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On two conjectures of Sun concerning Apéry-like series

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

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Steven Charlton

Li Lai

Ce Xu

Jianqiang Zhao


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.


Charlton, S., Gangl, H., Lai, L., Xu, C., & Zhao, J. (in press). On two conjectures of Sun concerning Apéry-like series. Forum Mathematicum,

Journal Article Type Article
Online Publication Date Feb 4, 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


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