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.
Categories: math.NT math.CA