$q$-Supercongruences Modulo the Third and Fifth Powers of a Cyclotomic Polynomial

Abstract: In this paper, we establish several $q$-supercongruences for basic hypergeometric series modulo the third and fifth powers of a cyclotomic polynomial. These results extend several existing supercongruences related to the $\mathrm{(A.2)}$ and $\mathrm{(C.2)}$ supercongruences of Van Hamme. Our proofs employ the creative microscoping method developed by Guo and Zudilin \cite{guo2019q}, together with Watson's ${}_{8}φ_7$ transformation formula.