Multiple harmonic sums $\mathcal{H}_{\lbrace s\rbrace^{2l}=1;p-1}$ modulo $p^4$ and applications
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arXiv
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| Format: | Preprint |
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2021
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| _version_ | 1866911743599443968 |
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| author | Levaillant, Claire I. |
| author_facet | Levaillant, Claire I. |
| contents | Wilson's theorem for the factorial got generalized to the moduli $p^2$ in 1900 and $p^3$ in 2000 by J.W.L. Glaisher and Z-H. Sun respectively. This paper which studies more generally the multiple harmonic sums $\mathcal{H}_{\lbrace s\rbrace^{2l}=1;p-1},2\leq 2l\leq p-1$ modulo $p^4$ in association with the Stirling numbers $\left[\begin{array}{l}\;\;\;p\\2s-1\end{array}\right], 2\leq 2s\leq p-1$ modulo $p^4$ is concerned with establishing a generalization of Wilson, Glaisher and Sun's results to the modulus $p^4$. We also break p-residues of convolutions of three divided Bernoulli numbers of respective orders $p-1$, $p-3$ and $p-5$ into smaller pieces and generalize some results of Sun for some of the generalized harmonic numbers of order $p-1$ modulo $p^4$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2104_12264 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Multiple harmonic sums $\mathcal{H}_{\lbrace s\rbrace^{2l}=1;p-1}$ modulo $p^4$ and applications Levaillant, Claire I. Number Theory Combinatorics 05A10, 05A05, 11B75, 11Y99 G.2.1 Wilson's theorem for the factorial got generalized to the moduli $p^2$ in 1900 and $p^3$ in 2000 by J.W.L. Glaisher and Z-H. Sun respectively. This paper which studies more generally the multiple harmonic sums $\mathcal{H}_{\lbrace s\rbrace^{2l}=1;p-1},2\leq 2l\leq p-1$ modulo $p^4$ in association with the Stirling numbers $\left[\begin{array}{l}\;\;\;p\\2s-1\end{array}\right], 2\leq 2s\leq p-1$ modulo $p^4$ is concerned with establishing a generalization of Wilson, Glaisher and Sun's results to the modulus $p^4$. We also break p-residues of convolutions of three divided Bernoulli numbers of respective orders $p-1$, $p-3$ and $p-5$ into smaller pieces and generalize some results of Sun for some of the generalized harmonic numbers of order $p-1$ modulo $p^4$. |
| title | Multiple harmonic sums $\mathcal{H}_{\lbrace s\rbrace^{2l}=1;p-1}$ modulo $p^4$ and applications |
| topic | Number Theory Combinatorics 05A10, 05A05, 11B75, 11Y99 G.2.1 |
| url | https://arxiv.org/abs/2104.12264 |