Proof of Two Conjectural $q$-Supercongruences Related to Van Hamme's $(G.2)$ Supercongruence

Abstract

In this paper, we prove two conjectures concerning parametric $q$- supercongruences related to Van Hamme's $(G.2)$ supercongruence. First, we confirm Tang\rq s conjecture \cite[Conjecture $4.1$]{Tang G2} by establishing the corresponding congruences modulo $[m]{\Phi }_m(q{)}^2$. Second, we prove a conjecture of Wang and Xu \cite[Conjecture $2$]{WangXu24}, extending their $q$-supercongruence modulo ${\Phi }_m(q{)}^5$ to modulo $[m]{\Phi }_m(q{)}^4$ when $m\equiv 1(\mathrm{mod}4)$.

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