Explicit transformations for generalized Lambert series associated with the divisor function ${\sigma }_a^{(N)}(n)$ and their applications

Let ${\sigma }_a^{(N)}(n)={\sum }_{d^N\mathrm{\mid }n}{d}^a$. An explicit transformation is obtained for the generalized Lambert series ${\sum }_{n=1}^{\mathrm{\infty }}{\sigma }_a^{(N)}(n)e^{-ny}$ for Re$(a)>-1$ using the recently established Voronoï 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 $q\to 1^{-}$ 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 ${\frac{\mathrm{\partial }{E}_{2N,\beta }(z^{2N})}{\mathrm{\partial }\beta }\mid }_{\beta =1}$, where $E_{\alpha ,\beta }(z)$ is a two-variable Mittag-Leffler function.