Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
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SOME RESULTS ON PARAMETRIC EULER SUMS

  • Xu, Ce (School of Mathematical Sciences Xiamen University)
  • Received : 2016.06.24
  • Accepted : 2017.02.02
  • Published : 2017.07.31
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Abstract

In this paper we present a new family of identities for parametric Euler sums which generalize a result of David Borwein et al. [2]. We then apply it to obtain a family of identities relating quadratic and cubic sums to linear sums and zeta values. Furthermore, we also evaluate several other series involving harmonic numbers and alternating harmonic numbers, and give explicit formulas.

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References

  1. D. H. Bailey, J. M. Borwein, and R. Girgensohn, Experimental evaluation of Euler sums, Experiment. Math. 3 (1994), no. 1, 17-30. https://doi.org/10.1080/10586458.1994.10504573
  2. D. Borwein, J. M. Borwein, and D. M. Bradley, Parametric Euler sum identities, J. Math. Anal. Appl. 316 (2006), no. 1, 328-338. https://doi.org/10.1016/j.jmaa.2005.04.040
  3. D. Borwein, J. M. Borwein, and R. Girgensohn, Explicit evaluation of Euler sums, Proc. Edinburgh Math. 38 (1995), no. 2, 277-294. https://doi.org/10.1017/S0013091500019088
  4. J. M. Borwein and D. M. Bradley, Thirty-two goldbach variations, Int. J. Number Theory 2 (2006), no. 1, 65-103. https://doi.org/10.1142/S1793042106000383
  5. J. M. Borwein, D. M. Bradley, and D. J. Brodhurst, Evaluations of k-fold Euler/Zagier sums: a compendium of results for arbitrary k, Electron. J. Combin. 4 (1997), no. 2, Research Paper 5, 21 pp.
  6. J. M. Borwein, D. M. Bradley, D. J. Broadhurst, and P. Lisonek, Special values of multiple polylogarithms, Trans. Amer. Math. Soc. 353 (2001), no. 3, 907-941. https://doi.org/10.1090/S0002-9947-00-02616-7
  7. J. M. Borwein and R. Girgensohn, Evaluation of triple Euler sums, Electron. J. Combin. 3 (1996), no. 1, Research Paper 23, 27 pp.
  8. K. Boyadzhiev, Evaluation of Euler-Zagier sums, Internat. J. Math. Sci. 27 (2001), no. 7, 407-412. https://doi.org/10.1155/S016117120100521X
  9. L. Comtet, Advanced Combinatorics, D. Reidel Publishing Company, Boston, 1974.
  10. M. Eie and C.-S. Wei, Evaluations of some quadruple Euler sums of even weight, Funct. Approx. Comment. Math. 46 (2012), no. 1, 63-67. https://doi.org/10.7169/facm/2012.46.1.5
  11. P. Flajolet and B. Salvy, Euler sums and contour integral representations, Experiment. Math. 7 (1998), no. 1, 15-35. https://doi.org/10.1080/10586458.1998.10504356
  12. P. Freitas, Integrals of Polylogarithmic Functions, Recurrence Relations, and Associated Euler Sums, Math. Compu. 74 (2005), no. 251, 1425-1440. https://doi.org/10.1090/S0025-5718-05-01747-3
  13. M. E. Hoffman, Multiple harmonic series, Pacific J. Math. 152 (1992), 275-290. https://doi.org/10.2140/pjm.1992.152.275
  14. I. Mezo, Nonlinear Euler sums, Pacific J. Math. 272 (2014), no. 1, 201-226. https://doi.org/10.2140/pjm.2014.272.201
  15. N. Nielsen, Handbuch der Theorie der Gammafunction and Theorie des Integralloga-rithmus and verwandter Transzendenten, Reprinted together as Die Gammafunction, Chelsea, New York, 1965.
  16. A. Sofo, Harmonic sums and integral representations, J. Appl. Anal. 16 (2010), no. 2, 265-277. https://doi.org/10.1515/JAA.2010.018
  17. A. Sofo, Quadratic alternating harmonic number sums, J. Number Theory 154 (2015), 144-159. https://doi.org/10.1016/j.jnt.2015.02.013
  18. A. Sofo and H. M. Srivastava, Identities for the harmonic numbers and binomial coefficients, Ramanujan J. 25 (2011), no. 1, 93-113. https://doi.org/10.1007/s11139-010-9228-3
  19. L. Tornheim, Harmonic double series, Amer. J. Math. 72 (1950), 303-314. https://doi.org/10.2307/2372034
  20. H. Tsumura, On some combinatorial relations for Tornheim's double series, Acta Arith. 105 (2002), no. 3, 239-252. https://doi.org/10.4064/aa105-3-3
  21. H. Tsumura, Evaluation formulas for Tornheim's type of alternating double series, Math. Comp. 73 (2004), no. 245, 251-258. https://doi.org/10.1090/S0025-5718-03-01572-2
  22. C. Xu and J. Cheng, Some results on Euler sums, Funct. Approx. Comment. Math. 54 (2016), no. 1, 25-37. https://doi.org/10.7169/facm/2016.54.1.3
  23. C. Xu, Y. Yan, and Z. Shi, Euler sums and integrals of polylogarithm functions, J. Number Theory 165 (2016), 84-108. https://doi.org/10.1016/j.jnt.2016.01.025
  24. C. Xu, Y. Yang, and J. Zhang, Explicit evaluation of quadratic Euler sums, Int. J. Number Theory 13 (2017), no. 3, 655-672. https://doi.org/10.1142/S1793042117500336
  25. C. Xu, M. Zhang, and W. Zhu, Some evaluation of q-analogues of Euler sums, Monatshefte Fur Mathematik 182 (2017), no. 4, 957-975. https://doi.org/10.1007/s00605-016-0915-z