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\begin{document}$$n\ge 3$$\end{document} be an integer and p be a prime with p≡1(modn)\documentclass[12pt]{minimal}
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\begin{document}$$p\equiv 1\pmod {n}$$\end{document}. In this paper, we show that nFn-1[n-1nn-1n…n-1n1…1|1]p-1≡-Γp(1n)n(modp3),\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} {}_nF_{n-1}\bigg [\begin{array}{llll} \frac{n-1}{n}&{}\frac{n-1}{n}&{}\ldots &{}\frac{n-1}{n}\\ &{}1&{}\ldots &{}1\end{array}\bigg | \, 1\bigg ]_{p-1}\equiv -\Gamma _p\bigg (\frac{1}{n}\bigg )^n\pmod {p^3}, \end{aligned}$$\end{document}where the truncated hypergeometric series nFn-1[x1x2…xny1⋯yn-1|z]m=∑k=0mzkk!∏j=0k-1(x1+j)⋯(xn+j)(y1+j)⋯(yn-1+j)\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} {}_nF_{n-1}\bigg [\begin{array}{llll} x_1&{}x_2&{}\ldots &{}x_n\\ &{}y_1&{}\cdots &{}y_{n-1}\end{array}\bigg | \, z\bigg ]_m=\sum _{k=0}^{m}\frac{z^k}{k!}\prod _{j=0}^{k-1}\frac{(x_1+j)\cdots (x_{n}+j)}{(y_1+j)\cdots (y_{n-1}+j)} \end{aligned}$$\end{document}and Γp\documentclass[12pt]{minimal}
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\begin{document}$$\Gamma _p$$\end{document} denotes the p-adic Gamma function. This confirms a conjecture of Deines et al. (Hypergeometric Series, Truncated Hypergeometric Series, and Gaussian Hypergeometric Functions, Directions in Number Theory, vol. 3, pp. 125–159. Assoc.WomenMath. Ser., Springer, New York, 2016). Furthermore, under the same assumptions, we also prove that pn·n+1Fn[11…1n+1n…n+1n|1]p-1≡-Γp1nn(modp3),\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} p^n\cdot {}_{n+1}F_{n}\bigg [\begin{matrix} 1&{}1&{}\ldots &{}1\\ &{}\frac{n+1}{n}&{}\ldots &{}\frac{n+1}{n}\end{matrix}\bigg | \, 1\bigg ]_{p-1} \equiv -\Gamma _p\left( \frac{1}{n}\right) ^n\pmod {p^3}, \end{aligned}$$\end{document}which solves another conjecture in [5].
