Quadratic relations for a q-analogue of multiple zeta values

The Ramanujan Journal volume 27, pages15–28(2012)Cite this article

Abstract

We obtain a class of quadratic relations for a q-analogue of multiple zeta values (qMZV’s). In the limit q→1, it turns into Kawashima’s relation for multiple zeta values. As a corollary we find that qMZV’s satisfy the linear relation contained in Kawashima’s relation. In the proof we make use of a q-analogue of Newton series and Bradley’s duality formula for finite multiple harmonic q-series.

This is a preview of subscription content, access via your institution.

References

  1. 1.

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

    Article  MATH  MathSciNet  Google Scholar 

  2. 2.

    Bradley, D.M.: Duality for finite multiple harmonic q-series. Discrete Math. 300(1–3), 44–56 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  3. 3.

    Hoffman, M.E., Ohno, Y.: Relations of multiple zeta values and their algebraic expression. J. Algebra 262(2), 332–347 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  4. 4.

    Kawashima, G.: A class of relations among multiple zeta values. J. Number Theory 129(4), 755–788 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  5. 5.

    Kawashima, G.: A generalization of the duality for finite multiple harmonic q-series. Ramanujan J. 21(3), 335–347 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  6. 6.

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

    Article  MATH  MathSciNet  Google Scholar 

  7. 7.

    Ohno, Y., Wakabayashi, N.: Cyclic sum of multiple zeta values. Acta Arith. 123(3), 289–295 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  8. 8.

    Ohno, Y., Zagier, D.: Multiple zeta values of fixed weight, depth, and height. Indag. Math. 12(4), 483–487 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  9. 9.

    Okuda, J., Takeyama, Y.: On relations for the multiple q-zeta values. Ramanujan J. 14(3), 379–387 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  10. 10.

    Tanaka, T., Wakabayashi, N.: An algebraic proof of the cyclic sum formula for multiple zeta values. J. Algebra 323(3), 766–778 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  11. 11.

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

    Article  MATH  MathSciNet  Google Scholar 

  12. 12.

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

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Affiliations

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

    Yoshihiro Takeyama

Authors
  1. Yoshihiro Takeyama
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yoshihiro Takeyama.

Additional information

The research of the author is supported by Grant-in-Aid for Young Scientists (B) No. 20740088.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Takeyama, Y. Quadratic relations for a q-analogue of multiple zeta values. Ramanujan J 27, 15–28 (2012). https://doi.org/10.1007/s11139-011-9328-8

Download citation

Keywords

  • Multiple zeta value
  • q-Analogue
  • Newton series
  • Finite multiple harmonic q-series
  • Kawashima’s relation

Mathematics Subject Classification (2000)

  • 05A30
  • 11M32