Skip to main content
SpringerLink
Log in

Multiple zeta star values on 3–2–1 indices

  • Published:
The Ramanujan Journal Aims and scope Submit manuscript

Abstract

In 2008, Muneta found explicit evaluation of the multiple zeta star value \(\zeta ^\star (\{3, 1\}^d)\), and in 2013, Yamamoto proved a sum formula for multiple zeta star values on 3–2–1 indices. In this paper, we provide another way of deriving the formulas mentioned above. It is based on our previous work on generating functions for multiple zeta star values and also on constructions of generating functions for restricted sums of alternating Euler sums. As a result, the formulas obtained are simpler and computationally more effective than the known ones. Moreover, we give explicit evaluations of \(\zeta ^\star (\{\{2\}^m, 3, \{2\}^m, 1\bigr \}^d)\) and \(\zeta ^\star (\{\{2\}^m, 3, \{2\}^m, 1\}^d, \{2\}^{m+1})\), which are new and have not appeared in the literature before.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Borwein, J.M., Bradley, D.M., Broadhurst, D.J.: Evaluations of \(k\)-fold Euler/Zagier sums: a compendium of results for arbitrary \(k\). Electron. J. Comb. 4(2), R5 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  2. Borwein, J.M., Bradley, D.M., Broadhurst, D.J., Lisoněk, P.: Combinatorial aspects of multiple zeta values. Electron. J. Comb. 5, R38 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  3. Borwein, J.M., Bradley, D.M., Broadhurst, D.J., Lisoněk, P.: Special values of multiple polylogarithms. Trans. Am. Math. Soc. 353(3), 907–941 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bowman, D., Bradley, D.M.: The algebra and combinatorics of shuffles and multiple zeta values. J. Comb. Theory Ser. A 97, 43–61 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  5. Charlton, S.: \(\zeta (\{\{2\}^m, 1, \{2\}^m, 3\}^n, \{2\}^m)/\pi ^{4n+2m(2n+1)}\) is rational. J. Number Theory 148, 463–477 (2015)

    Article  MathSciNet  Google Scholar 

  6. Charlton, S.: Analogues of cyclic insertion type identities for multiple zeta star values. Kyushu J. Math. 74, 337–352 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  7. Chen, K.-W., Chung, C.-L.: Some relations of multiple zeta values with even arguments. Mediterr. J. Math. 14, 110–122 (2017)

    Article  MATH  Google Scholar 

  8. Chen, K.-W., Chung, C.-L., Eie, M.: Sum formulas of multiple zeta values with arguments multiples of a common positive integer. J. Number Theory 177, 479–496 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  9. Coppo, M.-A., Candelpergher, B.: Inverse binomial series and values of Arakawa–Kaneko zeta functions. J. Number Theory 150, 98–119 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  10. Hessami Pilehrood, Kh., Hessami Pilehrood, T.: Generating functions for multiple zeta star values. J. Théor. Nombres Bordeaux 31, 343–360 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  11. Hoffman, M.E.: On multiple zeta values of even arguments. Int. J. Number Theory 13, 705–716 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  12. Imatomi, K., Tanaka, T., Tasaka, K., Wakabayashi, N.: On some combinations of multiple zeta star values. Nat. Sci. Ser. 42, 1–20 (2013)

  13. Kondo, H., Saito, S., Tanaka, T.: The Bowman–Bradley theorem for multiple zeta-star values. J. Number Theory 132, 1984–2002 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  14. Muneta, S.: On some explicit evaluations of multiple zeta-star values. J. Number Theory 128, 2538–2548 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  15. Yamamoto, S.: Explicit evaluation of certain sums of multiple zeta-star values. Funct. Approx. Comment. Math. 49, 283–289 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  16. Zagier, D.: Values of Zeta Functions and their Applications, First European Congress of Mathematics, Vol. II (Paris, 1992), Prog. Math., vol. 120, pp. 497–512. Birkhäuser, Basel-Boston (1994)

  17. Zhao, J.: Restricted sum formula of alternating Euler sums. Ramanujan J. 36, 375–401 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  18. Zhao, J.: Identity families of multiple harmonic sums and multiple zeta star values. J. Math. Soc. Jpn. 68, 1669–1694 (2016)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors would like to thank the anonymous referee for careful reading of the manuscript and useful comments.

Author information

Authors and Affiliations

  1. The Fields Institute for Research in Mathematical Sciences, 222 College St, Toronto, ON, M5T 3J1, Canada

    Kh. Hessami Pilehrood & T. Hessami Pilehrood

  2. Toronto Metropolitan University, 350 Victoria Street, Toronto, ON, M5T 3J1, Canada

    T. Hessami Pilehrood

Authors
  1. Kh. Hessami Pilehrood
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. T. Hessami Pilehrood
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to T. Hessami Pilehrood.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Hessami Pilehrood, K., Pilehrood, T.H. Multiple zeta star values on 3–2–1 indices. Ramanujan J 60, 259–285 (2023). https://doi.org/10.1007/s11139-022-00642-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11139-022-00642-9

Keywords

Mathematics Subject Classification

Search

Navigation