On the supercongruences involving harmonic numbers of order 2

Abstract

We prove several supercongruences involving the harmonic number of order two $H_n^{(2)}:={\sum }_{k=1}^{n}1/k^2$. For example, if $p>5$ is prime and $\alpha$ is $p$-integral, then we can completely determine $$\sum _{k=0}^{p-1}\frac{H_k^{(2)}}{k}\cdot \left(\genfrac{}{}{0ex}{}{\alpha }{k}\right)\left(\genfrac{}{}{0ex}{}{-1-\alpha }{k}\right)\text{and}\sum _{k=0}^{\frac{p-1}{2}}\frac{H_k^{(2)}}{k}\cdot \left(\genfrac{}{}{0ex}{}{\alpha }{k}\right)\left(\genfrac{}{}{0ex}{}{-1-\alpha }{k}\right)$$ modulo $p^3$. In particular, by setting $\alpha =-1/2$, we confirm two conjectured congruences of Z.-W. Sun.

The results of Theorem 1.2 in the first version have been removed