Streamlined WZ method proofs of Van Hamme supercongruences

TL;DR

This paper introduces a systematic procedure for finding WZ pairs to prove Van Hamme supercongruences, simplifying and unifying their proofs using the WZ method and $p$-adic gamma function approximations.

Contribution

It provides a new method for selecting WZ pairs to prove supercongruences, enabling uniform proofs and extending known results.

Findings

Unified proofs for multiple Van Hamme supercongruences.

Extension of G.2 and H.2 supercongruences to higher powers of p.

Identification of (I.2) as a special case where Gosper's algorithm applies.

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 $\Gamma_p$ 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.