Derivatives of theta functions as Traces of Partition Eisenstein series

TL;DR

This paper proves that Ramanujan's derivatives of theta functions can be explicitly expressed as traces of partition Eisenstein series, establishing their quasimodularity and providing new formulas for these series.

Contribution

It provides the first explicit proof that derivatives of theta functions are quasimodular by expressing them as traces of partition Eisenstein series.

Findings

Expressed derivatives of theta functions as traces of partition Eisenstein series.

Established the quasimodularity of these derivatives.

Derived explicit formulas for the series in terms of partition weights.

Abstract

In his "lost notebook'', Ramanujan used iterated derivatives of two theta functions to define sequences of $q$-series $\{U_{2t}(q)\}$ and $\{V_{2t}(q)\}$ that he claimed to be quasimodular. We give the first explicit proof of this claim by expressing them in terms of "partition Eisenstein series'', extensions of the classical Eisenstein series $E_{2k}(q)$ defined by $$\lambda=(1^{m_1}, 2^{m_2},\dots, n^{m_n}) \vdash n \ \ \ \ \ \longmapsto \ \ \ \ \ E_{\lambda}(q):= E_2(q)^{m_1} E_4(q)^{m_2}\cdots E_{2n}(q)^{m_n}. $$ For functions $\phi : \mathcal{P}\mapsto \mathbb{C}$ on partitions, the weight $2n$ partition Eisenstein trace is $$ \text{Tr}_n(\phi;q):=\sum_{\lambda \vdash n} \phi(\lambda)E_{\lambda}(q). $$ For all $t$, we prove that $U_{2t}(q)=\text{Tr}_t(\phi_U;q)$ and $V_{2t}(q)=\text{Tr}_t(\phi_V;q),$ where $\phi_U$ and $\phi_V$ are natural partition weights, giving the first explicit quasimodular formulas for these series.