Generalized Lambert series

Abstract

Let q = exp(2πiτ) with Im τ > 0, so 0 < |q| < 1. For any positive integer n, define

$$S_n = \sum\limits_{k \ne 0} {\frac{{( - 1)^k q^{(k^2 + k)/2} }}{{(1 - q^k )^n }},}$$

where the sum is over all nonzero integers k. Malcolm Perry needed to know the modular properties of S ₂, which arose in his work in quantum string theory. Ramanujan evaluated S ₂ in terms of Eisenstein series. We prove a general transformation formula that enables us to evaluate each sum S _n in terms of Eisenstein series, and to thus determine the modular properties of S _n . Moreover, our formula yields systematic proofs of related q-series identities of Ramanujan, proofs which are considerably simpler than those in the literature.