Legendre Functions, Spherical Rotations, and Multiple Elliptic Integrals

TL;DR

This paper derives new closed-form formulas involving Legendre functions, spherical rotations, and elliptic integrals, including proofs of conjectured identities and evaluations of Hilbert transforms, advancing mathematical understanding in special functions.

Contribution

It provides a closed-form expression for a generalized Clebsch-Gordan integral and proves a conjectured elliptic integral identity using analytic methods.

Findings

Closed-form formula for the generalized Clebsch-Gordan integral.

Evaluation of the finite Hilbert transform of Legendre functions.

Analytic proof of a conjectured elliptic integral identity.

Abstract

A closed-form formula is derived for the generalized Clebsch-Gordan integral $ \int_{-1}^1 {[}P_{\nu}(x){]}^2P_{\nu}(-x)\D x$, with $ P_\nu$ being the Legendre function of arbitrary complex degree $ \nu\in\mathbb C$. The finite Hilbert transform of $ P_{\nu}(x)P_{\nu}(-x),-1<x<1$ is evaluated. An analytic proof is provided for a recently conjectured identity $\int_0^1[\mathbf K(\sqrt{1-k^2})]^{3}\D k=6\int_0^1[\mathbf K(k)]^2\mathbf K(\sqrt{1-k^2})k\D k=[\Gamma(1/4)]^{8}/(128\pi^2) $ involving complete elliptic integrals of the first kind $ \mathbf K(k)$ and Euler's gamma function $ \Gamma(z)$.