Berndt, Bruce C.Bruce C.BerndtDixit, AtulAtulDixitKim, SunSunKimZaharescu, AlexandruAlexandruZaharescu2025-08-302025-08-302018-11-0710.1016/j.aim.2018.09.0012-s2.0-85053047863http://repository.iitgn.ac.in/handle/IITG2025/22711Let r<inf>k</inf>(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 r<inf>k</inf>(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 r<inf>k</inf>(n) and a product of two Bessel functions, and a series involving r<inf>k</inf>(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.trueAnalytic continuation | Bessel functions | Sums of squares | Voronoï summation formulaArticle10902082305-3387 November 20188arJournal7WOS:000447961200008