On the Atkin and Swinnerton-Dyer type congruences for some truncated hypergeometric ${}_1{F}_0$ series

Yong Zhang, Hao Pan

Let $p$ be an odd prime and let $n$ be a positive integer. For any positive integer $\alpha$ and $m\in \{1,2,3\}$, we have \begin{align*} \sum_{k=0}^{p^αn-1}\frac{(\frac12)_k}{k!}\cdot\f…