$q$-Analogues of some supercongruences related to generalized Van Hamme-type supercongruences

Abstract

Recently, Jana and Kalita (Res. Number Theory $\textbf{8}~ (2022),$ Article $54$) obtained certain supercongruences motivated by some generalized Van Hamme type supercongruences, specifically for an integer $\ell \geq 2$ and an odd prime $p$ with $p \equiv -1 \pmod{\ell},$ \begin{align*} \displaystyle \sum_{n=0}^{\frac{p^v + 1}{\ell}} (-1)^n (2 \ell n - 1) (\ell^2 n^2 - \ell n + 1) \frac{(-\frac{1}{\ell})_n^3}{(1)_n^3} \equiv (-1)^{\frac{p^v + p + 2}{\ell}} p^{3v} \pmod{p^{3v+1}}, \end{align*} and \begin{align*} \displaystyle \sum_{n=0}^{\frac{p^v + 1}{\ell}} (2 \ell n - 1) (2 \ell^2 n^2 - 2 \ell n + 1) \frac{(-\frac{1}{\ell})_n^4}{(1)_n^4} \equiv - p^{4v} \pmod{p^{4v+1}}. \end{align*} Employing the $q$-telescoping technique, similar to the $q$-WZ method, we here establish some supercongruences involving certain $q$-shifted factorials. As particular cases, we provide $q$-analogues of the above supercongruences.