On some conjectural hypergeometric congruences
Volume 161 / 2020
Colloquium Mathematicum 161 (2020), 251-261
MSC: Primary 11B65; Secondary 05A10, 11A07, 11B68.
DOI: 10.4064/cm7934-8-2019
Published online: 30 March 2020
Abstract
We first establish two congruences which extend Q. Gao’s results. Then we confirm some congruences conjectured by V. J. W. Guo and M. J. Schlos\-ser recently. For example, we show that for primes $p \gt 3$, $$ \sum _{k=0}^{p-1}(2pk-2k-1)\frac {\bigl (\frac {-1}{p-1}\big )_k^{2p-2}}{(k!)^{2p-2}}\equiv 0\pmod {p^5}. $$
