Two Definite Integrals Involving Products of Four Legendre Functions

The definite integrals ${\int }_{-1}^{1}x[P_{\nu }(x){]}^4\mathrm{d}x$ and ${\int }_0^{1}x[P_{\nu }(x){]}^2\{[P_{\nu }(x){]}^2-[P_{\nu }(-x){]}^2\}\mathrm{d}x$ are evaluated in closed form, where $P_{\nu }$ stands for the Legendre function of degree $\nu \in \mathbb{C}$. Special cases of these integral formulae have appeared in arithmetic studies of automorphic Green’s functions and Epstein zeta functions.