Derivatives of theta functions as Traces of Partition Eisenstein series
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arXiv
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| Format: | Preprint |
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2024
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| author | Amdeberhan, Tewodros Ono, Ken Singh, Ajit |
| author_facet | Amdeberhan, Tewodros Ono, Ken Singh, Ajit |
| contents | 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 $$λ=(1^{m_1}, 2^{m_2},\dots, n^{m_n}) \vdash n \ \ \ \ \ \longmapsto \ \ \ \ \ E_λ(q):= E_2(q)^{m_1} E_4(q)^{m_2}\cdots E_{2n}(q)^{m_n}. $$ For functions $ϕ: \mathcal{P}\mapsto \mathbb{C}$ on partitions, the weight $2n$ partition Eisenstein trace is $$ \text{Tr}_n(ϕ;q):=\sum_{λ\vdash n} ϕ(λ)E_λ(q). $$ For all $t$, we prove that $U_{2t}(q)=\text{Tr}_t(ϕ_U;q)$ and $V_{2t}(q)=\text{Tr}_t(ϕ_V;q),$ where $ϕ_U$ and $ϕ_V$ are natural partition weights, giving the first explicit quasimodular formulas for these series. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2407_08437 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Derivatives of theta functions as Traces of Partition Eisenstein series Amdeberhan, Tewodros Ono, Ken Singh, Ajit Number Theory Combinatorics 11F03, 05A17, 11M36 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 $$λ=(1^{m_1}, 2^{m_2},\dots, n^{m_n}) \vdash n \ \ \ \ \ \longmapsto \ \ \ \ \ E_λ(q):= E_2(q)^{m_1} E_4(q)^{m_2}\cdots E_{2n}(q)^{m_n}. $$ For functions $ϕ: \mathcal{P}\mapsto \mathbb{C}$ on partitions, the weight $2n$ partition Eisenstein trace is $$ \text{Tr}_n(ϕ;q):=\sum_{λ\vdash n} ϕ(λ)E_λ(q). $$ For all $t$, we prove that $U_{2t}(q)=\text{Tr}_t(ϕ_U;q)$ and $V_{2t}(q)=\text{Tr}_t(ϕ_V;q),$ where $ϕ_U$ and $ϕ_V$ are natural partition weights, giving the first explicit quasimodular formulas for these series. |
| title | Derivatives of theta functions as Traces of Partition Eisenstein series |
| topic | Number Theory Combinatorics 11F03, 05A17, 11M36 |
| url | https://arxiv.org/abs/2407.08437 |