Quasimodular forms arising from Jacobi's theta function and special symmetric polynomials
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2025
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| author | Amdeberhan, Tewodros Fel, Leonid G. Ono, Ken |
| author_facet | Amdeberhan, Tewodros Fel, Leonid G. Ono, Ken |
| contents | Ramanujan derived a sequence of even weight $2n$ quasimodular forms $U_{2n}(q)$ from derivatives of Jacobi's weight $3/2$ theta function. Using the generating function for this sequence, one can construct sequences of quasimodular forms of all nonnegative integer weights with minimal input: a weight 1 modular form and a power series $F(X)$. Using the weight 1 form $θ(q)^2$ and $F(X)=\exp(X/2)$, we obtain a sequence $\{Y_n(q)\}$ of weight $n$ quasimodular forms on $Γ_0(4)$ whose symmetric function avatars $\widetilde{Y}_n(\pmb{x}^k)$ are the symmetric polynomials $T_n(\pmb{x}^k)$ that arise naturally in the study of syzygies of numerical semigroups. With this information, we settle two conjectures about the $T_n(\pmb{x}^k).$ Finally, we note that these polynomials are systematically given in terms of the Borel-Hirzebruch $\widehat{A}$-genus for spin manifolds, where one identifies power sum symmetric functions $p_i$ with Pontryagin classes. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2507_12352 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Quasimodular forms arising from Jacobi's theta function and special symmetric polynomials Amdeberhan, Tewodros Fel, Leonid G. Ono, Ken Number Theory Algebraic Topology Combinatorics 11F11, 11F37, 11P81, 20M14 Ramanujan derived a sequence of even weight $2n$ quasimodular forms $U_{2n}(q)$ from derivatives of Jacobi's weight $3/2$ theta function. Using the generating function for this sequence, one can construct sequences of quasimodular forms of all nonnegative integer weights with minimal input: a weight 1 modular form and a power series $F(X)$. Using the weight 1 form $θ(q)^2$ and $F(X)=\exp(X/2)$, we obtain a sequence $\{Y_n(q)\}$ of weight $n$ quasimodular forms on $Γ_0(4)$ whose symmetric function avatars $\widetilde{Y}_n(\pmb{x}^k)$ are the symmetric polynomials $T_n(\pmb{x}^k)$ that arise naturally in the study of syzygies of numerical semigroups. With this information, we settle two conjectures about the $T_n(\pmb{x}^k).$ Finally, we note that these polynomials are systematically given in terms of the Borel-Hirzebruch $\widehat{A}$-genus for spin manifolds, where one identifies power sum symmetric functions $p_i$ with Pontryagin classes. |
| title | Quasimodular forms arising from Jacobi's theta function and special symmetric polynomials |
| topic | Number Theory Algebraic Topology Combinatorics 11F11, 11F37, 11P81, 20M14 |
| url | https://arxiv.org/abs/2507.12352 |