Proof of some congruence conjectures of Guo and Liu

Abstract

Let $n$ and $r$ be positive integers. Define the numbers $S_n^{(r)}$ by $S_n^{(r)}={\sum }_{k=0}^n{\left(\genfrac{}{}{0ex}{}{n}{k}\right)}^2\left(\genfrac{}{}{0ex}{}{2k}{k}\right)(2k+1{)}^r.$ 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 $0\le j<n$, $$(2j+1)\left(\genfrac{}{}{0ex}{}{2j}{j}\right)\sum _{k=j}^{n-1}(2k-j+1){\left(\genfrac{}{}{0ex}{}{k}{j}\right)}^2\equiv 0(\mathrm{mod}{n}^2).$$

arXiv comment11 pages, 0 figures, The Ramanujan Journal(2018)