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Abstract
Let $p$ be an odd prime and let $n$ be a positive integer with $p\nmid n$. For any positive integer $r$ and $\lambda \in \{1, 2, 3\}$ with $p\nmid \lambda $, we have $$ \sum _{k=0}^{p^{r}n-1}\frac {\left (\frac 12\right )_k}{k!}\cdot \frac {4^k}{\lambda ^
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