Skip to main content
SpringerLink
Log in

On \(q\)-analogues of two-one formulas for multiple harmonic sums and multiple zeta star values

  • Published:
Monatshefte für Mathematik Aims and scope Submit manuscript

Abstract

Recently, the present authors jointly with Tauraso found a family of binomial identities for multiple harmonic sums (MHS) on strings \((\{2\}^a,c,\{2\}^b)\) that appeared to be useful for proving new congruences for MHS as well as new relations for multiple zeta values. Very recently, Zhao generalized this set of MHS identities to strings with repetitions of the above patterns and, as an application, proved the two-one formula for multiple zeta star values conjectured by Ohno and Zudilin. In this paper, we extend our approach to \(q\)-binomial identities and prove \(q\)-analogues of two-one formulas for multiple zeta star values.

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. Bachmann, H., Kühn, U.: The algebra of generating functions for multiple divisor sums and applications to multiple zeta values (preprint). arXiv:1309.3920v2 [math.NT]

  2. Bachmann, H., Kühn, U.: A short note on a conjecture of Okounkov about a \(q\)-analogue of multiple zeta values (preprint). arXiv:1407.6796v1 [math.NT]

  3. Borwein, J.M., Bradley, D.M.: Thirty-two Goldbach variations. Int. J. Number Theory 2(1), 65–103 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  4. Bowman, D., Bradley, D.M.: Multiple polylogarithms: a brief survey. In: \(q\)-Series with Applications to Combinatorics, Number Theory, and Physics (Urbana, IL, 2000), pp. 71–92. Contemporary Mathematics, vol. 291. American Mathematical Society, Providence (2001)

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

    Article  MATH  MathSciNet  Google Scholar 

  6. Bradley, D.M.: Duality for finite multiple harmonic \(q\)-series. Discret. Math. 300, 44–56 (2005)

    Article  MATH  Google Scholar 

  7. Bradley, D.M.: On the sum formula for multiple \(q\)-zeta values. Rocky Mt. J. Math. 37(5), 1427–1434 (2007)

    Article  MATH  Google Scholar 

  8. Castillo Medina, J., Ebrahimi-Fard, K., Manchon, D.: Unfolding the double shuffle structure of qMZVs (preprint). arXiv:1310.1330v4 [math.NT]

  9. Cherednik, I.: On \(q\)-analogues of Riemann’s zeta function. Sel. Math. (N.S.) 7(4), 447–491 (2001)

  10. Dilcher, K., Hessami Pilehrood, Kh, Hessami Pilehrood, T.: On \(q\)-analogues of double Euler sums. J. Math. Anal. Appl. 410(2), 979–988 (2014)

    Article  MathSciNet  Google Scholar 

  11. Euler, L.: Meditationes circa singulare serierum genus. Novi Comm. Acad. Sci. Petropol. 20, 140–186 (1775); reprinted. In: Opera Omnia, Ser. 1, vol. 15, Teubner, Berlin, 1927, 217–267

  12. Hessami Pilehrood, Kh, Hessami Pilehrood, T., Tauraso, R.: New properties of multiple harmonic sums modulo \(p\) and \(p\)-analogues of Leshchiner’s series. Trans. Am. Math. Soc. 366(6), 3131–3159 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  13. Hessami Pilehrood, Kh., Hessami Pilehrood, T., Zhao, J.: On \(q\)-analogs of some families of multiple harmonic sum and multiple zeta star value identities (preprint) arXiv:1307.7985 [math.NT]

  14. Hoffman, M.E.: Multiple harmonic series. Pac. J. Math. 152(2), 275–290 (1992)

    Article  MATH  Google Scholar 

  15. Hoffman, M.E.: Algebraic aspects of multiple zeta values. In: Aoki, T., Kanemitsu, S., Nakahara, M., Ohno, Y. (eds) Zeta Functions, Topology and Quantum Physics. Developments in Mathematics, vol. 14, pp. 51–73. Springer, New York (2005)

  16. Hoffman, M.E.: Multiple zeta values: from Euler to the present. In: MAA Sectional Meeting, Annapolis, Maryland, November 10 (2007). http://www.usna.edu/Users/math/meh

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

    Article  MATH  MathSciNet  Google Scholar 

  18. Jouhet, F., Mosaki, E.: Irrationalité aux entiers impairs positifs d’un \(q\)-analogue de la fonction zêta de Riemann. Int. J. Number Theory 6(5), 959–988 (2010)

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

    Article  MATH  MathSciNet  Google Scholar 

  20. Ohno, Y., Okuda, J.: On the sum formula for the \(q\)-analogue of non-strict multiple zeta values. Proc. Am. Math. Soc. 135(10), 3029–3037 (2007)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  23. Ohno, Y., Zudilin, W.: Zeta stars. Commun. Number Theory Phys. 2(2), 325–347 (2008)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  26. Postelmans, K., Van Assche, W.: Irrationality of \(\zeta _q(1)\) and \(\zeta _q(2)\). J. Number Theory 126, 119–154 (2007)

  27. Sorokin, V.N.: On Apéry theorem (Russian). Vestnik Moskov. Univ. Ser. I Mat. Mekh. (3), 48–53 (1998); English transl. in: Moscow Univ. Math. Bull. 53(3), 48–52 (1998)

  28. Sorokin, V.N.: Cyclic graphs and Apéry’s theorem (Russian). Uspekhi Mat. Nauk 57(3(345)), 99–134 (2002); English transl. in: Russian Math. Surv. 57(3), 535–571 (2002)

  29. Vasil’ev, D.V.: Some formulas for the Riemann zeta function at integer points (Russian). Vestnik Moskov. Univ. Ser. I Mat. Mekh. (1996), (1), 81–84; English transl. in: Moscow Univ. Math. Bull. 51(1), 41–43 (1996)

  30. Waldschmidt, M.: Valeurs zêta multiples. Une introduction. J. Théor. Nombres Bordeaux 12(2), 581–595 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  31. Zagier, D.: Values of zeta functions and their applications. In: First European Congress of Mathematics, vol. II (Paris, 1992), pp. 497–512. Progress Mathematics, vol. 120. Birkhäuser, Basel (1994)

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

    Article  MATH  MathSciNet  Google Scholar 

  33. Zhao, J.: Identity families of multiple harmonic sums and multiple zeta (star) values (preprint). arXiv:1303.2227v1 [math.NT]

  34. Zlobin, S.A.: Generating functions for the values of a multiple zeta function (Russian). Vestnik Moskov. Univ. Ser. I Mat. Mekh. 60(2), 55–59 (2005); English transl. in: Moscow Univ. Math. Bull. 60(2), 44–48 (2005)

  35. Zudilin, V.V.: Diophantine problems for \(q\)-zeta values (Russian). Mat. Zametki 72(6), 936–940 (2002); translation in. Math. Notes 72(5–6), 858–862 (2002)

  36. Zudilin, W.: Algebraic relations for multiple zeta values. Uspekhi Mat. Nauk 58(2), 3–32 (2001); English transl. in: Russian Math. Surveys 58(1), 1–29 (2001)

Download references

Acknowledgments

We would like to thank Wadim Zudilin for drawing our attention to Zhao’s paper [33]. We also thank the referees of the paper for careful reading and valuable remarks that helped us to improve the presentation of the paper.

Author information

Authors and Affiliations

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

    Khodabakhsh Hessami Pilehrood & Tatiana Hessami Pilehrood

Authors
  1. Khodabakhsh Hessami Pilehrood
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Tatiana Hessami Pilehrood
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Khodabakhsh Hessami Pilehrood.

Additional information

Communicated by A. Constantin.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Hessami Pilehrood, K., Hessami Pilehrood, T. On \(q\)-analogues of two-one formulas for multiple harmonic sums and multiple zeta star values. Monatsh Math 176, 275–291 (2015). https://doi.org/10.1007/s00605-014-0715-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00605-014-0715-2

Keywords

Mathematics Subject Classification

Search

Navigation