Derivatives of theta functions as Traces of Partition Eisenstein series

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.