Supercongruences concerning truncated hypergeometric series

Let $$n\ge 3$$ be an integer and p a prime with $$p\equiv 1\pmod {n}$$ . In this paper, we show that $$\begin{aligned} {}_nF_{n-1}\bigg [\begin{array}{llll} \frac{n-1}{n}&{}\frac{n-1}{n}&{}\ldots &{}\frac{n-1}{n}\\ &{}1&{}\ldots &{}1\end{array}\bigg | \, 1\bigg ]_{p-1}\equiv -\Gamma _p\bigg (\frac{1}{n}\bigg )^n\pmod {p^3}, \end{aligned}$$ where the truncated hypergeometric series 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)} $$\Gamma _p$$ denotes p-adic Gamma function. This confirms conjecture of Deines et al. (Hypergeometric Series, Truncated Hypergeometric Gaussian Functions, Directions in Number Theory, vol. 3, pp. 125–159. Assoc.WomenMath. Ser., Springer, New York, 2016). Furthermore, under same assumptions, also prove 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 ]_{p-1} \equiv _p\left( \frac{1}{n}\right) ^n\pmod which solves another [5].