Proof of a supercongruence conjecture of (F.3) of Swisher using the WZ-method

Abstract

For a non-negative integer $m$, let $S(m)$ denote the sum given by $$S(m):=\sum _{n=0}^m\frac{(-1{)}^n(8n+1)}{n{!}^3}{\left(\frac{1}{4}\right)}_n^3.$$ Using the powerful WZ-method, for a prime $p\equiv 3$ $($mod $4)$ and an odd integer $r>1$, we here deduce a supercongruence relation for $S\left(\frac{p^r-3}{4}\right)$ in terms of values of $p$-adic gamma function. As a consequence, we prove one of the supercongruence conjectures of (F.3) posed by Swisher. This is the first attempt to prove supercongruences for a sum truncated at $\frac{p^r-(d-1)}{d}$ when $p^r\equiv -1$ $($mod $d)$.

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• Ramanujan-type supercongruences