Proof of a congruence concerning truncated hypergeometric series ${}_6F_5$ (1812.10324v1)

Abstract: In this paper, we mainly prove the following congruence conjectured by J.-C. Liu: $$ {}6F_5\bigg[\begin{matrix}\frac{5}{4}&\frac{1}{2}&\frac{1}{2}&\frac{1}{2}&\frac{1}{2}&\frac{1}{2}\&\frac{1}{4}&1&1&1&1\end{matrix}\bigg|\ -1\bigg]{\frac{p-1}{2}}\equiv-\frac{p^3}{16}\Gamma_p\left(\frac{1}{4}\right)^4\pmod{p^5}, $$ where $p\geq5$ are primes with $p\equiv3\pmod{4}$.