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arXiv · 2210.14704

On two conjectures of Sun concerning Ap\'ery-like series

Steven Charlton·Herbert Gangl·Li Lai·Ce Xu·Jianqiang Zhao
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Abstract

In this paper, we shall prove two conjectures of Z.-W. Sun concerning Ap\'ery-like series. One of the series is alternating whereas the other one is not. Our main strategy is to convert the series (resp.~the alternating series) to log-sine-cosine (resp.~log-sinh-cosh) integrals. Then we express all these integrals in terms of single-valued Bloch-Wigner-Ramakrishnan-Wojtkowiak-Zagier polylogarithms. The conjectures then follow from a few highly non-trivial functional equations of the polylogarithms of weight $3$ and $4$.

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Steven Charlton, Herbert Gangl, Li Lai, Ce Xu, Jianqiang Zhao. 2022-10-21. On two conjectures of Sun concerning Ap\'ery-like series. https://doi.org/10.1515/forum-2022-0325

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