Two Definite Integrals Involving Products of Four Legendre Functions

TL;DR

This paper derives closed-form expressions for two specific definite integrals involving products of four Legendre functions, which are relevant in advanced number theory and mathematical physics.

Contribution

It provides new closed-form evaluations of integrals involving Legendre functions of complex degree, extending previous special case results in automorphic forms and zeta functions.

Findings

Closed-form formulas for the integrals involving Legendre functions.

Connections to automorphic Green's functions and Epstein zeta functions.

Extension of known integral evaluations to complex degrees.

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.