Some Dwork-type $q$-supercongruences from a ${}_6{\varphi }_5$ summation formula

Abstract

With the help of a ${}_6{\varphi }_5$ summation formula and Guo and Zudilin's method, we shall establish some Dwork-type $q$-supercongruences in this paper. When $q\to 1$, these $q$-supercongruences are able to engender the corresponding supercongruences. One of them may be stated as follows: for any prime $p\ge 5$ and any positive integer $s$, \begin{align*} &\sum_{k=0}^{p^s-1}(6k-1)\frac{(-\frac{1}{3})_k^3}{(1)_k^3} \equiv 0\pmod{p^{3s}}. \end{align*}