New Clebsch-Gordan-type integrals involving threefold products of complete elliptic integrals

TL;DR

This paper derives new closed-form evaluations for complex elliptic integrals of Clebsch-Gordan type involving threefold products of complete elliptic integrals, expanding the mathematical tools available for physics and special functions.

Contribution

It introduces novel closed-form evaluations for CG-type integrals with threefold products of elliptic integrals using fractional derivative operators and semi-integration by parts techniques.

Findings

Proved new closed-form evaluations for threefold products of elliptic integrals.

Extended the class of known Clebsch-Gordan-type integrals.

Applied fractional derivatives to evaluate complex elliptic integrals.

Abstract

Multiple elliptic integrals related to the generalized Clebsch-Gordan (CG) integral are of importance in many areas in physics and special functions theory. Zhou has introduced and applied Legendre function-based techniques to prove symbolic evaluations for integrals of CG form involving twofold and threefold products of complete elliptic integral expressions, and this includes Zhou's remarkable proof of an open problem due to Wan. The foregoing considerations motivate the results introduced in this article, in which we prove closed-form evaluations for new CG-type integrals that involve threefold products of the complete elliptic integrals $K$ and $E$. Our methods are based on the use of fractional derivative operators, via a variant of a technique we had previously referred to as semi-integration by parts.