Sums of four polygonal numbers: precise formulas
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866918063342878720 |
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| author | Li, Jialin Wang, Haowu |
| author_facet | Li, Jialin Wang, Haowu |
| contents | In this paper we give unified formulas for the numbers of representations of positive integers as sums of four generalized $m$-gonal numbers, and as restricted sums of four squares under a linear condition, respectively. These formulas are given as $\mathbb{Z}$-linear combinations of Hurwitz class numbers. As applications, we prove several Zhi-Wei Sun's conjectures. As by-products, we obtain formulas for expressing the Fourier coefficients of $\vartheta(τ,z)^4$, $η(τ)^{12}$, $η(τ)^4$ and $η(τ)^8η(2τ)^8$ in terms of Hurwitz class numbers, respectively. The proof is based on the theory of Jacobi forms. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_14710 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Sums of four polygonal numbers: precise formulas Li, Jialin Wang, Haowu Number Theory Combinatorics 11E25, 11F50, 11F30, 11D85 In this paper we give unified formulas for the numbers of representations of positive integers as sums of four generalized $m$-gonal numbers, and as restricted sums of four squares under a linear condition, respectively. These formulas are given as $\mathbb{Z}$-linear combinations of Hurwitz class numbers. As applications, we prove several Zhi-Wei Sun's conjectures. As by-products, we obtain formulas for expressing the Fourier coefficients of $\vartheta(τ,z)^4$, $η(τ)^{12}$, $η(τ)^4$ and $η(τ)^8η(2τ)^8$ in terms of Hurwitz class numbers, respectively. The proof is based on the theory of Jacobi forms. |
| title | Sums of four polygonal numbers: precise formulas |
| topic | Number Theory Combinatorics 11E25, 11F50, 11F30, 11D85 |
| url | https://arxiv.org/abs/2405.14710 |