Explicit transformations of certain Lambert series

Abstract

An exact transformation, which we call the \emph{master identity}, is obtained for the first time for the series ${\sum }_{n=1}^{\infty }\ddot{I}{ƒ}_a(n)e^{-ny}$ 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 $\hat{I}\P (2m+1)$. 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 ${\text{SL}}_2\left(\mathbb{Z}\right)$, that of the Dedekind eta function as well as Ramanujan's formula for $\hat{I}\P (2m+1)$ 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 $K_{\hat{I}}½(z)$. 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.

The earlier title of the paper is modified to the current one. This paper has now been accepted for publication in 'Research in the Mathematical Sciences'