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.

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References in corpus (1)

• Ramanujan-type supercongruences