多重调和和 $\mathcal{H}_{\lbrace s\rbrace^{2l}=1;p-1}$ 模 $p^4$ 及其应用
Multiple harmonic sums $\mathcal{H}_{\lbrace s\rbrace^{2l}=1;p-1}$ modulo $p^4$ and applications
AI总结：
本文研究多重调和和模$p^4$，建立Wilson、Glaisher和Sun定理的推广，并分解Bernoulli数卷积的$p$-剩余，推广了Sun的结果。
摘要：
Wilson定理关于阶乘的推广分别于1900年和2000年由J.W.L. Glaisher和Z-H. Sun推广到模$p^2$和$p^3$。本文更一般地研究多重调和和$\mathcal{H}_{\lbrace s\rbrace^{2l}=1;p-1},2\leq 2l\leq p-1$模$p^4$，并与Stirling数$\left[\begin{array}{l}\\;\\;\\;p\\\\2s-1\end{array}\right], 2\leq 2s\leq p-1$模$p^4$相关联，旨在建立Wilson、Glaisher和Sun结果到模$p^4$的推广。我们还把三个分别阶为$p-1$、$p-3$和$p-5$的除以Bernoulli数的卷积的$p$-剩余分解成更小的部分，并推广了Sun关于某些阶为$p-1$的广义调和数模$p^4$的结果。
英文摘要：
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 $p^4$.
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