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arXiv · 2606.16495

Two $q$-congruences from Jackson's ${}_8\phi_7$ summation and Andrews' ${}_4\phi_3$ summation

Ji-Cai Liu·Qing-Yuan Tao
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Abstract

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 ${}_8\phi_7$ summation from the modulus $\Phi_n(q)^4$ to the modulus $\Phi_n(q)^5$ when $n\equiv3\pmod 5$. The second result proves a cyclotomic congruence modulo $\Phi_n(q)$ when $n\equiv1\pmod4$, by a specialization of Andrews' terminating ${}_4\phi_3$ summation.

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Ji-Cai Liu, Qing-Yuan Tao. 2026-06-15. Two $q$-congruences from Jackson's ${}_8\phi_7$ summation and Andrews' ${}_4\phi_3$ summation. https://arxiv.org/abs/2606.16495

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