Streamlined WZ method proofs of Van Hamme supercongruences

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.