Advertisement

On some explicit evaluations of nonlinear Euler sums

  • Jichao Zhang
  • Ce XuEmail author
Article
  • 13 Downloads

Abstract

In this paper, we extend tools developed in [26] to study harmonic numbers sums. We use these objects to compute Euler-related sums and to explore their connections with classical Euler sums. We establish some new relations between Euler sums and related sums. The relations obtained allow us to find some nice closed-form representations of nonlinear Euler sums through zeta values and linear sums.

Keywords

Euler sums Tornheim type series harmonic numbers Riemann zeta function 

MSC

40A05 33B99 33E20 11M06 11M32 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

References

  1. 1.
    G.E. Andrews, R. Askey, and R. Roy, Special Functions, Cambridge Univ. Press, Cambridge, 2000.zbMATHGoogle Scholar
  2. 2.
    D.H. Bailey, J.M. Borwein, and R. Girgensohn, Experimental evaluation of Euler sums, Exp. Math., 3(1):17–30, 1994.MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    B.C. Berndt, Ramanujan’s Notebooks, Springer, New York, 1989.CrossRefzbMATHGoogle Scholar
  4. 4.
    J. Blümlein, D.J. Broadhurst, and J.A.M. Vermaseren, The multiple zeta value data mine, Comput. Phys. Commun., 181(3):582–625, 2010.MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    D. Borwein, J.M. Borwein, and R. Girgensohn, Explicit evaluation of Euler sums, Proc. Edinb. Math., Ser. II, 38(2):277–294, 1995.MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    J.M. Borwein, D.M. Bradley, and D.J. Broadhurst, Evaluations of k-fold Euler/Zagier sums: A compendium of results for arbitrary k, Electron. J. Combin., 4(2):R5, 1997.zbMATHGoogle Scholar
  7. 7.
    J.M. Borwein, D.M. Bradley, D.J. Broadhurst, and P. Lisonǒk, Special values of multiple polylogarithms, Trans. Am. Math. Soc., 353(3):907–941, 2001.MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    J.M. Borwein and R. Girgensohn, Evaluation of triple Euler sums, Electron. J. Comb., 3(1):R23, 1996.zbMATHGoogle Scholar
  9. 9.
    J.M. Borwein, I.J. Zucker, and J. Boersma, The evaluation of character Euler double sums, Ramanujan J., 15(3): 377–405, 2008.MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    R.E. Crandall and J.P. Buhler, On the evaluation of Euler sums, Exp. Math., 3(4):275–285, 1994.MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    P. Flajolet and B. Salvy, Euler sums and contour integral representations, Exp. Math., 7(1):15–35, 1998.MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    P. Freitas, Integrals of polylogarithmic functions, recurrence relations, and associated Euler sums, Math. Comput., 74(251):1425–1440, 2005.MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    M.E. Hoffman, Multiple harmonic series, Pacific J. Math., 152(2):275–290, 1992.MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    I. Mező, Nonlinear Euler sums, Pacific J. Math., 272(1):201–226, 2014.MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Kh. Pilehrood, T. Pilehrood, and R. Tauraso, New properties of multiple harmonic sums modulo p and p-analogues of Leshchiner’s series, Trans. Am. Math. Soc., 366(6):3131–3159, 2014.MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    A. Sofo and H.M. Srivastava, Identities for the harmonic numbers and binomial coefficients, Ramanujan J., 25(1): 93–113, 2011.MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    H.M. Srivastava and J. Choi, Zeta and q-Zeta Functions and Associated Series and Integrals, Elsevier, Amsterdam, 2012.zbMATHGoogle Scholar
  18. 18.
    L. Tornheim, Harmonic double series, Am. J. Math., 72(2):303–314, 1950.MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    W. Wang and Xu. C, Euler sums of weights 10 and 11, and some special types, 2017, available from:  https://doi.org/10.13140/RG.2.2.20636.08326/1.
  20. 20.
    W. Wang and Y. Lyu, Euler sums and Stirling sums, J. Number Theory, 185:160–193, 2018.MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    C. Xu, Multiple zeta values and Euler sums, J. Number Theory, 177:443–478, 2017.MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    C. Xu, Computation and theory of Euler sums of generalized hyperharmonic numbers, C. R., Math., Acad. Sci. Paris, 356(3):243–252, 2018.MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    C. Xu, Some evaluation of cubic Euler sums, J. Math. Anal. Appl., 466(1):789–805, 2018.MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    C. Xu, Evaluations of nonlinear Euler sums of weight ten, Appl. Math. Comput., 346:594–611, 2019.MathSciNetCrossRefGoogle Scholar
  25. 25.
    C. Xu and J. Cheng, Some results on Euler sums, Funct. Approximatio, Comment. Math., 54(1):25–37, 2016.MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    C. Xu and Zh. Li, Tornheim type series and nonlinear Euler sums, J. Number Theory, 174:40–67, 2017.MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    C. Xu andW. Wang, Explicit formulas of Euler sums via multiple zeta values, 2018, arXiv:1805.08056.Google Scholar
  28. 28.
    C. Xu, Y. Yan, and Z. Shi, Euler sums and integrals of polylogarithm functions, J. Number Theory, 165:84–108, 2016.MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    C. Xu, Y. Yang, and J. Zhang, Explicit evaluation of quadratic Euler sums, Int. J. Number Theory, 13(3):655–672, 2017.MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    D. Zagier, Values of zeta functions and their applications, in A. Joseph et al., First European Congress of Mathematics. Round Tables, Paris, July 6–10, 1992, Prog. Math., Vol. 121, Birkhäuser, Basel, 1994, pp. 497–512.Google Scholar
  31. 31.
    D. Zagier, Evaluation of the multiple zeta values ζ(2, . . . , 2, 3, 2, . . . , 2), Ann. Math., 2(2):977–1000, 2012.MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    J. Zhao, Multiple Zeta Functions, Multiple Polylogarithms and Their Special Values, Ser. Number Theory Appl., Vol. 12, World Scientific, Hackensack, NJ, 2016.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.School of ScienceHubei University of TechnologyWuhanChina
  2. 2.School of Mathematical SciencesXiamen UniversityXiamenChina

Personalised recommendations