An exact integral-to-sum relation for products of Bessel functions

Abstract

A useful identity relating the infinite sum of two Bessel functions to their infinite integral was discovered in Dominici et al. (Dominici et al. 2012 Proc. R. Soc. A 468 , 2667–2681). 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.