An analogue of a formula of Popov

Let $r_{k}(n)$ denote the number of representations of the positive integer $n$ as the sum of $k$ squares. We prove a new summation formula involving $r_{k}(n)$ and the Bessel functions of the first kind, which constitutes an analogue of a result due to the Russian mathematician A. I. Popov.
Comment: V2: We have added a final section containing a proof of a generalization of the Ramanujan-Guinand formula; Reference [13] of the previous version corrected