Proof of a congruence concerning truncated hypergeometric series ${}_6{F}_5$

Abstract

In this paper, we mainly prove the following congruence conjectured by J.-C. Liu: $${}_6{F}_5\left[\begin{array}{cccccc}\frac{5}{4}& \frac{1}{2}& \frac{1}{2}& \frac{1}{2}& \frac{1}{2}& \frac{1}{2}\\ & \frac{1}{4}& 1& 1& 1& 1\end{array}\right|-1{]}_{\frac{p-1}{2}}\equiv -\frac{p^3}{16}{\Gamma }_p{\left(\frac{1}{4}\right)}^4(\mathrm{mod}{p}^5),$$ where $p\ge 5$ are primes with $p\equiv 3(\mathrm{mod}4)$.

Cited by in corpus (1)

• $p$-adic analogues of hypergeometric identities and their applications