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

Source abstract

In this paper, we prove two conjectures concerning parametric $q$- supercongruences related to Van Hamme's $\mathrm{(G.2)}$ supercongruence. First, we confirm Tang\rq s conjecture \cite[Conjecture $4.1$]{Tang G2} by establishing the corresponding congruences modulo $[m]Φ_m(q)^2$. Second, we prove a conjecture of Wang and Xu \cite[Conjecture $2$]{WangXu24}, extending their $q$-supercongruence modulo $Φ_m(q)^5$ to modulo $[m]Φ_m(q)^4$ when $m\equiv1\pmod4$.