A Generalisation For The Infinite Integral Over Three Spherical Bessel Functions

TL;DR

This paper derives a new, generalized formula for the infinite integral over three spherical Bessel functions, involving a finite sum over associated Legendre functions that accommodates broader parameter ranges.

Contribution

It introduces a generalized analytical expression for the integral over three spherical Bessel functions, extending previous results to include larger values of the associated Legendre function order.

Findings

New formula involving finite sum over associated Legendre functions

Extension to cases where |m| > l for the Legendre functions

Generalization to rational m values for specific l

Abstract

A new formula is derived that generalises an earlier result for the infinite integral over three spherical Bessel functions. The analytical result involves a finite sum over associated Legendre functions, $P_l^m(x)$, of degree $l$ and order $m$. The sum allows for values of $|m|$ that are greater than $l$. A generalisation for the associated Legendre functions to allow for any rational $m$ for a specific $l$ is also shown