Asymptotics of the shifted finite differences of the overpatition function and a problem of Wang--Xie--Zhang
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| Format: | Preprint |
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2025
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| _version_ | 1866908737804959744 |
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| author | Mukherjee, Gargi |
| author_facet | Mukherjee, Gargi |
| contents | Let $\overline{p}(n)$ denote the overpartition function, and for $j\in \mathbb{N}$, $Δ^r_j$ denote the $r$-fold applications of the shifted difference operator $Δ_j$ defined by $Δ_j(a)(n):=a(n)-a(n-j)$. The main goal of this paper is to derive an asymptotic expansion of $Δ^r_j(\overline{p})(n)$ with an effective error bound which subsequently gives an answer to a problem of Wang, Xie, and Zhang. In order to get the asymptotics of $Δ^r_j(\overline{p})(n)$, we derive an asymptotic expansion of the shifted overpartition function $\overline{p}(n+k)$ for any integer $k\neq 0$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_23679 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Asymptotics of the shifted finite differences of the overpatition function and a problem of Wang--Xie--Zhang Mukherjee, Gargi Number Theory 05A16, 05A20, 11P82 Let $\overline{p}(n)$ denote the overpartition function, and for $j\in \mathbb{N}$, $Δ^r_j$ denote the $r$-fold applications of the shifted difference operator $Δ_j$ defined by $Δ_j(a)(n):=a(n)-a(n-j)$. The main goal of this paper is to derive an asymptotic expansion of $Δ^r_j(\overline{p})(n)$ with an effective error bound which subsequently gives an answer to a problem of Wang, Xie, and Zhang. In order to get the asymptotics of $Δ^r_j(\overline{p})(n)$, we derive an asymptotic expansion of the shifted overpartition function $\overline{p}(n+k)$ for any integer $k\neq 0$. |
| title | Asymptotics of the shifted finite differences of the overpatition function and a problem of Wang--Xie--Zhang |
| topic | Number Theory 05A16, 05A20, 11P82 |
| url | https://arxiv.org/abs/2512.23679 |