Some Congruences Involving Binomial Coefficient and Fermat Quotient

Abstract

In this paper, we investigate some congruences involving sums of $\frac{d^{-k}\left(\genfrac{}{}{0ex}{}{x}{k}\right)\left(\genfrac{}{}{0ex}{}{x+k}{k}\right)}{\left(\genfrac{}{}{0ex}{}{2k}{k}\right)}$, where $x$ be a $p$-adic integer, $k$ be a non-negative integer, and $d$ $(d\ne 0)$ be a rational number.