An Exact Integral-to-Sum Relation for Products of Bessel Functions

TL;DR

This paper extends a known identity linking sums and integrals of Bessel functions to products of N Bessel functions, providing simpler proofs and broader applicability using classical summation theorems.

Contribution

It generalizes an existing integral-to-sum relation for Bessel functions to products of multiple functions, with simplified proofs and expanded validity.

Findings

Extended the identity to products of N Bessel functions.

Provided simpler proofs using Abel-Plana and Poisson summation.

Broadened the range of validity for the relation.

Abstract

A useful identity relating the infinite sum of two Bessel functions to their infinite integral was discovered in Dominici et al. (2012). Here, we extend this result to products of $N$ Bessel functions, and show it can be straightforwardly proven using the Abel-Plana theorem, or the Poisson summation formula. For $N=2$, the proof is much simpler than that of Dominici et al., and significantly enlarges the range of validity.