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

Abstract

Closed-form evaluations of certain integrals of $J_{0}(ξ)$, 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 $J_s(ξ)$ in the integrand, encompasses several important integrals in the literature including Sonine's integral. Here we derive an analogous integral identity where $J_{s}(ξ)$ is replaced by a kernel consisting of a combination of $J_{s}(ξ)$, $K_{s}(ξ)$ and $Y_{s}(ξ)$ 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 $I_λ(ξ)$ and $K_λ(ξ)$ 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.

30 pages, submitted for publication