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

TL;DR

This paper proves a supercongruence conjecture related to a specific sum involving factorials and Pochhammer symbols, using the WZ-method and p-adic gamma functions, marking a novel approach for sums truncated at certain indices.

Contribution

It introduces a new supercongruence relation for sums truncated at rac{p^r-3}{4} when p^r ≡ -1 mod 4, using the WZ-method and p-adic gamma functions, and proves a conjecture of Swisher.

Findings

Established a supercongruence relation for S((p^r-3)/4).

Connected the sum to p-adic gamma function values.

First proof of supercongruences for sums truncated at (p^r-(d-1))/d when p^r ≡ -1 mod d.

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 $d)$.