It is significant to study congruences involving multiple harmonic sums. Let $p$ be an odd prime, in recent years, the following curious congruence $$\sum_{ \substack{ i+j+k=p \\ i, j, k>0}} \frac{ 1}{ i j k} \equiv-2 B_{ p-3}\pmod p$$ 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 $$\sum_{ \substack{ k_1+k_2+\cdots+k_n=p \\ k_i> 0, 1 \le i \le n}} \dfrac{ (-1)^{ k_1}\left(\dfrac{ k_1}{ 3}\right)}{ k_1 \cdots k_n} \equiv \dfrac{ (n-1)!}{ n}\dfrac{ 2^{ n-1}+1}{ 3\cdot6^{ n-1}}B_{ p-n}\left(\dfrac{ 1}{ 3}\right)\pmod p,$$ where $n$ is even, $B_n(x)$ denote the Bernoulli polynomials.
Categories: math.NT