The WZ pair and finite congruences

Abstract

We construct a meromorphic Wilf--Zeilberger pair for the hypergeometric series for \(3\ln 2\), completing the WZ proof left open by Guillera and Rogers. The same pair yields a one-parameter identity and, after a half-integer shift and differentiation, an inverse-binomial series for \(\zeta(3)\). We also study the finite truncations of the logarithmic series. For every prime \(p\ge 5\), we prove that the lower truncation is congruent to \(q_p(2)-1\) modulo \(p\), while the upper truncation is congruent to \(-\frac{7}{16}p^2B_{p-3}\) modulo \(p^3\). The proofs combine the WZ method with terminating hypergeometric identities, the Pfaff--Saalsch\"utz identity, and classical congruences for harmonic sums.