Weighted divisor sums and Bessel function series, IV

One fragment (p. 335) published with Ramanujan's Lost Notebook contains two formulas, each involving a finite trigonometric sum and a doubly infinite series of Bessel functions. The identities are connected with the classical circle and divisor problems, respectively. This paper is devoted to the first identity. First, we obtain a generalization in the setting of Riesz sums. Second, we prove a trigonometric analogue.