Extension of a conjectural supercongruence of (G.3) of Swisher using Zeilberger's algorithm

Abstract

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.