Proof of some congruences conjectured by Guo and Liu

Guo-Shuai Mao

Published 2015-11-17Version 1

Let $n$ and $r$ be positive integers and $p$ a prime. Define the numbers $S_n^{(r)}$ and $T_n^{(r)}$ by $$S_n^{(r)}=\sum_{k=0}^n\binom{n}{k}^2\binom{2k}{k}(2k+1)^r\ \mbox{,}\ T_n^{(r)}=\sum_{k=0}^n\binom{n}{k}^2\binom{2k}{k}(2k+1)^r(-1)^k.$$ In this paper we first prove the following congruences: $$\sum_{k=0}^{n-1}S_k^{(2r)}\equiv0\pmod{n^2},$$ $$\sum_{k=0}^{n-1}T_k^{(2r)}\equiv0\pmod{n^2}$$ and $$\sum_{k=0}^{p-1}T_k^{(2)}\equiv\frac{p^2}{2}\left(5-3\big(\frac{p}{5}\big)\right)\pmod{p^3}.$$ We also show that there exist integers $a_{2r-1}$ and $b_r$, independent of n, such that $$a_{2r-1}\sum_{k=0}^{n-1}S_k^{(2r-1)}\equiv0\pmod{n^2},$$ $$b_r\sum_{k=0}^{n-1}kS_k^{(r)}\equiv0\pmod{n^2}.$$ This confirms several recent conjectures of Guo and Liu.