Analogue of a Fock-type integral arising from electromagnetism and its applications in number theory

Closed-form evaluations of certain integrals of J(ξ) , the Bessel function of the first kind, have been crucial in the studies on the electromagnetic field of alternating current in a circuit with two groundings, as can be seen from the works of Fock and Bursian, Schermann, etc. Koshliakov’s generalization of one such integral, which contains J<inf>s</inf>(ξ) in the integrand, encompasses several important integrals in the literature including Sonine’s integral. Here, we derive an analogous integral identity where J<inf>s</inf>(ξ) is replaced by a kernel consisting of a combination of J<inf>s</inf>(ξ) , K<inf>s</inf>(ξ) and Y<inf>s</inf>(ξ). This kernel is important in number theory because of its role in the Voronoï summation formula for the sum-of-divisors function σ<inf>s</inf>(n). Using this identity and the Voronoï summation formula, we derive a general transformation relating infinite series of products of Bessel functions I<inf>λ</inf>(ξ) and K<inf>λ</inf>(ξ) with those involving the Gaussian hypergeometric function. As applications of this transformation, several important results are derived, including what we believe to be a corrected version of the first identity found on page 336 of Ramanujan’s Lost Notebook.