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

Abstract

Closed-form evaluations of certain integrals of $J_{0}(\xi)$, 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 $J_s(\xi)$ in the integrand, encompasses several important integrals in the literature including Sonine's integral. Here we derive an analogous integral identity where $J_{s}(\xi)$ is replaced by a kernel consisting of a combination of $J_{s}(\xi)$, $K_{s}(\xi)$ and $Y_{s}(\xi)$ that is of utmost importance in number theory. Using this identity and the Vorono\"{\dotlessi} summation formula, we derive a general transformation relating infinite series of products of Bessel functions $I_{\lambda}(\xi)$ and $K_{\lambda}(\xi)$ 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.