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

Research paper by **Chen Wang, Hao Pan**

Indexed on: **07 Jun '18**Published on: **07 Jun '18**Published in: **arXiv - Mathematics - Number Theory**

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#### 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.