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

Liton Karmakar; Arijit Jana

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arXiv · 2610.15050

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

Liton Karmakar·Arijit Jana

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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 $m\equiv1\pmod4$.

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BibTeX RIS

Liton Karmakar, Arijit Jana. 2026-10-10. Proof of Two Conjectural $q$-Supercongruences Related to Van Hamme's $\mathrm{(G.2)}$ Supercongruence. https://arxiv.org/abs/2610.15050

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