Proof of some congruence conjectures of Guo and Liu

TL;DR

This paper proves conjectures related to certain combinatorial sums involving binomial coefficients and their congruences modulo squares of integers, extending previous conjectures by Sun and others.

Contribution

It establishes new congruence relations for sums involving binomial coefficients, confirming conjectures of Guo and Liu that extend earlier work by Sun.

Findings

Confirmed conjectures on divisibility properties of sums involving binomial coefficients.

Derived new congruences modulo n^2 for sums of specific combinatorial sequences.

Used Zeilberger algorithm to prove key congruence relations.

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 $0\leq j<n$, $$(2j+1)\binom{2j}j\sum_{k=j}^{n-1}(2k-j+1)\binom kj^2\equiv0\pmod{n^2}.$$