Balanced derivatives, identities, and bounds for trigonometric and Bessel series

Motivated by two identities published with Ramanujan's lost notebook and connected, respectively, the Gauss circle problem Dirichlet divisor problem, in an earlier paper, three of present authors derived representations for certain sums products trigonometric functions as double series Bessel [8] . These are generalized paper introducing novel notion balanced derivatives, leading to further theorems. As we will see below, regions convergence unbalanced case entirely different than those case. From this viewpoint, it is remarkable that Ramanujan had intuition formulate entries are, our new terminology, “balanced”. If x denotes number appearing sums, addition proving mentioned above, theorems conjectures upper lower bounds → ∞ established.