Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

EULER SUMS OF GENERALIZED HYPERHARMONIC NUMBERS

  • Xu, Ce (School of Mathematical Sciences Xiamen University)
  • Received : 2017.10.01
  • Accepted : 2018.01.30
  • Published : 2018.09.01
Download PDF
⟨ Previous Next ⟩

Abstract

The generalized hyperharmonic numbers $h^{(m)}_n(k)$ are defined by means of the multiple harmonic numbers. We show that the hyperharmonic numbers $h^{(m)}_n(k)$ satisfy certain recurrence relation which allow us to write them in terms of classical harmonic numbers. Moreover, we prove that the Euler-type sums with hyperharmonic numbers: $$S(k,m;p):=\sum\limits_{n=1}^{{\infty}}\frac{h^{(m)}_n(k)}{n^p}(p{\geq}m+1,\;k=1,2,3)$$ can be expressed as a rational linear combination of products of Riemann zeta values and harmonic numbers. This is an extension of the results of Dil [10] and $Mez{\ddot{o}}$ [19]. Some interesting new consequences and illustrative examples are considered.

Keywords

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. A. T. Benjamin, D. Gaebler, and R. Gaebler, A combinatorial approach to hyperharmonic numbers, Integers 3 (2003), A15, 9 pp.
  3. D. Borwein, J. M. Borwein, and R. Girgensohn, Explicit evaluation of Euler sums, Proc. Edinburgh Math. Soc. (2) 38 (1995), no. 2, 277-294. https://doi.org/10.1017/S0013091500019088
  4. J. Borwein, P. Borwein, R. Girgensohn, and S. Parnes, Making sense of experimental mathematics, Math. Intelligencer 18 (1996), no. 4, 12-18. https://doi.org/10.1007/BF03027288
  5. 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
  6. J. M. Borwein and R. Girgensohn, Evaluation of triple Euler sums, Electron. J. Combin. 3 (1996), no. 1, Research Paper 23, approx. 27 pp.
  7. J. M. Borwein, I. J. Zucker, and J. Boersma, The evaluation of character Euler double sums, Ramanujan J. 15 (2008), no. 3, 377-405. https://doi.org/10.1007/s11139-007-9083-z
  8. L. Comtet, Advanced Combinatorics, revised and enlarged edition, D. Reidel Publishing Co., Dordrecht, 1974.
  9. J. H. Conway and R. K. Guy, The Book of Numbers, Copernicus, New York, 1996.
  10. A. Dil and K. N. Boyadzhiev, Euler sums of hyperharmonic numbers, J. Number Theory 147 (2015), 490-498. https://doi.org/10.1016/j.jnt.2014.07.018
  11. A. Dil and V. Kurt, Polynomials related to harmonic numbers and evaluation of harmonic number series II, Appl. Anal. Discrete Math. 5 (2011), no. 2, 212-229. https://doi.org/10.2298/AADM110615015D
  12. A. Dil and I. Mezo, A symmetric algorithm for hyperharmonic and Fibonacci numbers, Appl. Math. Comput. 206 (2008), no. 2, 942-951. https://doi.org/10.1016/j.amc.2008.10.013
  13. 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
  14. P. Freitas, Integrals of polylogarithmic functions, recurrence relations, and associated Euler sums, Math. Comp. 74 (2005), no. 251, 1425-1440. https://doi.org/10.1090/S0025-5718-05-01747-3
  15. Kh. Hessami Pilehrood, T. Hessami Pilehrood, and R. Tauraso, New properties of multiple harmonic sums modulo p and p-analogues of Leshchiner's series, Trans. Amer. Math. Soc. 366 (2014), no. 6, 3131-3159. https://doi.org/10.1090/S0002-9947-2013-05980-6
  16. M. Kaneko and Y. Ohno, On a kind of duality of multiple zeta-star values, Int. J. Number Theory 6 (2010), no. 8, 1927-1932. https://doi.org/10.1142/S179304211000385X
  17. C. Markett, Triple sums and the Riemann zeta function, J. Number Theory 48 (1994), no. 2, 113-132. https://doi.org/10.1006/jnth.1994.1058
  18. I. Mezo, Nonlinear Euler sums, Pacific J. Math. 272 (2014), no. 1, 201-226. https://doi.org/10.2140/pjm.2014.272.201
  19. I. Mezo and A. Dil, Hyperharmonic series involving Hurwitz zeta function, J. Number Theory 130 (2010), no. 2, 360-369. https://doi.org/10.1016/j.jnt.2009.08.005
  20. A. Sofo, Quadratic alternating harmonic number sums, J. Number Theory 154 (2015), 144-159. https://doi.org/10.1016/j.jnt.2015.02.013
  21. 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
  22. 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
  23. C. Xu, M. Zhang, and W. Zhu, Some evaluation of harmonic number sums, Integral Transforms Spec. Funct. 27 (2016), no. 12, 937-955. https://doi.org/10.1080/10652469.2016.1231675
  24. C. Xu, M. Zhang, and W. Zhu, Some evaluation of q-analogues of Euler sums, Monatsh. Math. 182 (2017), no. 4, 957-975. https://doi.org/10.1007/s00605-016-0915-z
  25. D. Zagier, Values of zeta functions and their applications, in First European Congress of Mathematics, Vol. II (Paris, 1992), 497-512, Progr. Math., 120, Birkhauser, Basel, 1994.
  26. D. Zagier, Evaluation of the multiple zeta values ${\zeta}$(2, . . . , 2, 3, 2, . . . , 2), Ann. of Math. (2) 175 (2012), no. 2, 977-1000. https://doi.org/10.4007/annals.2012.175.2.11