Super Congruences Involving Multiple Harmonic Sums and Bernoulli Numbers

zbMATH Open1422.11001arXiv1702.08401MaRDI QIDQ5274914

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Publication date: 7 July 2017

Abstract: Let $m$, $r$ and $n$ be positive integers. We denote by $$ any tuple of odd positive integers $$ such that $k_1+dots+k_t=n$ and $k_{j}ge3$ for all $j$. In this paper we prove that for every sufficiently large prime $p$ ${\displaystyle }$ sum_{substack{l_1+l_2+cdots+l_n=mp^r p mid l_1 l_2 cdots l_n }} frac1{l_1l_2cdots l_n} equiv p^{r-1} sum_{{�f k}vdash n} C_{m,{�f k}} B_{p-{�f k}} pmod{p^r} ${\displaystyle }$ where $$ are products of Bernoulli numbers and the coefficients $$ are polynomials of $m$ independent of $p$ and $r$. This generalizes previous results by many different authors and confirms a conjecture by the authors and their collaborators.

Full work available at URL: https://arxiv.org/abs/1702.08401

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