# $p$-adic analogues of two ${}_4F_3$ hypergeometric identities and their
applications

Research paper by **Chen Wang, Zhi-Wei Sun**

Indexed on: **16 Oct '19**Published on: **15 Oct '19**Published in: **arXiv - Mathematics - Number Theory**

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#### Abstract

In this paper, we mainly study the congruence properties of the following
truncated hypergeometric series $$
{}_4F_3\bigg[\begin{matrix}\alpha&1+\frac{\alpha}{2}&\alpha&\alpha\\
&\frac{\alpha}{2}&1&1\end{matrix}\bigg|\ \lambda\bigg]_{p-1}, $$ where $p$ is
an odd prime, $\lambda=\pm1$ and $\alpha\in\mathbb{Z}_p^{\times}$. For example,
for $\lambda=-1$ we obtain $$
{}_4F_3\bigg[\begin{matrix}\alpha&1+\frac{\alpha}{2}&\alpha&\alpha\\
&\frac{\alpha}{2}&1&1\end{matrix}\bigg|-1\bigg]_{p-1}\equiv\frac{\alpha+\langle-\alpha\rangle_p}{\Gamma_p(1+\alpha)\Gamma_p(1-\alpha)}\pmod{p^3},
$$ where $\langle x\rangle_p$ is the least nonnegative residue of $x$ modulo
$p$ and $\Gamma_p$ is the well-known $p$-adic gamma function. As applications,
we confirm several conjectures posed by Sun recently; for example, we determine
$$\sum_{k=0}^{p-1}(-1)^k(2k+1)\sum_{j=0}^k\binom {-x}j^3\binom{x-1}{k-j}^3$$
modulo $p^2$ for any prime $p>3$ and $p$-adic integer $x$.