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Proving Two Conjectural Series for \(\zeta (7)\) and Discovering More Series for \(\zeta (7)\)

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Mathematical Aspects of Computer and Information Sciences (MACIS 2019)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 11989))

Abstract

We give a proof of two identities involving binomial sums at infinity conjectured by Zhi-Wei Sun. In order to prove these identities, we use a recently presented method i.e., we view the series as specializations of generating series and derive integral representations. Using substitutions, we express these integral representations in terms of cyclotomic harmonic polylogarithms. Finally, by applying known relations among the cyclotomic harmonic polylogarithms, we derive the results. These methods are implemented in the computer algebra package HarmonicSums.

J. Ablinger—This work was supported by the Austrian Science Fund (FWF) grant SFB F50 (F5009-N15) and has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 764850 “SAGEX”.

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Notes

  1. 1.

    The package HarmonicSums (Version 1.0 19/08/19) together with a Mathematica notebook containing the computations described here can be downloaded at https://risc.jku.at/sw/harmonicsums.

  2. 2.

    The Mathematica built-in differential equation solver was not sufficient to solve these differential equations. The implemented solver does not rely on the Mathematica built-in DSolve.

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Authors and Affiliations

  1. Research Institute for Symbolic Computation, Johannes Kepler University, Linz, Austria

    Jakob Ablinger

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  1. Jakob Ablinger
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Correspondence to Jakob Ablinger .

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Editors and Affiliations

  1. AIT Austrian Institute of Technology, Vienna, Austria

    Daniel Slamanig

  2. IMJ-PRG, Sorbonne University, Paris, France

    Elias Tsigaridas

  3. Institute of Information Technologies, Gebze Technical University, Gebze, Turkey

    Zafeirakis Zafeirakopoulos

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Ablinger, J. (2020). Proving Two Conjectural Series for \(\zeta (7)\) and Discovering More Series for \(\zeta (7)\). In: Slamanig, D., Tsigaridas, E., Zafeirakopoulos, Z. (eds) Mathematical Aspects of Computer and Information Sciences. MACIS 2019. Lecture Notes in Computer Science(), vol 11989. Springer, Cham. https://doi.org/10.1007/978-3-030-43120-4_5

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  • DOI: https://doi.org/10.1007/978-3-030-43120-4_5

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  • Print ISBN: 978-3-030-43119-8

  • Online ISBN: 978-3-030-43120-4

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