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On some conjectural hypergeometric congruences

Volume 161 / 2020

Chen Wang 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 $p \gt 3$, $$ \sum _{k=0}^{p-1}(2pk-2k-1)\frac {\bigl (\frac {-1}{p-1}\big )_k^{2p-2}}{(k!)^{2p-2}}\equiv 0\pmod {p^5}. $$

Authors

  • Chen WangDepartment of Mathematics
    Nanjing University
    Nanjing 210093, People’s Republic of China
    e-mail

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