An analogue of a formula of Popov

TL;DR

This paper introduces a new summation formula connecting the number of representations of integers as sums of squares with Bessel functions, extending Popov's classical results.

Contribution

It presents a novel summation formula involving $r_k(n)$ and Bessel functions, providing an analogue to Popov's formula.

Findings

Derived a new summation formula involving $r_k(n)$ and Bessel functions

Extended Popov's classical results to a new mathematical context

Provides tools for further research in number theory and special functions

Abstract

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.