Some new curious congruences involving multiple harmonic sums

Abstract

It is significant to study congruences involving multiple harmonic sums. Let $p$ be an odd prime, in recent years, the following curious congruence $$\sum _{\begin{array}{c}i+j+k=p\\ i,j,k>0\end{array}}\frac{1}{ijk}\equiv -2B_{p-3}(\mathrm{mod}p)$$ has been generalized along different directions, where $B_n$ denote the $n$th Bernoulli number. In this paper, we obtain several new generalizations of the above congruence by applying congruences involving multiple harmonic sums. For example, we have $$\sum _{\begin{array}{c}{k}_1+k_2+\cdots +k_n=p\\ k_i>0,1\le i\le n\end{array}}\frac{(-1{)}^{k_1}\left(\frac{k_1}{3}\right)}{k_1\cdots k_n}\equiv \frac{(n-1)!}{n}\frac{2^{n-1}+1}{3\cdot 6^{n-1}}{B}_{p-n}\left(\frac{1}{3}\right)(\mathrm{mod}p),$$ where $n$ is even, $B_n(x)$ denote the Bernoulli polynomials.

arXiv comment12 pages