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

## TL;DR

This paper derives a new integral identity involving Bessel functions, applies it to number theory, and corrects a Ramanujan identity using advanced summation formulas.

## Contribution

It introduces an analogue of Fock-type integrals with a kernel of Bessel functions, connecting them to hypergeometric functions and correcting a Ramanujan identity.

## Key findings

- Derived a new integral identity involving Bessel functions and their combinations.
- Established a transformation linking series of Bessel functions with hypergeometric functions.
- Corrected a known Ramanujan identity using the new integral and summation formula.

## 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.