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

TL;DR

The paper advances Ramanujan-type supercongruences by deriving $p^4$-level refinements of Van Hamme's (E.2) and (F.2) cases, as well as two Swisher congruences, using a generalized Wilf–Zeilberger (WZ) framework. The refinements connect truncated hypergeometric sums to Euler polynomials via the term $E_{p-3}(\alpha)$, providing a unified $p$-adic formula across cases and parameter choices. It builds on prior $p^4$ results (e.g., Sun) and introduces a robust WZ-based approach with case analyses, culminating in a detailed congruence modulo $p^4$. The work also proposes a $q$-analogue conjecture to spur further research in $q$-congruences, bridging classical supercongruences and their $q$-deformations.

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.

Theorems & Definitions (20)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Lemma 2.1: Lehmer Lehmer
• Lemma 2.2: MOS
• Lemma 2.3
• proof
• Lemma 2.4