Legendre Functions, Spherical Rotations, and Multiple Elliptic Integrals

Abstract

A closed-form formula is derived for the generalized Clebsch-Gordan integral $ \int_{-1}^1 {[}P_ν(x){]}^2P_ν(-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=[Γ(1/4)]^{8}/(128π^2) $ involving complete elliptic integrals of the first kind $\mathbf{K}(k)$ and Euler's gamma function $\Gamma (z)$.

32 pages, revised according to referees' reports, some proofs simplified, conclusions intact

References in corpus (2)

• Elliptic integral evaluations of Bessel moments
• Moments of elliptic integrals and critical $L$-values

Cited by in corpus (10)

• Kontsevich-Zagier Integrals for Automorphic Green's Functions. I
• $\mathbb{Q}$-linear dependence of certain Bessel moments
• On Some Integrals Over the Product of Three Legendre Functions
• Legendre Functions of Fractional Degree: Transformations and Evaluations
• Kontsevich-Zagier Integrals for Automorphic Green's Functions. II
• Ramanujan Series for Epstein Zeta Functions
• Two Definite Integrals Involving Products of Four Legendre Functions
• Symmetries of certain double integrals related to Hall effect devices
• Ferrers functions of arbitrary degree and order and related functions
• Spherical Couplings and Multiple Elliptic Integrals