본문 바로가기
KERD
논문KCI 등재

Parametric Euler Sums of Harmonic Numbers

Ce Xu|Junjie Quan|Xiyu Wang|Xiaoxue Wei
대한수학회 (The Korean Mathematical Society)|2024.07|대한수학회보 (Bulletin of the KMS)|제61권 제4호|pp. 1033-1051|DOI 10.4134/BKMS.b230591

0

조회

0

다운로드

0

인용

0

참고문헌

키워드

contour integralharmonic numbersParametric Euler sumsresidue theorem
주제분류수학

초록

In this paper, we define a parametric variant of generalized Euler sums and construct contour integration to give some explicit evaluations of these parametric Euler sums. In particular, we establish several explicit formulas of (Hurwitz) zeta functions, linear and quadratic parametric Euler sums. Furthermore, we also give an explicit evaluation of alternating double zeta values $\ze(\overline{2j};2m+1)$ in terms of a combination of alternating Riemann zeta values by using the parametric Euler sums.

Abstract

In this paper, we define a parametric variant of generalized Euler sums and construct contour integration to give some explicit evaluations of these parametric Euler sums. In particular, we establish several explicit formulas of (Hurwitz) zeta functions, linear and quadratic parametric Euler sums. Furthermore, we also give an explicit evaluation of alternating double zeta values $\ze(\overline{2j};2m+1)$ in terms of a combination of alternating Riemann zeta values by using the parametric Euler sums.

참고문헌 (25)

  1. H. Alzer, "Four parametric linear Euler sums", J. Math. Anal. Appl, 484(1), pp.123661~ (2020)[DOI]
  2. D. H. Bailey, "Experimental evaluation of Euler sums", Experiment. Math, 3(1), pp.17~30 (1994)
  3. D. Borwein, "Parametric Euler sum identities", J. Math. Anal. Appl, 316(1), pp.328~338 (2006)[DOI]
  4. J. Choi, "Certain summation formulas involving harmonic numbers and generalized harmonic numbers", Appl. Math. Comput, 218(3), pp.734~740 (2011)[DOI]
  5. J. Choi, "Finite summation formulas involving binomial coefficients, harmonic numbers and generalized harmonic numbers", J. Inequal. Appl, pp.49~ (2013)[DOI]
  6. J. Choi, "Some summation formulas involving harmonic numbers and generalized harmonic numbers", Math. Comput. Modelling, 54(9-10), pp.2220~2234 (2011)[DOI]
  7. L. Euler, "Meditationes circa singulare serierum genus", Novi Comm. Acad. Sci. Petropol, 20, pp.140~186
  8. P. Flajolet, "Euler sums and contour integral representations", Experiment. Math, 7(1), pp.15~35 (1998)
  9. M. E. Hoffman, "Multiple harmonic series", Pacific J. Math, 152(2), pp.275~290 (1992)
  10. M. E. Hoffman, "An odd variant of multiple zeta values", Commun. Number Theory Phys, 13(3), pp.529~567 (2019)
  11. M. Kaneko, "On multiple zeta values of level two", Tsukuba J. Math, 44(2), pp.213~234 (2020)[DOI]
  12. F. Luo, "A note on Arakawa-Kaneko zeta values and Kaneko-Tsumura η-values", Bull. Malays. Math. Sci. Soc, 46(1), pp.21~ (2023)[DOI]
  13. I. Mezo, "Nonlinear Euler sums", Pacific J. Math, 272(1), pp.201~226 (2014)[DOI]
  14. X. Si, "Euler-type sums involving multiple harmonic sums and binomial coefficients", Open Math, 19(1), pp.1612~1619 (2021)[DOI]
  15. A. Sofo, "Quadratic alternating harmonic number sums", J. Number Theory, 154, pp.144~159 (2015)[DOI]
  16. A. Sofo, "Extension of the four Euler sums being linear with parameters and series involving the zeta functions", J. Math. Anal. Appl, 515(1), pp.126370~ (2022)[DOI]
  17. A. Sofo, "Identities for the harmonic numbers and binomial coefficients", Ramanujan J, 25(1), pp.93~113 (2011)[DOI]
  18. W. Wang, "Euler sums and Stirling sums", J. Number Theory, 185, pp.160~193 (2018)[DOI]
  19. C. Xu, "Some evaluation of parametric Euler sums", J. Math. Anal. Appl, 451(2), pp.954~975 (2017)[DOI]
  20. C. Xu, "Some evaluations of infinite series involving parametric harmonic numbers", Int. J. Number Theory, 15(7), pp.1531~1546 (2019)[DOI]
  21. C. Xu, "Explicit formulas of Euler sums via multiple zeta values", J. Symbolic Comput, 101, pp.109~127 (2020)[DOI]
  22. C. Xu, "Variants of multiple zeta values with even and odd summation indices", Math. Z, 300(3), pp.3109~3142 (2022)[DOI]
  23. H. Yuan, "Double shuffle relations of double zeta values and the double Eisenstein series at level N", J. Lond. Math. Soc, 92(3), pp.520~546 (2015)[DOI]
  24. D. Zagier, "Values of Zeta Functions and Their Applications, First European Congress of Mathematics, Vol. II", pp.497~512 (1994), Birkhauser
  25. J. Zhao, "Multiple Zeta Functions, Multiple Polylogarithms and Their Special Values" (2016), World Sci. Publ

관련 논문AI추천

Some results on parametric Euler sumsCe Xu대한수학회 · 대한수학회보2017 · 제54권 제4호Explicit evaluation of harmonic sumsCe Xu대한수학회 · 대한수학회논문집2018 · 제33권 제1호Evaluations of some quadratic Euler sumsCe Xu, Xin Si대한수학회 · 대한수학회보2020 · 제57권 제2호Some evaluations of infinite series involving Dirichlet type parametric harmonic numbersCe Xu, Hongyuan Rui, Xiaobin Yin대한수학회 · 대한수학회보2024 · 제61권 제3호SOME RELATIONS ON PARAMETRIC LINEAR EULER SUMSCe Xu, Jianing Zhou, Weiguo Lu대한수학회 · 대한수학회보2023 · 제60권 제4호Special values and integral representations for the Hurwitz-type Euler zeta functions김대열, 김민수, Su Hu대한수학회 · 대한수학회지2018 · 제55권 제1호

이 간행물의 다른 논문 (16건)

Schatten classes of composition operators on Dirichlet Type Spaces with Superharmonic weightsZuoling Liupp. 875-895The Chern Sectional Curvature of a Hermitian ManifoldPandeng Cao, Hongjun Lipp. 897-906A note on representation numbers of quadratic forms modulo prime powersRan Xiongpp. 907-915Convergence of the Euler-Maruyama method for stochastic differential equations driven by $G$-Brownian motionCunxia Liu, Wen Lupp. 917-932Rigidity results for compact V-static spaceBotao Wang, Guangyue Huangpp. 933-948
원문 보기