# Sums of squares and products of Bessel functions

## TL;DR

This paper proves a rigorous Voronoi summation formula for the representation of integers as sums of squares, establishing new transformations involving Bessel functions and hypergeometric functions, generalizing classical results.

## Contribution

It provides the first rigorous proof of a Voronoi summation formula for sums of squares and introduces new transformations linking these sums to special functions.

## Key findings

- Established a new summation formula for r_k(n)
- Derived a transformation connecting series of r_k(n) with Bessel and hypergeometric functions
- Generalized classical results of Hardy, Dixon, Ferrar, and Popov.

## Abstract

Let $r_k(n)$ denote the number of representations of the positive integer $n$ as the sum of $k$ squares. We rigorously prove for the first time a Voronoi summation formula for $r_k(n), k\geq2,$ proved incorrectly by A. I. Popov and later rediscovered by A. P. Guinand, but without proof and without conditions on the functions associated in the transformation. Using this summation formula we establish a new transformation between a series consisting of $r_k(n)$ and a product of two Bessel functions, and a series involving $r_k(n)$ and the Gaussian hypergeometric function. This transformation can be considered as a massive generalization of well-known results of G. H. Hardy, and of A. L. Dixon and W. L. Ferrar, as well as of a classical result of A. I. Popov that was completely forgotten. An analytic continuation of this transformation yields further useful results that generalize those obtained earlier by Dixon and Ferrar.