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

Abstract

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 $\text {Re}(a)>-1$ 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 $q\rightarrow 1^{-}$ 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 $\left. \frac{\partial }{\partial \beta }E_{2N, \beta }(z^{2N})\right| _{\beta =1}$, where $E_{\alpha , \beta }(z)$ is the two-variable Mittag–Leffler function.