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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2411.16968 |
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| _version_ | 1866915250944606208 |
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| author | Gomez, Kevin Ono, Ken Saad, Hasan Singh, Ajit |
| author_facet | Gomez, Kevin Ono, Ken Saad, Hasan Singh, Ajit |
| contents | We revisit Euler's partition function recurrence, which asserts, for integers $n\geq 1,$ that $$ p(n)=p(n-1)+p(n-2)-p(n-5)-p(n-7)+\dots = \sum_{k\in \mathbb{Z}\setminus \{0\}} (-1)^{k+1} p(n-ω(k)), $$ where $ω(m):=(3m^2+m)/2$ is the $m$th pentagonal number. We prove that this classical result is the $ν=0$ case of an infinite family of ``pentagonal number'' recurrences. For each $ν\geq 0,$ we prove for positive $n$ that
$$ p(n)=\frac{1}{g_ν(n,0)}\left(α_ν\cdot σ_{2ν-1}(n)+ \mathrm{Tr}_{2ν}(n) +\sum_{k\in \mathbb{Z}\setminus \{0\}} (-1)^{k+1} g_ν(n,k)\cdot p(n-ω(k))\right), $$ where $σ_{2ν-1}(n)$ is a divisor function, $\mathrm{Tr}_{2ν}(n)$ is the $n$th weight $2ν$ Hecke trace of values of special twisted quadratic Dirichlet series, and each $g_ν(n,k)$ is a polynomial in $n$ and $k.$ The $ν=6$ case can be viewed as a partition theoretic formula for Ramanujan's tau-function, as we have $$ \mathrm{Tr}_{12}(n)=-\frac{33108590592}{691}\cdot τ(n). $$ |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_16968 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Pentagonal number recurrence relations for $p(n)$ Gomez, Kevin Ono, Ken Saad, Hasan Singh, Ajit Number Theory Combinatorics 11P82, 05A17 We revisit Euler's partition function recurrence, which asserts, for integers $n\geq 1,$ that $$ p(n)=p(n-1)+p(n-2)-p(n-5)-p(n-7)+\dots = \sum_{k\in \mathbb{Z}\setminus \{0\}} (-1)^{k+1} p(n-ω(k)), $$ where $ω(m):=(3m^2+m)/2$ is the $m$th pentagonal number. We prove that this classical result is the $ν=0$ case of an infinite family of ``pentagonal number'' recurrences. For each $ν\geq 0,$ we prove for positive $n$ that $$ p(n)=\frac{1}{g_ν(n,0)}\left(α_ν\cdot σ_{2ν-1}(n)+ \mathrm{Tr}_{2ν}(n) +\sum_{k\in \mathbb{Z}\setminus \{0\}} (-1)^{k+1} g_ν(n,k)\cdot p(n-ω(k))\right), $$ where $σ_{2ν-1}(n)$ is a divisor function, $\mathrm{Tr}_{2ν}(n)$ is the $n$th weight $2ν$ Hecke trace of values of special twisted quadratic Dirichlet series, and each $g_ν(n,k)$ is a polynomial in $n$ and $k.$ The $ν=6$ case can be viewed as a partition theoretic formula for Ramanujan's tau-function, as we have $$ \mathrm{Tr}_{12}(n)=-\frac{33108590592}{691}\cdot τ(n). $$ |
| title | Pentagonal number recurrence relations for $p(n)$ |
| topic | Number Theory Combinatorics 11P82, 05A17 |
| url | https://arxiv.org/abs/2411.16968 |