\Gamma(1/4)]^{8}/(128\pi^2) $ involving complete elliptic integrals of the first kind $\mathbf{K}(k)$ and Euler’s gamma function $\mathrm{\Gamma }(z)$.