On a theorem of A. I. Popov on sums of squares

Let r _k (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 _k (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 ₂(n) from Ramanujan’s lost notebook.