Refinements of Van Hamme's (E.2) and (F.2) supercongruences and two supercongruences by Swisher

TL;DR

This paper generalizes several of Van Hamme's and Swisher's supercongruences to the modulus p^4 case using a more general WZ pair, linking results to Euler polynomials and proposing a related q-congruence conjecture.

Contribution

It introduces a new general WZ pair to extend Van Hamme's (E.2) and (F.2) supercongruences, as well as two by Swisher, to higher modulus p^4, connecting to Euler polynomials.

Findings

Generalized supercongruences to modulus p^4

Linked supercongruences to Euler polynomials

Proposed a conjecture on q-congruences

Abstract

In 1997, Van Hamme proposed 13 supercongruences on truncated hypergeometric series. Van Hamme's (B.2) supercongruence was first confirmed by Mortenson and received a WZ proof by Zudilin later. In 2012, using the WZ method again, Sun extended Van Hamme's (B.2) supercongruence to the modulus $p^4$ case, where $p$ is an odd prime. In this paper, by using a more general WZ pair, we generalize Hamme's (E.2) and (F.2) supercongruences, as well as two supercongruences by Swisher, to the modulus $p^4$ case. Our generalizations of these supercongruences are related to Euler polynomials. We also put forward a relevant conjecture on $q$-congruences for further study.