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

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 ${}_8ϕ_7$ 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.