On two conjectural supercongruences of Z.-W. Sun

Abstract

In this paper, we mainly prove two conjectural supercongruences of Sun by using the following identity:

$$\begin{aligned} \sum _{k=0}^n\left( {\begin{array}{c}2k\\ k\end{array}}\right) ^2\left( {\begin{array}{c}2n-2k\\ n-k\end{array}}\right) ^2=16^n\sum _{k=0}^n \frac{\left( {\begin{array}{c}n+k\\ k\end{array}}\right) \left( {\begin{array}{c}n\\ k\end{array}}\right) \left( {\begin{array}{c}2k\\ k\end{array}}\right) ^2}{(-16)^k}, \end{aligned}$$

which arises from a ${}_4F_3$ hypergeometric transformation. For any prime $p>3$, we prove that

$$\begin{aligned}&\sum _{n=0}^{p-1}\frac{n+1}{8^n}\sum _{k=0}^n\left( {\begin{array}{c}2k\\ k\end{array}}\right) ^2 \left( {\begin{array}{c}2n-2k\\ n-k\end{array}}\right) ^2\equiv (-1)^{(p-1)/2}p+5p^3E_{p-3}\pmod {p^4},\\&\sum _{n=0}^{p-1}\frac{2n+1}{(-16)^n} \sum _{k=0}^n\left( {\begin{array}{c}2k\\ k\end{array}}\right) ^2\left( {\begin{array}{c}2n-2k\\ n-k\end{array}}\right) ^2\equiv (-1)^{(p-1)/2}p +3p^3E_{p-3}\pmod {p^4}, \end{aligned}$$

where $E_{p-3}$ is the $(p-3)\hbox {th}$ Euler number.