Two $q$-congruences from Jackson's ${}_8{\varphi }_7$ summation and Andrews' ${}_4{\varphi }_3$ 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 ${}_8{\varphi }_7$ summation from the modulus ${\Phi }_n(q{)}^4$ to the modulus ${\Phi }_n(q{)}^5$ when $n\equiv 3(\mathrm{mod}5)$. The second result proves a cyclotomic congruence modulo ${\Phi }_n(q)$ when $n\equiv 1(\mathrm{mod}4)$, by a specialization of Andrews' terminating ${}_4{\varphi }_3$ summation.

12 pages