Eisenstein-type series associated to partition ranks

Abstract

In this paper, we introduce a class of functions that behave like classical Eisenstein series in many ways, but with a key distinction: only their non-holomorphic completions transform like (quasi)modular forms. We show how the partition rank generating function can be expressed in terms of partition traces of these functions. A key feature of our construction is that the completions satisfy a holomorphic anomaly equation - a phenomenon typically seen in the context of quantum field theory and string theory. We also show that the Fourier coefficients of these Eisenstein-type series are integral.