Sums of squares and products of Bessel functions

Abstract

Let rk(n) denote the number of representations of the positive integer n as the sum of k squares. We rigorously prove for the first time a Voronoï summation formula for rk(n),k≥2, proved incorrectly by A.I. Popov and later rediscovered by A.P. Guinand, but without proof and without conditions on the functions associated in the transformation. Using this summation formula we establish a new transformation between a series consisting of rk(n) and a product of two Bessel functions, and a series involving rk(n) and the Gaussian hypergeometric function. This transformation can be considered as a massive generalization of well-known results of G.H. Hardy, and of A.L. Dixon and W.L. Ferrar, as well as of a classical result of A.I. Popov that was completely forgotten. An analytic continuation of this transformation yields further useful results that generalize those obtained earlier by Dixon and Ferrar.