The WZ pair and finite congruences

Source 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 $ζ(3)$. 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--Saalschütz identity, and classical congruences for harmonic sums.