Proof of some conjectural congruences involving Domb numbers

Abstract

In this paper, we mainly prove the following conjectures of Z.-H. Sun \cite{SH2}: Let $p>3$ be a prime. If $p\equiv 1(\mathrm{mod}3)$ and $p=x^2+3y^2$, then we have $$\sum _{k=0}^{p-1}\frac{D_k}{4^k}\equiv \sum _{k=0}^{p-1}\frac{D_k}{16^k}\equiv 4x^2-2p-\frac{p^2}{4x^2}(\mathrm{mod}{p}^3),$$ and if $p\equiv 2(\mathrm{mod}3)$, then $$\sum _{k=0}^{p-1}\frac{D_k}{4^k}\equiv -2\sum _{k=0}^{p-1}\frac{D_k}{16^k}\equiv \frac{p^2}{2}{\left(\genfrac{}{}{0ex}{}{\frac{p-1}{2}}{\frac{p-5}{6}}\right)}^{-2}(\mathrm{mod}{p}^3),$$ where $D_n={\sum }_{k=0}^n{\left(\genfrac{}{}{0ex}{}{n}{k}\right)}^2\left(\genfrac{}{}{0ex}{}{2k}{k}\right)\left(\genfrac{}{}{0ex}{}{2n-2k}{n-k}\right)$ stands for the $n$th Domb number.

26 pages. (1.2) in Theorem 1.3 is added. arXiv admin note: text overlap with arXiv:2111.08775