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arXiv · 2606.12037

Proofs of two $q$-congruence conjectures of Guo

Ji-Cai Liu·Qing-Yuan Tao
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Abstract

We prove two conjectural $q$-congruences proposed by Guo. The first is Conjecture 7.2 in Guo's work on $q$-analogues of two ``divergent'' Ramanujan-type supercongruences; it asserts a square-cyclotomic congruence for a truncated $q$-analogue of a Ramanujan-type sum when $n\equiv1\pmod4$. The second is Conjecture 4.1 in Guo's extension of Van Hamme's $(A.2)$ supercongruence; it gives divisibility modulo $[n]$ for a family of truncated basic hypergeometric sums with a parameter $s$. The proof of the first result relies on a known Watson-transformation congruence obtained by Guo. The proof of the second result is based on period decomposition at primitive roots of unity and a reflection cancellation inside residue blocks.

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Ji-Cai Liu, Qing-Yuan Tao. 2026-06-10. Proofs of two $q$-congruence conjectures of Guo. https://arxiv.org/abs/2606.12037

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