  Let $\sigma_a^{(N)}(n)=\sum_{d^{N}|n}d^a$. An explicit transformation is
obtained for the generalized Lambert series
$\sum_{n=1}^{\infty}\sigma_{a}^{(N)}(n)e^{-ny}$ for Re$(a)>-1$ using the
recently established Vorono\"i summation formula for $\sigma_a^{(N)}(n)$, and
is extended to a wider region by analytic continuation. For $N=1$, this Lambert
series plays an important role in string theory scattering amplitudes as can be
seen in the recent work of Dorigoni and Kleinschmidt. These transformations
exhibit several identities - a new generalization of Ramanujan's formula for
$\zeta(2m+1)$, an identity associated with extended higher Herglotz functions,
generalized Dedekind eta-transformation, Wigert's transformation etc., all of
which are derived in this paper, thus leading to their uniform proofs. A
special case of one of these explicit transformations naturally leads us to
consider generalized power partitions with ``$n^{2N-1}$ copies of $n^{N}$''.
Asymptotic expansion of their generating function as $q\to1^{-}$ is also
derived which generalizes Wright's result on the plane partition generating
function. In order to obtain these transformations, several new intermediate
results are required, for example, a new reduction formula for Meijer
$G$-function and an almost closed-form evaluation of $\left.\frac{\partial
E_{2N, \beta}(z^{2N})}{\partial\beta}\right|_{\beta=1}$, where $E_{\alpha,
\beta}(z)$ is a two-variable Mittag-Leffler function.
