Mock modular forms from the k-rank moments
Kilian Rausch
TL;DR
This work develops a comprehensive framework linking higher k-rank moments to mock modular objects by constructing a family of mock Eisenstein series whose moments are captured by partition traces. It defines holomorphic and completed objects $f_{k,j}$ and $\hat{f}_{k,j}$, proves modularity and a finite-derivative algebra structure under $D$, and expresses k-rank moments as traces through Pólya cycle indices and Appell-type PDEs. A key innovation is the quasi-completion analysis and the divisor-like sum formulas $g_{a,b,\ell}$ that uniquely determine the holomorphic parts, yielding integrality of Fourier coefficients and answering a related question about quasi-completions for odd parameters. The results extend the mock-modular framework to higher $k$-ranks, providing explicit recurrence relations and structural results that culminate in explicit formulas for $f_{k,j}$ in terms of divisor-like sums, with potential applications to partition statistics and modular-type partitions.
Abstract
In this paper, the generating functions of Garvans so-called $k$-ranks are used, to define a family of mock Eisenstein series. The $k$-rank moments are then expressed as partition traces of these functions. We explore the modular properties of this new family, give recursive formulas for them involving divisor like sums, and prove that their Fourier coefficient are integral. Furthermore, we show that these functions lie in an algebra that is generated only by derivatives up to a finite order but is nevertheless closed under differentiation. In the process, we also answer a question raised by Bringmann, Pandey and van Ittersum by showing that the divisor like sum $$\left(1-2^{\ell-1} \right) \frac{B_\ell}{2\ell}+ \sum_{2n-1 \geq bm \geq b} (2n-bm)^{\ell-1} q^{mn} - \sum_{m-1 \geq 2bn \geq 2b} (m-2bn)^{\ell-1} q^{mn},$$ has a quasi-completion, when $b\geq 3$ is odd.