Dixit, AtulAtulDixitBerndt, Bruce C.Bruce C.BerndtKim, SunSunKimZaharescu, AlexandruAlexandruZaharescu2025-08-282025-08-282016-10-01http://arxiv.org/abs/1610.05840https://d8.irins.org/handle/IITG2025/20028Let rk(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 rk(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 r2(n) from Ramanujan�s lost notebook.en-USe-Printe-Print123456789/555