Two Definite Integrals Involving Products of Four Legendre Functions

Abstract: The definite integrals $ \int_{-1}^1x[P_\nu(x)]^4\,\mathrm{d} x$ and $ \int_{0}^1x[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.