  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.
