PreprintCC BY 4.0arXiv
The WZ pair and finite congruences
Abstract
We construct a meromorphic Wilf--Zeilberger pair for the hypergeometric series for 3ln 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≥ 5, we prove that the lower truncation is congruent to qp(2)−1 modulo p, while the upper truncation is congruent to −7/16p2Bp−3 modulo p3. The proofs combine the WZ method with terminating hypergeometric identities, the Pfaff--Saalschütz identity, and classical congruences for harmonic sums.
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BibTeX
@misc{chen2026pair,
title = {{The WZ pair and finite congruences}},
author = {Xiang Chen and Junyan Liu and Hengfei Lu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.16844},
eprint = {2610.16844},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.16844},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The WZ pair and finite congruences
AU - Xiang Chen
AU - Junyan Liu
AU - Hengfei Lu
PY - 2026
DA - 2026/10/12
PB - arXiv
DO - 10.48550/arxiv.2610.16844
UR - https://arxiv.org/abs/2610.16844
AB - We construct a meromorphic Wilf--Zeilberger pair for the hypergeometric series for 3ln 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≥ 5, we prove that the lower truncation is congruent to q_p(2)−1 modulo p, while the upper truncation is congruent to −7/16p^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.
ER -