Extensions of Watson’s theorem and the Ramanujan–Guinand formula

Abstract

Ramanujan provided several results involving the modified Bessel function [Formula: see text] in his Lost Notebook. One of them is the famous Ramanujan–Guinand formula, equivalent to the functional equation of the non-holomorphic Eisenstein series on SL[Formula: see text]. Recently, this formula was generalized by Dixit, Kesarwani, and Moll. In this paper, 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 [Formula: see text] which is itself a generalization of [Formula: see text]. Analytic continuation of all these results are also given.