Qing-Yuan Tao | Searcharxiv

arXiv2026

Two $q$-congruences from Jackson's ${}_8ϕ_7$ summation and Andrews' ${}_4ϕ_3$ summation

We prove two $q$-congruence conjectures of Guo on truncated basic hypergeometric series. The first result strengthens a congruence obtained from Jackson's terminating very-well-poised ${}_8ϕ_7$ summation from the modulus $Φ_n(q)^4$ to the modulus $Φ_n(q)^5$ when $n\equiv3\pmod 5$. The second result proves a cyclotomic congruence modulo $Φ_n(q)$ when $n\equiv1\pmod4$, by a specialization of Andrews' terminating ${}_4ϕ_3$ summation.

math.NT↗

arXiv2026

Proofs of two $q$-congruence conjectures of Guo

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.

math.NT↗