Two Definite Integrals Involving Products of Four Legendre Functions

Abstract

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.

12 pages, accepted version. Proof of two integral formulae stated in arXiv:1301.1735v4 and arXiv:1506.00318v2

References in corpus (1)

• Elliptic integral evaluations of Bessel moments

Cited by in corpus (1)

• Some algebraic and arithmetic properties of Feynman diagrams