论文标题
使用Zeilberger的算法扩展了Swisher（G.3）的猜想超轻
Extension of a conjectural supercongruence of (G.3) of Swisher using Zeilberger's algorithm
论文作者
论文摘要
使用Zeilberger的算法，我们在这里给出了超级美食$$的证明\ sum_ {n = 0}^{\ frac {p^r -3} {4}}}（8n+1）\ frac {\ left（\ frac {1} {4} {4} \ right）_nn^4} {（1） \ sum_ {n = 0}^{\ frac {p^{r-2} -3} {4}}}}（8n+1）\ frac {\ left（\ frac {1} {4} {4} {4} \ right）_n^4} {（ } p^{\ frac {3r-1} {2}}），对于任何奇数整数$ r> 3 $。这扩展了Swisher（G.3）（G.3）的第三个猜想的超级企业，比Swisher所期望的要高的素数更高。
Using Zeilberger's algorithm, we here give a proof of the supercongruence $$ \sum_{n=0}^{\frac{p^r-3}{4}}(8n+1)\frac{\left(\frac{1}{4}\right)_n^4}{(1)_n^4}\equiv -p^3 \sum_{n=0}^{\frac{p^{r-2}-3}{4}}(8n+1)\frac{\left(\frac{1}{4}\right)_n^4}{(1)_n^4} ~~(\text{mod }p^{\frac{3r-1}{2}}),$$ for any odd integer $r>3$. This extends the third conjectural supercongruence of (G.3) of Swisher to modulo higher prime powers than that expected by Swisher.
