Divisibility results concerning truncated hypergeometric series

In this paper, using the well-known Karlsson-Minton formula, we mainly establish two divisibility results concerning truncated hypergeometric series. Let $n>2$ and $q>0$ be integers with $2\mid n$ or $2\nmid q$. We show that $\sum_{k=0}^{p-1}\frac{(q-\frac{p}{n})_k^n}{(1)_k^n}\equiv0\pmod{p^3}$ and $p^n\sum_{k=0}^{p-1}\frac{(1)_k^n}{(\frac{p}{n}-q+2)_k^n}\equiv0\pmod{p^3}$ for any prime $p>\max\{n,(q-1)n+1\}$, where $(x)_k$ denotes the Pochhammer symbol defined by $(x)_k=\begin{cases}1,\quad &k=0,\\ x(x+1)\cdots(x+k-1),\quad &k>0.\end{cases}$ Let $n\geq4$ be an even integer. Then for any prime $p$ with $p\equiv-1\pmod{n}$, the first congruence above implies that $\sum_{k=0}^{p-1} \frac{(\frac{1}{n})_k^n}{(1)_k^n}\equiv0\pmod{p^3}.$ This confirms a recent conjecture of Guo.

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