Explicit transformations of certain Lambert series

TL;DR

This paper introduces a new exact transformation called the master identity for Lambert series involving divisor functions, deriving various modular transformations, identities, and generalizations, including a novel Bessel function extension and a sum-of-squares transformation.

Contribution

The paper presents the first explicit master identity for Lambert series, deriving new modular transformations, identities related to zeta values, and a novel Bessel function generalization.

Findings

Derived the master identity for Lambert series with divisor functions.

Obtained new modular transformations for even and odd integer parameters.

Proved a new two-variable generalization of the modified Bessel function.

Abstract

An exact transformation, which we call the \emph{master identity}, is obtained for the first time for the series $\sum_{n=1}^{\infty}\sigma_{a}(n)e^{-ny}$ for $a\in\mathbb{C}$ and Re$(y)>0$. New modular-type transformations when $a$ is a non-zero even integer are obtained as its special cases. The precise obstruction to modularity is explicitly seen in these transformations. These include a novel companion to Ramanujan's famous formula for $\zeta(2m+1)$. The Wigert-Bellman identity arising from the $a=0$ case of the master identity is derived too. When $a$ is an odd integer, the well-known modular transformations of the Eisenstein series on $\textup{SL}_{2}\left(\mathbb{Z}\right)$, that of the Dedekind eta function as well as Ramanujan's formula for $\zeta(2m+1)$ are derived from the master identity. The latter identity itself is derived using Guinand's version of the Vorono\"{\dotlessi} summation formula and an integral evaluation of N.~S.~Koshliakov involving a generalization of the modified Bessel function $K_{\nu}(z)$. Koshliakov's integral evaluation is proved for the first time. It is then generalized using a well-known kernel of Watson to obtain an interesting two-variable generalization of the modified Bessel function. This generalization allows us to obtain a new modular-type transformation involving the sums-of-squares function $r_k(n)$. Some results on functions self-reciprocal in the Watson kernel are also obtained.