On a kind of generalized multi-harmonic sums

Abstract

Let $p$ be an odd prime, Jianqiang Zhao has established a curious congruence, which is $$\sum _{\genfrac{}{}{0ex}{}{i+j+k=p}{i,j,k>0}}\frac{1}{ijk}\equiv -2B_{p-3}(\mathrm{mod}p),$$ where $B_n$ denotes the $n$-th Bernoulli number. In this paper, we will generalize this kind of sums and prove a family of similar congruences modulo prime powers $p^r$.

11 pages, no figures