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 $$\lambda =(1^{m_1},2^{m_2},\dots ,n^{m_n})⊢n⟼E_{\lambda }(q):=E_2(q{)}^{m_1}{E}_4(q{)}^{m_2}\cdots E_{2n}(q{)}^{m_n}.$$ For functions $\varphi :\mathcal{P}\mapsto \mathbb{C}$ on partitions, the weight $2n$ partition Eisenstein trace is $${\text{Tr}}_n(\varphi ;q):=\sum _{\lambda ⊢n}\varphi (\lambda )E_{\lambda }(q).$$ For all $t$, we prove that $U_{2t}(q)={\text{Tr}}_t({\varphi }_U;q)$ and $V_{2t}(q)={\text{Tr}}_t({\varphi }_V;q),$ where ${\varphi }_U$ and ${\varphi }_V$ are natural partition weights, giving the first explicit quasimodular formulas for these series.

12 pages, 0 figures. Replaced due to typos correction and the LaTex codes in the Abstract needed fixing. Minor revisions that address referee's comment. Additional but minor typographical errors corrected in the new version