A Sextuple Integral Containing the Product of Associated Legendre polynomials $P_v^u(x) P_{\nu }^{\mu }(y)$: Derivation and Evaluation

TL;DR

This paper derives a new six-dimensional integral involving the product of associated Legendre polynomials with different indices, expressed through special functions, and explores its special cases and applications.

Contribution

It introduces a novel derivation of a complex six-dimensional integral involving associated Legendre polynomials with general indices, expressed via Hurwitz-Lerch zeta functions.

Findings

Derived a new six-dimensional integral involving associated Legendre polynomials.

Expressed the integral in terms of Hurwitz-Lerch zeta functions and constants.

Obtained special cases involving fundamental constants and special functions.

Abstract

In this present paper we derive a six dimensional integral containing the product of the Associated Legendre Polynomials $P_v^u(x) P_{\nu }^{\mu }(y)$ where the indices are different and general. Included in the kernel of this integral is the generalized logarithmic function and coefficient logarithmic functions. The derivation of this integral is written in terms of the Hurwitz-Lerch zeta function and constant coefficients raised to a power. Special cases of this integral are derived in terms of fundamental constants and other special functions. All the results in this work are new.