Proof of some conjectural congruences involving products of two binomial coefficients

Abstract

In this paper, we mainly prove the following conjectures of Z.-W. Sun: Let $p\equiv 3\left(\mathrm{mod}4\right)$ be a prime. Then$\sum _{k=0}^{p-1}\frac{{\left(\begin{array}{c}2k\\ k\end{array}\right)}^2}{(2k-1)8^k}\equiv -\left(\frac{2}{p}\right)\frac{p+1}{2^{p-1}+1}\left(\begin{array}{c}(p+1)/2\\ (p+1)/4\end{array}\right)\left(\mathrm{mod}{p}^2\right),3\sum _{k=0}^{p-1}\frac{\left(\begin{array}{c}2k\\ k\end{array}\right)\left(\begin{array}{c}2k\\ k+1\end{array}\right)}{(2k-1)8^k}\equiv p+\left(\frac{2}{p}\right)\frac{2p}{\left(\begin{array}{c}(p+1)/2\\ (p+1)/4\end{array}\right)}\left(\mathrm{mod}{p}^2\right),$ where $\left(\frac{\cdot }{p}\right)$ stands for the Legendre symbol. The necessary proofs are provided by the computer algebra software Sigma to find and verify the underlying hypergeometric sum identities.