Proof of a conjecture of Garvan and Jennings-Shaffer on the nonnegativity of M_{C1}(m,n) and M_{C5}(m,n)
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908540913844224 |
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| author | He, Bing Liu, Shuming |
| author_facet | He, Bing Liu, Shuming |
| contents | In their 2016 paper on exotic Bailey--Slater SPT-functions, Garvan and Jennings-Shaffer introduced many new spt-crank-type functions and proposed a conjecture that the spt-crank-type functions $M_{C1}(m,n)$ and $M_{C5}(m,n)$ are both nonnegative for all $m\in\mathbb{Z}$ and $n\in\mathbb{N}.$ Applying Wright\textquoteright s circle method, Jang and Kim showed that $M_{C1}(m,n)$ and $M_{C5}(m,n)$ are both positive for a fixed integer $m$ and large enough integers $n.$ Up to now, no complete proof of this conjecture has been given. In this paper, we provide a complete proof for this conjecture by using the theory of lattice points. Our proof is quite different from that of Jang and Kim. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_12561 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Proof of a conjecture of Garvan and Jennings-Shaffer on the nonnegativity of M_{C1}(m,n) and M_{C5}(m,n) He, Bing Liu, Shuming Number Theory Combinatorics In their 2016 paper on exotic Bailey--Slater SPT-functions, Garvan and Jennings-Shaffer introduced many new spt-crank-type functions and proposed a conjecture that the spt-crank-type functions $M_{C1}(m,n)$ and $M_{C5}(m,n)$ are both nonnegative for all $m\in\mathbb{Z}$ and $n\in\mathbb{N}.$ Applying Wright\textquoteright s circle method, Jang and Kim showed that $M_{C1}(m,n)$ and $M_{C5}(m,n)$ are both positive for a fixed integer $m$ and large enough integers $n.$ Up to now, no complete proof of this conjecture has been given. In this paper, we provide a complete proof for this conjecture by using the theory of lattice points. Our proof is quite different from that of Jang and Kim. |
| title | Proof of a conjecture of Garvan and Jennings-Shaffer on the nonnegativity of M_{C1}(m,n) and M_{C5}(m,n) |
| topic | Number Theory Combinatorics |
| url | https://arxiv.org/abs/2509.12561 |