In this paper we prove three results conjectured by Z.-W. Sun. Let $p$ be an odd prime and let $h\in \mathbb{ Z}$ with $2h-1\equiv0\pmod{ p^{ }}$. For $a\in\mathbb{ Z}^{ +}$ and $p^a>3$, we show that \begin{ align}\notag \sum_{ k=0}^{ p^a-1}\binom{ hp^a-1}{ k}\binom{ 2k}{ k}\bigg(-\frac{ h}{ 2}\bigg)^k\equiv0\pmod{ p^{ a+1}}. \end{ align} Also, for any $n\in \mathbb{ Z}^{ +}$ we have \begin{ align} \notag \nu_{ p}\bigg(\sum_{ k=0}^{ n-1}\binom{ hn-1}{ k}\binom{ 2k}{ k}\bigg(-\frac{ h}{ 2}\bigg)^k\bigg)\geq\nu_{ p}(n)\notag, \end{ align} where $\nu_p(n)$ denotes the $p$-adic order of $n$. For any integer $m\not\equiv 0\pmod{ p^{ }}$ and positive integer $n$, we have \begin{ align*} \frac{ 1}{ pn}\bigg(\sum_{ k=0}^{ pn-1}\binom{ pn-1}{ k}\frac{ \binom{ 2k}{ k}}{ (-m)^k}-\bigg(\frac{ m(m-4)}{ p}\bigg)\sum_{ k=0}^{ n-1}\binom{ n-1}{ k}\frac{ \binom{ 2k}{ k}}{ (-m)^k}\bigg)\in \mathbb{ Z}_{ p}, \end{ align*} where $(\frac{ .}{ })$ is the Legendre symbol and $\mathbb{ Z}_p$ is the ring of $p$-adic integers.
Categories: math.CO