Koshliakov's formula and Guinand's formula in Ramanujan's lost notebook

Abstract

On two pages in his lost notebook, Ramanujan recorded several theorems involving the modified Bessel function Kν (z). These include Koshliakov's formula and Guinand's formula, both connected with the functional equation of nonanalytic Eisenstein series, and both discovered by these authors several years after Ramanujan's death. Other formulas, including one by K. Soni and two particularly elegant new results, are also stated without proof by Ramanujan. In this paper, we prove all the formulas claimed by Ramanujan on these two pages and conclude with a survey of related results. © Springer Science+Business Media, LLC 2008.