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

Abstract

© 2023 World Scientific Publishing Company.Ramanujan provided several results involving the modified Bessel function Kz(x) 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 SL2(ℤ). 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 muλ) which is itself a generalization of Kz(x). Analytic continuation of all these results are also given.