Skip to main content
SpringerLink
Log in

Some evaluation of q-analogues of Euler sums

  • Published:
Monatshefte für Mathematik Aims and scope Submit manuscript

Abstract

In this paper, we discuss the analytic representations of q-Euler sums which involve q-harmonic numbers through q-polylogarithms, either linearly or nonlinearly, and give explicit formulae for several classes of q-Euler sums in terms of q-polylogarithms and q-special functions. Furthermore, we develop new closed form representations of sums of quadratic and cubic parametric q-Euler sums. Finally, we can find that the q-Euler sums are reducible to the classical Euler sums when q approaches 1.

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. Andrews, G.E., Askey, R., Roy, R.: Special Functions. University Press, Cambridge (2000)

    MATH  Google Scholar 

  2. Annaby, M.H., Mansour, Z.S.: q-Fractional Calculus and Equations. Springer, Heidelberg (2012)

    Book  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  4. Bailey, D.H., Borwein, J.M., Girgensohn, R.: Experimental evaluation of Euler sums. Exp Math. 3(1), 17–30 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bailey, D.H., Borwein, J.M., Crandall, R.E.: Computation and theory of extended Mordell–Tornheim–Witten sums. Math. Comp. 83(288), 1795–1821 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bangerezako, G.: Variational q-calculus. J. Math. Anal. 289, 650–665 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  7. Berndt, B.C.: Ramanujans Notebooks, Part I. Springer-Verlag, New York. (1985)

    Book  MATH  Google Scholar 

  8. Berndt, B.C.: Ramanujans Notebooks, Part II. Springer-Verlag, New York. (1989)

    Book  MATH  Google Scholar 

  9. Borwein, J.M., Bradley, D.M.: Thirty-two goldbach variations. Int. J. Num. Theory. 2(1), 65–103 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  10. Borwein, D., Borwein, J.M., Bradley, D.M.: Parametric Euler sum identities. J. Math. Anal. Appl. 316(1), 328–338 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  11. Borwein, D., Borwein, J.M., Girgensohn, R.: Explicit evaluation of Euler sums. Proc. Edinburgh Math. 38, 277–294 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  12. Borwein, J., Borwein, P., Girgensohn, R., Parnes, S.: Making sense of experimental mathematics. Math Intell. 18(4), 12–18 (1996)

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  14. Borwein, J.M., Zucker, I.J., Boersma, J.: The evaluation of character Euler double sums. Ramanujan J. 15(3), 377–405 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  15. Borwein, J.M., Girgensohn, R.: Evaluation of triple Euler sums. Electron. J. Combin 3, 2–7 (1996)

    MathSciNet  MATH  Google Scholar 

  16. Boyadzhiev, K.: Evaluation of Euler–Zagier sums. Internat. J. Math. Sci. 27(7), 407–412 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  17. Bradley, D.M.: A q-analog of Eulers decomposition formula for the double zeta function. Int J Math Math Sci. 21, 3453C–3458 (2005)

    Article  MathSciNet  Google Scholar 

  18. Eie, M., Chuan-Sheng, W.: Evaluations of some quadruple Euler sums of even weight. Functions et Approx. 46(1), 63–67 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  19. Flajolet, P., Salvy, B.: Euler sums and contour integral representations. Exp Math. 7(1), 15–35 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  20. Freitas, P.: Integrals of polylogarithmic functions, recurrence relations, and associated Euler sums. Math. Comput. 74(251), 1425–1440 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  21. Jackson, H.F.: q-Difference equations. Am. J. Math. 32, 305–314 (1910)

    Article  MATH  Google Scholar 

  22. Comtet, L.: Advanced combinatorics. D Reidel Publishing Company, Boston (1974)

    Book  MATH  Google Scholar 

  23. Lorente, A.S.: Some q-representations of the q-analogue of the Hurwitz zeta function. Lecturas Matemticas. 36(1), 13–20 (2015)

    MathSciNet  Google Scholar 

  24. Salem, A.: Two classes of bounds for the q-gamma and the q-digamma functions in terms of the q-zeta functions. Banach J Math Anal. 8(1), 109–117 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  25. Tomita, Y.: Hermite’s formulas for q-analogues of Hurwitz zeta functions. Funct. Approx. Comment. Math. 45(2), 289–301 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  26. Wakayama, M., Yamasaki, Y.: Integral Representations of q-analogues of the Hurwitz Zeta Function. Monatshefte Fr Mathematik 149(2), 141–154 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  27. Zhao, J.: q-Multiple zeta functions and q-multiple polylogarithms. Ramanujan J. 14(2), 189–221 (2003)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

The authors would like to thank the anonymous referees for their valuable suggestions for improving the original version of this paper.

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, Xiamen University, Xiamen, 361005, People’s Republic of China

    Ce Xu, Mingyu Zhang & Weixia Zhu

Authors
  1. Ce Xu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mingyu Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Weixia Zhu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ce Xu.

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

Xu, C., Zhang, M. & Zhu, W. Some evaluation of q-analogues of Euler sums. Monatsh Math 182, 957–975 (2017). https://doi.org/10.1007/s00605-016-0915-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00605-016-0915-z

Keywords

Mathematics Subject Classification

Search

Navigation