On the restricted partition function via determinants with Bernoulli polynomials
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2018
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| _version_ | 1866929330394759168 |
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| author | Cimpoeas, Mircea |
| author_facet | Cimpoeas, Mircea |
| contents | Let $r\geq 1$ be an integer, $\mathbf a=(a_1,\ldots,a_r)$ a vector of positive integers and let $D\geq 1$ be a common multiple of $a_1,\ldots,a_r$. We prove that, if a determinant $Δ_{r,D}$, which depends only on $r$ and $D$, with entries consisting in values of Bernoulli polynomials is nonzero, then the restricted partition function $p_{\mathbf a}(n): = $ the number of integer solutions $(x_1,\dots,x_r)$ to $\sum_{j=1}^r a_jx_j=n$ with $x_1\geq 0, \ldots, x_r\geq 0$ can be computed in terms of values of Bernoulli polynomials and Bernoulli Barnes numbers. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1806_08996 |
| institution | arXiv |
| publishDate | 2018 |
| record_format | arxiv |
| spellingShingle | On the restricted partition function via determinants with Bernoulli polynomials Cimpoeas, Mircea Number Theory 11P81, 11B68, 11P82 Let $r\geq 1$ be an integer, $\mathbf a=(a_1,\ldots,a_r)$ a vector of positive integers and let $D\geq 1$ be a common multiple of $a_1,\ldots,a_r$. We prove that, if a determinant $Δ_{r,D}$, which depends only on $r$ and $D$, with entries consisting in values of Bernoulli polynomials is nonzero, then the restricted partition function $p_{\mathbf a}(n): = $ the number of integer solutions $(x_1,\dots,x_r)$ to $\sum_{j=1}^r a_jx_j=n$ with $x_1\geq 0, \ldots, x_r\geq 0$ can be computed in terms of values of Bernoulli polynomials and Bernoulli Barnes numbers. |
| title | On the restricted partition function via determinants with Bernoulli polynomials |
| topic | Number Theory 11P81, 11B68, 11P82 |
| url | https://arxiv.org/abs/1806.08996 |