Proof of some congruence conjectures of Guo and Liu

Abstract

Let $n$ and $r$ be positive integers. Define the numbers $S_n^{(r)}$ by $S_n^{(r)}=\sum_{k=0}^n\binom{n}{k}^2\binom{2k}{k}(2k+1)^r.$ In this paper we prove some conjectures of Guo and Liu which extend some conjectures of Z.-W. Sun \cite{Su1}, such as: There exist integers $a_{2r-1}$ and $b_r$, independent of $n$, such that $$a_{2r-1}\sum_{k=0}^{n-1}S_k^{(2r-1)}\equiv0\pmod{n^2}\ \mbox{and}\ b_r\sum_{k=0}^{n-1}kS_k^{(r)}\equiv0\pmod{n^2}.$$ By Zeilberger algorithm, we find that for all $0\leq j<n$, $$(2j+1)\binom{2j}j\sum_{k=j}^{n-1}(2k-j+1)\binom kj^2\equiv0\pmod{n^2}.$$