Dwork-type $q$-congruences through the $q$-Lucas theorem

Abstract

Employing the $q$-Lucas theorem and some known $q$-supercongruences, we give some Dwork-type $q$-congruences, confirming three conjectures in [J. Combin. Theory, Ser. A 178 (2021), Art.~105362]. As conclusions, we obtain the following supercongruences: for any prime $p\equiv 1\pmod{4}$ and positive integer $r$, \begin{align*} \sum_{k=0}^{(p^r-1)/2} \frac{(\frac{1}{2})_k^3}{k!^3} &\equiv -\Gamma_p(\tfrac{1}{4})^4 \sum_{k=0}^{(p^{r-1}-1)/2} \frac{(\frac{1}{2})_k^3}{k!^3} \pmod{p^{r+1}}, \\ \sum_{k=0}^{p^r-1} \frac{(\frac{1}{2})_k^3}{k!^3} &\equiv -\Gamma_p(\tfrac{1}{4})^4 \sum_{k=0}^{p^{r-1}-1} \frac{(\frac{1}{2})_k^3}{k!^3} \pmod{p^{r+1}}, \end{align*} where $\Gamma_p(x)$ stands for the $p$-adic Gamma function. The first one confirms a weaker form of Swisher's (H.3) conjecture for $p\equiv 1\pmod{4}$, which originally predicts that the supercongruence is true modulo $p^{3r}$.