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On q-analogues of quadratic Euler sums

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Abstract

In this paper we study q-analogues of Euler sums and present a new family of identities by using the method of Jackson q-integral representations of series. We then apply it to obtain a family of identities relating quadratic Euler sums to linear sums and q-polylogarithms. Furthermore, we use certain stuffle products to evaluate several q-series with q-harmonic numbers. Some interesting new results and illustrative examples are considered. Finally, if q tends to 1, we obtain some explicit relations for the classical Euler sums.

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References

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

    MathSciNet  MATH  Google Scholar 

  2. J. Blumlein, S. Kurth, Harmonic sums and Mellin transforms up to two loop order. Phys. Rev. D. 60, 014018 (1999)

    Google Scholar 

  3. D. Borwein, J.M. Borwein, R. Girgensohn, Explicit evaluation of Euler sums. Proc. Edinb. Math. Soc. 38, 277–294 (1995)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  6. M.W. Coffey, On one-dimensional digamma and polygamma series related to the evaluation of Feynman diagrams. J. Comput. Appl. Math. 183, 84–100 (2005)

    MathSciNet  MATH  Google Scholar 

  7. M.W. Coffey, On a three-dimensional symmetric Ising tetrahedron and contributions to the theory of the dilogarithm and Clausen functions. J. Math. Phys. 49(4), 542–555 (2008)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  9. L. Euler, 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)

    Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  11. K. Hessami Pilehrood, T. Hessami Pilehrood, On \(q\)-analogues of two-one formulas for multiple harmonic sums and multiple zeta star values. Monatsh. Math. 176, 275–291 (2015)

    MathSciNet  MATH  Google Scholar 

  12. K. Hessami Pilehrood, T. Hessami Pilehrood, 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)

    MathSciNet  MATH  Google Scholar 

  13. M.E. Hoffman, The algebra of multiple harmonic series. J. Algebra 194(2), 477–495 (1997)

    MathSciNet  MATH  Google Scholar 

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

  15. H.F. Jackson, \(q\)-difference equations. Am. J. Math. 32, 305–314 (1910)

    MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  17. Z. Li, On harmonic sums and alternating Euler sums. arXiv:1012.5192

  18. A.S. Lorente, Some \(q\)-representations of the \(q\)-analogue of the Hurwitz zeta function. Lect. Mat. 36(1), 13–20 (2015)

    MathSciNet  MATH  Google Scholar 

  19. S. Muneta, Algebraic setup of non-strict multiple zeta values. Acta Arith. 136(1), 7–18 (2009)

    MathSciNet  MATH  Google Scholar 

  20. A. Salem, 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)

    MathSciNet  MATH  Google Scholar 

  21. X. Si, Some results on \(q\)-harmonic number sums. Adv. Differ. Equ. (2018). https://doi.org/10.1186/s13662-018-1480-7

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  23. M. Wakayama, Y. Yamasaki, Integral representations of \(q\)-analogues of the Hurwitz zeta function. Monatsh. Math. 149(2), 141–154 (2006)

    MathSciNet  MATH  Google Scholar 

  24. W. Wang, Y. Lyu, Euler sums and Stirling sums. J. Number Theory 185, 160–193 (2018)

    MathSciNet  MATH  Google Scholar 

  25. C. Xu, Multiple zeta values and Euler sums. J. Number Theory 177, 443–478 (2017)

    MathSciNet  MATH  Google Scholar 

  26. C. Xu, Some evaluation of quadratic Euler sums. arXiv:1701.03724 [math.NT]

  27. C. Xu, Some evaluation of parametric Euler sums. J. Math. Anal. Appl. 451, 954–975 (2017)

    MathSciNet  MATH  Google Scholar 

  28. C. Xu, Z. Li, Tornheim type series and nonlinear Euler sums. J. Number Theory 174, 40–67 (2017)

    MathSciNet  MATH  Google Scholar 

  29. C. Xu, Y. Yan, Z. Shi, Euler sums and integrals of polylogarithm functions. J. Number Theory 165, 84–108 (2016)

    MathSciNet  MATH  Google Scholar 

  30. C. Xu, M. Zhang, W. Zhu, Some evaluation of \(q\)-analogues of Euler sums. Monatsh. Math. 182(4), 957–975 (2017)

    MathSciNet  MATH  Google Scholar 

  31. D. Zagier, Values of zeta functions and their applications, in First European Congress of Mathematics, Volume II, 120 (Birkhäuser, Boston, 1994), pp. 497–512

  32. D. Zagier, Evaluation of the multiple zeta values \(\zeta (2,\ldots,2,3,2,\ldots,2)\). Ann. Math. 2(2), 977–1000 (2012)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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Acknowledgements

Funding was provided by National Natural Science Foundation of China (Grant No. 11471245).

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Authors and Affiliations

  1. School of Mathematical Sciences, Tongji University, Shanghai, 200092, People’s Republic of China

    Zhonghua Li

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

    Ce Xu

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  1. Zhonghua Li
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  2. Ce Xu
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Correspondence to Ce Xu.

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The first author is supported by the National Natural Science Foundation of China (Grant No. 11471245) and the Natural Science Foundation of Shanghai (Grant No. 14ZR1443500). We thank the anonymous referee for suggestions which led to improvements in the exposition.

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Li, Z., Xu, C. On q-analogues of quadratic Euler sums. Period Math Hung 81, 1–19 (2020). https://doi.org/10.1007/s10998-020-00312-7

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