A supercongruence concerning truncated hypergeometric series \({}_nF_{n-1}\)

Abstract

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.