On two conjectures of Sun concerning Apéry-like series-Reference-Cited by-同舟云学术

On two conjectures of Sun concerning Apéry-like series

Published:2023-02-04 Issue:0 Volume:0 Page:
ISSN:0933-7741
Container-title:Forum Mathematicum
language:en
Short-container-title:

Author:

Charlton Steven¹^ORCID,Gangl Herbert²^ORCID,Lai Li³^ORCID,Xu Ce⁴^ORCID,Zhao Jianqiang⁵^ORCID

Affiliation:

1. Fachbereich Mathematik (AZ) , Universität Hamburg , Bundesstraße 55, 20146 Hamburg , Germany

2. Department of Mathematical Sciences , Durham University , Durham DH1 3LE , United Kingdom

3. Department of Mathematical Sciences , Tsinghua University , Beijing 100084 , P. R. China

4. School of Mathematics and Statistics , Anhui Normal University , Wuhu 241002 , P. R. China

5. Department of Mathematics , The Bishop’s School , La Jolla , CA 92037 , USA

Abstract

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.

Publisher

Walter de Gruyter GmbH

Subject

Applied Mathematics,General Mathematics

Link

https://www.degruyter.com/document/doi/10.1515/forum-2022-0325/pdf

Reference17 articles.

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2. K. C. Au, Evaluation of one-dimensional polylogarithmic integral, with applications to infinite series, preprint (2020), https://arxiv.org/abs/2007.03957.

3. D. Borwein, J. M. Borwein, A. Straub and J. Wan, Log-sine evaluations of Mahler measures, II, Integers 12 (2012), no. 6, 1179–1212.

4. J. M. Borwein and A. Straub, Mahler measures, short walks and log-sine integrals, Theoret. Comput. Sci. 479 (2013), 4–21.

5. J. M. Campbell, P. Levrie, C. Xu and J. Zhao, On a problem involving the squares of odd harmonic numbers, preprint (2022), https://arxiv.org/abs/2206.05026.

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