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

TL;DR

This paper rigorously proves a series transformation involving the number of representations of integers as sums of squares, originally stated by A. I. Popov, and extends it to related identities connected to Ramanujan's work.

Contribution

The paper provides the first rigorous proof of Popov's series transformation involving sums of squares and Bessel functions, and extends it to related identities.

Findings

Proved Popov's series transformation rigorously.

Extended identities to include Ramanujan's sum of squares identity.

Established new connections between sums of squares and Bessel functions.

Abstract

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.