  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.
