Streamlined WZ method proofs of Van Hamme supercongruences
Andres Valloud
TL;DR
The paper develops a systematic WZ-based framework to prove Van Hamme supercongruences by constructing generalized WZ pairs and employing a degree-elimination strategy that yields linear telescoping via a cleverly chosen hypergeometric $q(k)$. By coupling this with the Long–Ramakrishna $p$-adic gamma approximations, the author obtains uniform proofs for several Van Hamme congruences, extends known mod-$p$ powers for select cases, and demonstrates that even $(I.2)$ falls under Gosper’s algorithm when viewed through the WZ lens. A key innovation is the concept of a WZ device, which mechanically produces a linear $ ext{Delta}$ and enables degree-collapse across multiple cases, thereby broadening the applicability of the WZ method to $p$-adic supercongruences. Collectively, these methods unify and streamline proofs for $B.2, C.2, D.2, E.2, F.2, G.2, H.2$, with additional extensions and special-case interpretations, and offer a practical blueprint for tackling analogous congruences using WZ techniques.
Abstract
Using the WZ method to prove supercongruences critically depends on an inspired WZ pair choice. This paper demonstrates a procedure for finding WZ pair candidates to prove a given supercongruence. When suitable WZ pairs are thus obtained, coupling them with the $p$-adic approximation of $Γ_p$ by Long and Ramakrishna enables uniform proofs for the Van Hamme supercongruences B.2, C.2, D.2, E.2, F.2, G.2, and H.2. This approach also yields the known extensions of G.2 modulo $p^4$, and of H.2 modulo $p^3$ when $p$ is $3$ modulo $4$. Finally, the Van Hamme supercongruence I.2 is shown to be a special case of the WZ method where Gosper's algorithm itself succeeds.