On three conjectural congruences of Z.-H. Sun involving Apéry and Apéry-like numbers

Abstract

In this paper, we mainly prove the following conjectures of Sun Z-H (New congruences involving Apéry-like numbers. Preprint at arXiv:2004.07172v2): Let $p>3$ be a prime. Then

$$\begin{aligned}&A'_{2p-1}\equiv A'_1+\frac{200}{3}p^3B_{p-3}\ \left( \mathrm{{mod}}\ p^4\right) ,\\&\quad D_{2p-1}\equiv 64^{2(p-1)}D_1-\frac{44}{3}p^3B_{p-3}\ \left( \mathrm{{mod}}\ p^4\right) , \end{aligned}$$

where $A'_n=\sum _{k=0}^n\left( {\begin{array}{c}n\\ k\end{array}}\right) ^2\left( {\begin{array}{c}n+k\\ k\end{array}}\right)$, $D_n=\sum _{k=0}^n\left( {\begin{array}{c}n\\ k\end{array}}\right) ^2\left( {\begin{array}{c}2k\\ k\end{array}}\right) \left( {\begin{array}{c}2n-2k\\ n-k\end{array}}\right)$ and $B_n$ stands for the nth Bernoulli number. We also give a generalization of a conjecture of Sun involving Franel numbers $f_n=\sum _{k=0}^n\left( {\begin{array}{c}n\\ k\end{array}}\right) ^3$.