Derivations on the algebra of multiple harmonic q-series and their applications

The Ramanujan Journal volume 52, pages41–65(2020)Cite this article

Abstract

We introduce derivations on the algebra of multiple harmonic q-series and show that they generate linear relations among the q-series which contain the derivation relations for a q-analogue of multiple zeta values due to Bradley. As a byproduct, we obtain Ohno-type relations for finite multiple harmonic q-series at a root of unity.

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References

  1. 1.

    Bachmann, H., Kühn, U.: The algebra of generating functions for multiple divisor sums and applications to multiple zeta values. Ramanujan J. 40(3), 605–648 (2016)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Bachmann, H., Takeyama, Y., Tasaka, K.: Cyclotomic analogues of finite multiple zeta values. Compos. Math. 154(12), 2701–2721 (2018)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Bradley, D.M.: Multiple \(q\)-zeta values. J. Algebra 283(2), 752–798 (2005)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Castillo-Medina, J., Ebrahimi-Fard, K., Manchon, D.: Unfolding the double shuffle structure of \(q\)-multiple zeta values. Bull. Aust. Math. Soc. 91(3), 368–388 (2015)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Hoffman, M.E.: Multiple harmonic series. Pac. J. Math. 152, 275–290 (1992)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Hoffman, M.E., Ohno, Y.: Relations of multiple zeta values and their algebraic expression. J. Algebra 262, 332–347 (2003)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Ihara, K., Kaneko, M., Zagier, D.: Derivation and double shuffle relations for multiple zeta values. Compos. Math. 142(2), 307–338 (2006)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Kaneko, M.: Finite multiple zeta values, (in Japanese) Various aspects of multiple zeta values. RIMS Kôkyûroku Bessatsu B 68, 175–190 (2017)

    MATH  Google Scholar 

  9. 9.

    Kaneko, M., Kurokawa, N., Wakayama, M.: A variation of Euler’s approach to values of the Riemann zeta function. Kyushu J. Math. 57(1), 175–192 (2003)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Kaneko, M., Zagier, D.: Finite multiple zeta values, in preparation

  11. 11.

    Ohno, Y.: A generalization of the duality and sum formulas on the multiple zeta values. J. Number Theory 74(1), 39–43 (1999)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Ohno, Y., Okuda, J., Zudilin, W.: Cyclic \(q\)-MZSV sum. J. Number Theory 132(1), 144–155 (2012)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Okunkov, A.Y.: Hilbert schemes and multiple \(q\)-zeta values. Funct. Anal. Appl. 48(2), 138–144 (2014)

    MathSciNet  Article  Google Scholar 

  14. 14.

    Oyama, K.: Ohno-type relation for finite multiple zeta values. Kyushu J. Math. 72, 277–285 (2018)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Schlesinger K-G.: Some remarks on \(q\)-deformed multiple polylogarithms, arXiv:math/0111022

  16. 16.

    Seki, S., Yamamoto, S.: A new proof of the duality of multiple zeta values and its generalizations, arXiv:1806.04679

  17. 17.

    Seki, S., Yamamoto, S.: Ohno-type identities for multiple harmonic sums, arXiv:1806.04785

  18. 18.

    Takeyama, Y.: The algebra of a \(q\)-analogue of multiple harmonic series. SIGMA Symmetry Integr. Geom. Methods Appl. 9, 15 (2013). Paper 061

    MathSciNet  MATH  Google Scholar 

  19. 19.

    Zhao, J.: Multiple \(q\)-zeta functions and multiple \(q\)-polylogarithms. Ramanujan J. 14(2), 189–221 (2007)

    MathSciNet  Article  Google Scholar 

  20. 20.

    Zhao, J.: Uniform approach to double shuffle and duality relations of various \(q\)-analogs of multiple zeta values via Rota-Baxter algebras, arXiv:1412.8044

  21. 21.

    Zudilin, V.V.: Algebraic relations for multiple zeta values. Russ. Math. Surv. 58(1), 1–29 (2003)

    MathSciNet  Article  Google Scholar 

Download references

Acknowledgements

The author is deeply grateful to Kojiro Oyama for an explanation of [14]. He also wishes to express his gratitude to Professor Masanobu Kaneko for valuable information on [8].

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  1. Department of Mathematics, Faculty of Pure and Applied Sciences, University of Tsukuba, Tsukuba, Ibaraki, 305-8571, Japan

    Yoshihiro Takeyama

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  1. Yoshihiro Takeyama
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Correspondence to Yoshihiro Takeyama.

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The research of the author is supported by JSPS KAKENHI Grant Number 18K03233.

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Takeyama, Y. Derivations on the algebra of multiple harmonic q-series and their applications. Ramanujan J 52, 41–65 (2020). https://doi.org/10.1007/s11139-019-00139-y

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Keywords

  • Multiple harmonic q-series
  • Multiple zeta values
  • Roots of unity

Mathematics Subject Classification

  • 05A30
  • 11M32
  • 33E20