Supercongruences concerning truncated hypergeometric series

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. Furthermore, under the same assumptions, we also prove that $$p^n\cdot {}_{n+1} F_n \bigg[ \begin{matrix} 1 &1 &\ldots &1\\ &\frac{n+1}{n} &\ldots &\frac{n+1}{n} \end{matrix}\bigg | \, 1\bigg]_{p-1} \equiv -\Gamma_p \Bigl(\frac{1}{n} \Bigr)^n \quad(\mathrm{mod}\ {p^3}),$$ which solves another conjecture of Deines, Fuselier, Long, Swisher and Tu.