  An exact transformation, which we call a \emph{master identity}, is obtained
for the series $\sum_{n=1}^{\infty}\sigma_{a}(n)e^{-ny}$ for $a\in\mathbb{C}$
and Re$(y)>0$. As corollaries when $a$ is an odd integer, we derive the
well-known transformations of the Eisenstein series on
$\textup{SL}_{2}\left(\mathbb{Z}\right)$, that of the Dedekind eta function as
well as Ramanujan's famous formula for $\zeta(2m+1)$. Corresponding new
transformations when $a$ is a non-zero even integer are also obtained as
special cases of the master identity. These include a novel companion to
Ramanujan's formula for $\zeta(2m+1)$. Although not modular, it is surprising
that such explicit transformations exist. The Wigert-Bellman identity arising
from the $a=0$ case of the master identity is derived too. 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 transformation involving the sums-of-squares function $r_k(n)$.
Some results on functions self-reciprocal in the Watson kernel are also
obtained.
