  Motivated by two identities published with Ramanujan's lost notebook and
connected, respectively, with the Gauss circle problem and the Dirichlet
divisor problem, in an earlier paper, three of the present authors derived
representations for certain sums of products of trigonometric functions as
double series of Bessel functions. These series are generalized in the present
paper by introducing the novel notion of balanced derivatives, leading to
further theorems. As we will see below, the regions of convergence in the
unbalanced case are entirely different than those in the balanced case. From
this viewpoint, it is remarkable that Ramanujan had the intuition to formulate
entries that are, in our new terminology, "balanced". If $x$ denotes the number
of products of the trigonometric functions appearing in our sums, in addition
to proving the identities mentioned above, theorems and conjectures for upper
and lower bounds for the sums as $x\to\infty$ are established.
