Berndt, Bruce C.Bruce C.BerndtDixit, AtulAtulDixitKim, SunSunKimZaharescu, AlexandruAlexandruZaharescu2025-08-302025-08-302017-01-0110.1090/proc/135472-s2.0-85021435530http://repository.iitgn.ac.in/handle/IITG2025/22592Let r<inf>k</inf> (n) denote the number of representations of the positive integer n as the sum of k squares. In 1934, the Russian mathematician A. I. Popov stated, but did not rigorously prove, a beautiful series transformation involving r<inf>k</inf> (n) and certain Bessel functions. We provide a proof of this identity for the first time, as well as for another identity, which can be regarded as both an analogue of Popov’s identity and an identity involving r<inf>2</inf>(n) from Ramanujan’s lost notebook.trueBessel functions | Dirichlet characters | Dirichlet series | Ramanujan’s lost notebook | Sums of squares | Voronoï summation formulaArticlehttps://www.ams.org/proc/2017-145-09/S0002-9939-2017-13547-3/S0002-9939-2017-13547-3.pdf108868263795-3808201755WOS:000404113200014