Explicit transformations for generalized Lambert series associated with the divisor function σ(N)a(n) and their applications

Abstract

Let σ(N)a(n)=∑dN|nda. An explicit transformation is obtained for the generalized Lambert series ∑∞n=1σ(N)a(n)e-ny for Re(a)>-1 using the recently established Voronoï summation formula for σ(N)a(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 ζ(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 ''n2N-1 copies of nN''. Asymptotic expansion of their generating function as q→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 ∂E2N,β(z2N)∂β∣∣β=1, where Eα,β(z) is a two-variable Mittag-Leffler function.