论文标题
有关截断超几何序列的划分结果
Divisibility results concerning truncated hypergeometric series
论文作者
论文摘要
在本文中，使用众所周知的Karlsson-Minton公式，我们主要建立了两个有关截短的超几何序列的划分结果。令$ n> 2 $和$ q> 0 $为整数，$ 2 \中间n $或$ 2 \ nmid Q $。我们表明$$ \ sum_ {k = 0}^{p-1} \ frac {（q- \ frac {p} {n} {n}）_ k^n} {（1）_k^n} \ equiv0 $$ p^n \ sum_ {k = 0}^{p-1} \ frac {（1）_k^n} {（\ frac {p} {n} {n} -q+2）_k^n} \ equiv0 \ equiv0 \ equiv0 \ pmod { $（x）_k $表示由$$（x）_k = \ begin {cases} 1，\ quad＆k = 0，\\ x（x+1）\ cdots（x+k-1），\ quad＆k> 0. \ 0. \ end eNd {cases} $ n \ n \ geq4 $均匀。然后，对于任何Prime $ p $，带有$ p \ equiv-1 \ pmod {n} $，上面的第一个一致性意味着$$ \ sum_ {k = 0}^{p-1} \ frac {（\ frac {1} {1} {n} {n} {n} {n} {n} {n}）_ k^n} {1） $$这证实了最近对郭的猜想。
In this paper, using the well-known Karlsson-Minton formula, we mainly establish two divisibility results concerning truncated hypergeometric series. Let $n>2$ and $q>0$ be integers with $2\mid n$ or $2\nmid q$. We show that $$\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.
