Super Congruences Involving Multiple Harmonic Sums and Bernoulli Numbers

Abstract

Let $m$, $r$ and $n$ be positive integers. We denote by $\mathbf{k}⊢n$ any tuple of odd positive integers $\mathbf{k}=(k_1,\dots ,k_t)$ such that $k_1+\cdots +k_t=n$ and $k_j\ge 3$ for all $j$. In this paper we prove that for every sufficiently large prime $p$ $$\sum _{\begin{array}{c}{l}_1+l_2+\cdots +l_n=m{p}^{r}p\nmid l_1{l}_2\cdots l_n\end{array}}\frac{1}{l_1{l}_2\cdots l_n}\equiv p^{r-1}\sum _{\mathbf{k}⊢n}{C}_{m,\mathbf{k}}{B}_{p-\mathbf{k}}(\mathrm{mod}{p}^r)$$ where $B_{p-\mathbf{k}}=B_{p-k_1}{B}_{p-k_2}\cdots B_{p-k_t}$ are products of Bernoulli numbers and the coefficients $C_{m,\mathbf{k}}$ 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.

19 pages. We updated the proof of Lemma 5.1 and corrected some misprints

Cited by in corpus (1)

• Some Families of Super Congruences Involving Alternating Multiple Harmonic Sums