A supercongruence concerning truncated hypergeometric series ${}_nF_{n-1}$

Chen Wang, Hao Pan

Published 2018-06-07Version 1

Let $n\geq 3$ be an integer and $p$ be a prime with $p\equiv 1\pmod{n}$. In this paper, we show that $${}_nF_{n-1}\bigg[\begin{matrix} \frac{n-1}{n}&\frac{n-1}{n}&\ldots&\frac{n-1}{n} &1&\ldots&1\end{matrix}\bigg | \, 1\bigg]_{p-1}\equiv -\Gamma_p\bigg(\frac{1}{n}\bigg)^n\pmod{p^3}, $$ where the truncated hypergeometric series $$_nF_{n-1}\bigg[\begin{matrix} x_1&x_2&\ldots&x_n &y_1&\cdots&y_{n-1}\end{matrix}\bigg | \, z\bigg]_m=\sum_{k=0}^{m}\frac{z^k}{k!}\prod_{j=0}^{k-1}\frac{(x_1+j)\cdots(x_{n}+j)}{(y_1+j)\cdots(y_{n-1}+j)} $$ and $\Gamma_p$ denotes the $p$-adic gamma function. This confirms a conjecture of Deines, Fuselier, Long, Swisher and Tu.