On congruences with binomial coecients and harmonic numbers

Abstract

In this paper, we obtain super congruences%
\begin{equation*}
\sum\limits_{k=1}^{\left[ (p-1)/r\right] }\left( -1\right) ^{k+1}\binom{%
\alpha p-1}{k}H_{k}^{2}\pmod {p^{2}}\text{ and}\sum\limits_{k=1}^{\left[
(p-1)/r\right] }\left( -1\right) ^{k}k\binom{\alpha p-1}{k}H_{k}^{2}%
\pmod
{p^{2}},
\end{equation*}%
where $r\in \left\{ 1,2,3\right\} $ and $\alpha $ is a $p-$adic integer.
Also, we give new congruences involving binomial coefficients and harmonic
numbers.