Extensions of Watson's theorem and the Ramanujan-Guinand formula

TL;DR

This paper generalizes Watson's theorem and the Ramanujan-Guinand formula, providing new proofs and extensions involving modified Bessel functions, with implications for automorphic forms and special functions.

Contribution

It introduces new generalizations of Watson's theorem and the Ramanujan-Guinand formula, expanding their applicability and providing alternative proofs.

Findings

Generalized Watson's theorem using ${}_ u K_z(x,eta)$

Derived a new proof of the Ramanujan-Guinand formula extension

Established analytic continuation for the generalized results

Abstract

Ramanujan provided several results involving the modified Bessel function $K_z(x)$ in his Lost Notebook. One of them is the famous Ramanujan-Guinand formula, equivalent to the functional equation of the non-holomorphic Eiesenstien series on $SL_2(z)$. Recently, this formula was generalized by Dixit, Kesarwani, and Moll. In this article, we first obtain a generalization of a theorem of Watson and, as an application of it, give a new proof of the result of Dixit, Kesarwani, and Moll. Watson's theorem is also generalized in a different direction using ${}_\mu K_z(x,\lambda)$ which is itself a generalization of $K_z(x)$. Analytic continuation of all these results are also given.