Traces of partition Eisenstein series
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arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| author | Amdeberhan, Tewodros Griffin, Michael Ono, Ken Singh, Ajit |
| author_facet | Amdeberhan, Tewodros Griffin, Michael Ono, Ken Singh, Ajit |
| contents | We study "partition Eisenstein series", extensions of the Eisenstein series $G_{2k}(τ),$ defined by $$λ=(1^{m_1}, 2^{m_2},\dots, k^{m_k}) \vdash k \ \ \ \ \ \longmapsto \ \ \ \ \ G_λ(τ):= G_2(τ)^{m_1} G_4(τ)^{m_2}\cdots G_{2k}(τ)^{m_k}. $$ For functions $ϕ: \mathcal{P}\rightarrow \mathbb{C}$ on partitions, the weight $2k$ "partition Eisenstein trace" is the quasimodular form $$ {\mathrm{Tr}}_k(ϕ;τ):=\sum_{λ\vdash k} ϕ(λ)G_λ(τ). $$ These traces give explicit formulas for some well-known generating functions, such as the $k$th elementary symmetric functions of the inverse points of 2-dimensional complex lattices $\mathbb{Z}\oplus \mathbb{Z}τ,$ as well as the $2k$th power moments of the Andrews-Garvan crank function. To underscore the ubiquity of such traces, we show that their generalizations give the Taylor coefficients of generic Jacobi forms with torsional divisor. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_08807 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Traces of partition Eisenstein series Amdeberhan, Tewodros Griffin, Michael Ono, Ken Singh, Ajit Number Theory We study "partition Eisenstein series", extensions of the Eisenstein series $G_{2k}(τ),$ defined by $$λ=(1^{m_1}, 2^{m_2},\dots, k^{m_k}) \vdash k \ \ \ \ \ \longmapsto \ \ \ \ \ G_λ(τ):= G_2(τ)^{m_1} G_4(τ)^{m_2}\cdots G_{2k}(τ)^{m_k}. $$ For functions $ϕ: \mathcal{P}\rightarrow \mathbb{C}$ on partitions, the weight $2k$ "partition Eisenstein trace" is the quasimodular form $$ {\mathrm{Tr}}_k(ϕ;τ):=\sum_{λ\vdash k} ϕ(λ)G_λ(τ). $$ These traces give explicit formulas for some well-known generating functions, such as the $k$th elementary symmetric functions of the inverse points of 2-dimensional complex lattices $\mathbb{Z}\oplus \mathbb{Z}τ,$ as well as the $2k$th power moments of the Andrews-Garvan crank function. To underscore the ubiquity of such traces, we show that their generalizations give the Taylor coefficients of generic Jacobi forms with torsional divisor. |
| title | Traces of partition Eisenstein series |
| topic | Number Theory |
| url | https://arxiv.org/abs/2408.08807 |