Generalizations (in the spirit of Koshliakov) of some formulas from Ramanujan's Lost Notebook

TL;DR

This paper generalizes several of Ramanujan's identities from his Lost Notebook within a modern framework related to Koshliakov's theory, expanding the understanding of divisor functions and eta-function transformations.

Contribution

It introduces new generalizations of Ramanujan's formulas based on recent developments in Koshliakov's theory, bridging historical identities with contemporary mathematical frameworks.

Findings

Generalized formulas for divisor functions

Extended transformation formulas for Dedekind eta-function

Connections between Ramanujan's identities and Koshliakov's theory

Abstract

In his lost notebook, Ramanujan recorded beautiful identities. These include earlier versions of Koshliakov's formula for the divisor function and the transformation formula for the logarithm of Dedekind's $\eta-$function. In this paper we establish some generalizations of these formulas of Ramanujan in a setting that only recently reemerged in the literature and which concerns a beautiful theory due to Koshliakov.