Supercongruences concerning truncated hypergeometric series

Abstract

Let $n\ge 3$ be an integer and p be 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

$$\begin{aligned} {}_nF_{n-1}\bigg [\begin{array}{llll} x_1&{}x_2&{}\ldots &{}x_n\\ &{}y_1&{}\cdots &{}y_{n-1}\end{array}\bigg | \, z\bigg ]_m=\sum _{k=0}^{m}\frac{z^k}{k!}\prod _{j=0}^{k-1}\frac{(x_1+j)\cdots (x_{n}+j)}{(y_1+j)\cdots (y_{n-1}+j)} \end{aligned}$$

and $\Gamma _p$ 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

$$\begin{aligned} p^n\cdot {}_{n+1}F_{n}\bigg [\begin{matrix} 1&{}1&{}\ldots &{}1\\ &{}\frac{n+1}{n}&{}\ldots &{}\frac{n+1}{n}\end{matrix}\bigg | \, 1\bigg ]_{p-1} \equiv -\Gamma _p\left( \frac{1}{n}\right) ^n\pmod {p^3}, \end{aligned}$$

which solves another conjecture in [5].