Weighted Divisor Sums and Bessel Function Series

On page 335 in his lost notebook, Ramanujan records without proof an identity involving a finite trigonometric sum and a doubly infinite series of ordinary Bessel functions. We provide the first published proof of this result. The identity yields as corollaries representations of weighted divisor sums, in particular, the summatory function for r2(n), the number of representations of the positive integer n as a sum of two squares.