On two conjectures of Sun concerning Apéry-like series

Charlton, Steven; Gangl, Herbert; Lai, Li; Xu, Ce; Zhao, Jianqiang

Abstract

We prove two conjectural identities of Z.-W. Sun concerning Apéry-like series. One of the series is alternating, whereas the other one is not. Our main strategy is to convert the series and the alternating series to log-sine-cosine and log-sinh-cosh integrals, respectively. Then we express all these integrals using single-valued Bloch–Wigner–Ramakrishnan–Wojtkowiak–Zagier polylogarithms. The conjectures then follow from a few rather non-trivial functional equations of those polylogarithms in weights 3 and 4.

Citation

Charlton, S., Gangl, H., Lai, L., Xu, C., & Zhao, J. (2023). On two conjectures of Sun concerning Apéry-like series. Forum Mathematicum, 35(6), 1533-1547. https://doi.org/10.1515/forum-2022-0325

• Journal Article Type: Article
• Online Publication Date: Feb 4, 2023
• Publication Date: Nov 1, 2023
• Deposit Date: Feb 8, 2023
• Publicly Available Date: Feb 8, 2023
• Journal: Forum Mathematicum
• Print ISSN: 0933-7741
• Electronic ISSN: 1435-5337
• Publisher: De Gruyter
• Peer Reviewed: Peer Reviewed
• Volume: 35
• Issue: 6
• Pages: 1533-1547
• DOI: https://doi.org/10.1515/forum-2022-0325
• Keywords: log-sine-cosine integrals, 11M32, Sun’s conjectures, Apéry-like series, colored multiple zeta values
• Public URL: https://durham-repository.worktribe.com/output/1180932