$q$-Analogues of some supercongruences related to Van Hamme's (E.2) and (F.2) supercongruences

Abstract: In this paper, we present a one-parameter extension of some supercongruences related to Van Hamme's $\mathrm{(E.2)}$ and $\mathrm{(F.2)}$ supercongruences. To deduce these results, we first establish the corresponding $q$-supercongruences modulo the third power of a cyclotomic polynomial. Our proofs rely on the \emph{creative microscoping} method devised by Guo and Zudilin~\cite{guo2019q}, together with Jackson's ${}<em>{6}φ</em>{5}$ summation formula.