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On two conjectured supercongruences involving truncated hypergeometric series

Research paper by Guo-Shuai Mao, Hao Pan

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



Abstract

In this paper, we prove two conjectured supercongruences of Sun. For example, let $p$ be an odd prime and $r\geq 1$. Suppose that $x$ is a $p$-adic integer with $x\equiv-2k\pmod p$ for some $1\leq k\leq (p+1)/(2r+1)$. Then we have $${}_{2r+1}F_{2r}\bigg[\begin{matrix}-x&-x&\ldots&-x\\ &1&\ldots&1\end{matrix}\bigg|\,1\bigg]_{p-1}\equiv0\pmod{p^2},$$ where $$ {}_{q+1}F_{q}\bigg[\begin{matrix}x_0&x_1&\ldots&x_{q}\\ &y_1&\ldots&y_q\end{matrix}\bigg|\,z\bigg]_{n}=\sum_{k=0}^n\frac{(x_0)_k(x_1)_k\cdots(x_q)_k}{(y_1)_k\cdot (y_q)_k}\cdot\frac{z^k}{k!}. $$