Supercongruences concerning truncated hypergeometric series

Abstract

Let $n\ge 3$ be an integer and $p$ be a prime with $p\equiv 1(\mathrm{mod}n)$. In this paper, we show that $${}_n{F}_{n-1}\left[\begin{array}{cccc}\frac{n-1}{n}& \frac{n-1}{n}& \dots & \frac{n-1}{n}\\ & 1& \dots & 1\end{array}\right|1{]}_{p-1}\equiv -{\Gamma }_p(\frac{1}{n}{)}^n(\mathrm{mod}{p}^3),$$ where the truncated hypergeometric series $${}_n{F}_{n-1}\left[\begin{array}{cccc}{x}_1& x_2& \dots & x_n\\ & y_1& \cdots & y_{n-1}\end{array}\right|z{]}_m=\sum _{k=0}^m\frac{z^k}{k!}\prod _{j=0}^{k-1}\frac{(x_1+j)\cdots (x_n+j)}{(y_1+j)\cdots (y_{n-1}+j)}$$ 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 $$p^n\cdot {}_{n+1}{F}_n\left[\begin{array}{cccc}1& 1& \dots & 1\\ & \frac{n+1}{n}& \dots & \frac{n+1}{n}\end{array}\right|1{]}_{p-1}\equiv -{\Gamma }_p(\frac{1}{n}{)}^n(\mathrm{mod}{p}^3),$$ which solves another conjecture of Deines, Fuselier, Long, Swisher and Tu.

This is a preliminary manuscript. Theorem 1.2 is added

References in corpus (2)

• A modular supercongruence for ${}_6{F}_5$: an Apéry-like story
• $p$-adic analogues of hypergeometric identities

Cited by in corpus (4)

• A parametric congruence motivated by Orr's identity
• Divisibility results concerning truncated hypergeometric series
• A further $q$-analogue of Van Hamme's (H.2) supercongruence for $p\equiv 1(\mathrm{mod}4)$
• $q$-Supercongruences from the $q$-Saalschütz identity