Trigonometric analogues of the identities associated with twisted sums of divisor functions (Q6579271)

In a previous work, the authors [Adv. Appl. Math. 153, Article ID 102601, 44 p. (2024; Zbl 1541.11069)] established the Cohen type identities for some twisted sums of the divisor functions. In that same paper, they also derived the Voronoï summation formulas by appealing to their Cohen-type identities. The present paper aims to derive an equivalent version of the results offered in the previous article in terms of identities involving finite sums of trigonometric functions and the doubly infinite series. More precisely, the authors establish the identities associated with the \(K\)-Bessel function and the following weighted sums of divisor functions such as \N\[\N\sum_{d|n} d^z \sin (2\pi d \theta),\: \sum_{d|n} d^z \sin \left(\frac{2\pi n \theta}{d}\right),\: \sum_{d|n} d^z \cos (2\pi d \theta),\:\sum_{d|n} d^z \cos \left(\frac{2\pi n \theta}{d}\right).\N\]\NThey prove that these identities are equivalent to their previous results. Moreover, they present formulas for the following two infinite series \N\[\N\sum_{n=1}^\infty r_6(n) n^{\nu/2}K_\nu(a\sqrt{nx}), \: \sum_{n=1}^\infty r_6(n) e^{-4\pi \sqrt{nx}},\N\]\Nwhere \(K_\nu (z)\) denotes the modified Bessel function of order \(\nu\) and \(r_6(n)\) denotes the number of representations of a natural number \(n\) as a sum of six squares.