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

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.

• The paper generalizes Van Hamme's and Swisher's supercongruences for truncated hypergeometric series, extending them to modulus p^4 using an advanced Wilf-Zeilberger pairing method.
• Theorems 1.1-1.4 provide concrete p^4 refinements to Van Hamme’s (E.2), (F.2) and two of Swisher’s supercongruences, incorporating Euler numbers E_{p-3} for odd primes p.
• The study enhances understanding of p-adic properties of hypergeometric series by linking these refined supercongruences to Euler polynomials, opening avenues for research in q-series and computational number theory.

Analysis of Refinements of Van Hamme's and Swisher's Supercongruences

The paper by Victor J. W. Guo and Chen Wang presents significant advancements in the understanding of supercongruences associated with truncated hypergeometric series. These supercongruences are extensions and refinements of earlier work by Van Hamme and Swisher, specifically targeting Van Hamme's conjectures (E.2) and (F.2), and two additional supercongruences proposed by Swisher.

The authors achieve their generalizations through the utilization of a more comprehensive Wilf-Zeilberger (WZ) pairing method. This approach allows the extension of these supercongruences to the modulus $p^4$, where $p$ is an odd prime. The linkage between these supercongruences and the Euler polynomials represents a novel integration of classical number theory and analysis with modern computational techniques, underlining the theoretical elegance and technical depth of the study.

Key Numerical Results and Claims

The main contributions of the paper are encapsulated in several new theorems that provide refinements to known supercongruences. Specifically, the paper delivers the following:

• Theorems 1.1 and 1.2 generalize Van Hamme’s (E.2) and (F.2) supercongruences to the modulus $p^4$, incorporating the Euler numbers $E_{p-3}$.
• Theorems 1.3 and 1.4 provide analogous modulus $p^4$ refinements to Swisher’s previously established supercongruences.

These theorems demonstrate that for specified conditions related to the prime $p$, the previously verified (mod $p^3$) congruences hold true with adjustments involving Euler numbers at the $p^4$ level.

Implications and Speculation

The implications of this study are both significant and multifaceted. Practically, these results enhance our understanding of p-adic properties of hypergeometric series and their arithmetic, enriching the toolkit available for computational number theorists. Moreover, the connection to Euler polynomials broadens the theoretical landscape for future explorations regarding q-series and other complex structures in number theory.

The groundwork laid by this study encourages further exploration in the field of q-congruences, as posited by the conjecture presented at the conclusion of the paper. This stimulates future research directions to bridge the gap between existing q-congruences and these refined supercongruences, potentially impacting the development of new algorithms in computational mathematics.

Conclusion

Guo and Wang's paper, through rigorous theoretical development and innovative use of WZ pairs, makes a substantial contribution to the field of number theory. Their work on refining Van Hamme's and Swisher's supercongruences not only advances existing computational techniques but also fosters a deeper understanding of modular arithmetic and hypergeometric identities. This scholarly endeavor sets a foundation for future research into more intricate structures and identities within the domain of p-adic numbers and Eulerian mathematics.