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

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.