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

Research paper by Chen Wang, Zhi-Wei Sun

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



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 $x$.