Hierarchies of sum rules for squares of spherical Bessel functions

TL;DR

This paper develops hierarchical sum rules for squared spherical Bessel functions using recurrence relations, enabling efficient numerical evaluation of complex sums with multiple terms.

Contribution

It introduces three independent hierarchies of sum rules derived from a four-term recurrence relation, advancing the computational methods for spherical Bessel functions.

Findings

Closed-form expressions for finite weighted sums

Three independent hierarchies of sum rules

Enhanced numerical evaluation efficiency

Abstract

A four-term recurrence relation for squared spherical Bessel functions is shown to yield closed-form expressions for several types of finite weighted sums of these functions. The resulting sum rules, which may contain an arbitrarily large number of terms, are found to constitute three independent hierarchies. Their use leads to an efficient numerical evaluation of these sums.