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

Abstract

Closed-form evaluations of certain integrals of J0(?), 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 Js(?) in the integrand, encompasses several important integrals in the literature including Sonine�s integral. Here, we derive an analogous integral identity where Js(?) is replaced by a kernel consisting of a combination of Js(?), Ks(?) and Ys(?). This kernel is important in number theory because of its role in the Vorono� summation formula for the sum-of-divisors function ?s(n). Using this identity and the Vorono� summation formula, we derive a general transformation relating infinite series of products of Bessel functions I?(?) and K?(?) 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.