On Some Integrals Over the Product of Three Legendre Functions

TL;DR

This paper evaluates several definite integrals involving products of Legendre functions and their variants in closed form, revealing connections to elliptic integrals with arithmetic significance.

Contribution

It provides explicit closed-form evaluations of complex integrals involving Legendre functions, extending known results and linking to elliptic integrals.

Findings

Explicit closed-form formulas for integrals involving Legendre functions.

Connections established between Legendre integrals and elliptic integrals.

Results have implications for arithmetic properties of special functions.

Abstract

The definite integrals $ \int_{-1}^1(1-x^2)^{(\nu-1)/2}[P_\nu(x)]^3\D x$, $ \int_{-1}^1(1-x^2)^{(\nu-1)/2}[P_\nu(x)]^2P_{\nu}(-x)\D x$, $\int_{-1}^1x(1-x^2)^{(\nu-1)/2}[P_{\nu+1}(x)]^3\D x $ and $\int_{-1}^1x(1-x^2)^{(\nu-1)/2}[P_{\nu+1}(x)]^2P_{\nu+1}(-x)\D x $ are evaluated in closed form, where $P_\nu$ is the Legendre function of degree $\nu$, and $ \R\nu>-1$. Special cases of these formulae are related to certain integrals over elliptic integrals that have arithmetic interest.