The p -Adic Valuations of Sums of Binomial Coefficients

Abstract

In this paper, we prove three supercongruences on sums of binomial coefficients conjectured by Z.-W. Sun. Let $p$ be an odd prime and let $h\in \mathbb{Z}$ with $2h-1\equiv 0\left(\mathrm{mod}p\right)$ . For $a\in {\mathbb{Z}}^{+}$ and $p^a>3$ , we show that ${\displaystyle {\sum }_{k=0}^{p^a-1}\left(\begin{array}{c}h{p}^a-1\hfill \\ k\hfill \end{array}\right)\left(\begin{array}{c}2k\hfill \\ k\hfill \end{array}\right){\left(-\left(h/2\right)\right)}^k}\equiv 0\left(\mathrm{mod}{p}^{a+1}\right)$ . Also, for any $n\in {\mathbb{Z}}^{+}$ , we have ${\nu }_p\left({\displaystyle {\sum }_{k=0}^{n-1}\left(\begin{array}{c}hn-1\hfill \\ k\hfill \end{array}\right)\left(\begin{array}{c}2k\hfill \\ k\hfill \end{array}\right){\left(-\left(h/2\right)\right)}^k}\right)\ge {\nu }_p\left(n\right)$ , where ${\nu }_p\left(n\right)$ denotes the $p$ -adic order of $n$ . For any integer $m\cancel{\equiv }0\left(\mathrm{mod}p\right)$ and positive integer $n$ , we have $\left(1/pn\right)\left({\displaystyle {\sum }_{k=0}^{pn-1}\left(\begin{array}{c}pn-1\hfill \\ k\hfill \end{array}\right)\left(\left(\begin{array}{c}2k\hfill \\ k\hfill \end{array}\right)/{\left(-m\right)}^k\right)}-\left(m\left(m-4\right)/p\right){\displaystyle {\sum }_{k=0}^{n-1}\left(\begin{array}{c}n-1\hfill \\ k\hfill \end{array}\right)\left(\left(\begin{array}{c}2k\hfill \\ k\hfill \end{array}\right)/{\left(-m\right)}^k\right)}\right)\in {\mathbb{Z}}_p$ , where $\left(\dot{-}\right)$ is the Legendre symbol and ${\mathbb{Z}}_p$ is the ring of $p$ -adic integers.