Two parametric $q$-supercongruences from Andrews' multi-series transformation

Source abstract

Recently, Guo and Guo (Results Math $76:37, ~2021$) obtained curious $q$-analogues of the following two supercongruences conjectured by Guo and Schlosser \cite[Conjectures $12.6$ and $12.7$]{GuoSclssr ConstApprox 21}: $$\begin{align*} \sum_{n=0}^{(p-1)/3} (6n+1) \frac{\left(\frac{1}{3}\right)_n^{4}(2n)!} {(n!)^{4}\left(\frac{2}{3}\right)_{2n}} \equiv p \pmod{p^{3}} \qquad \text{for primes } p\equiv 1 \pmod{3}, \end{align*}$$ $$\begin{align*} \sum_{n=0}^{(p+1)/3} (6n-1) \frac{\left(-\frac{1}{3}\right)_n^{4}(2n)!} {(n!)^{4}\left(-\frac{2}{3}\right)_{2n}} \equiv p \pmod{p^{3}} \qquad \text{for primes } p\equiv 2 \pmod{3}, \end{align*}$$ where $(z)_n=z(z+1)\cdots(z+n-1)$ denotes the shifted factorial. In this paper, we establish a one-parameter extension of them. The main ingredients of our proofs are the creative microscoping method developed by Guo and Zudilin \cite{guo2019q} and Andrews' multi-series transformation.