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EULER SUMS EVALUATABLE FROM INTEGRALS

  • Jung, Myung-Ho (Department of Mathematics College of Natural Sciences Pusan National University) ;
  • Cho, Young-Joon (Department of Mathematics Education Pusan National University) ;
  • Choi, June-Sang (Department of Mathematics College of Natural Sciences Dongguk University)
  • Published : 2004.07.01

Abstract

Ever since the time of Euler, the so-called Euler sums have been evaluated in many different ways. We give here a proof of the classical Euler sum by following Lewin's method. We also consider some related formulas involving Euler sums, which are evaluatable from some known definite integrals.

Keywords

References

  1. Analysis v.18 Some series of the Zeta and related functions V. S. Adamchik;H. M. Srivastava
  2. Experiment. Math. v.3 Experimental evaluation of Euler sums D. H. Bailey;J. M. Borwein;R. Girgensohn https://doi.org/10.1080/10586458.1994.10504573
  3. Ramanujan J. v.4 A new method for investigating Euler sums A. Basu;T. M. Apostol https://doi.org/10.1023/A:1009868016412
  4. Ramanujan's Notebooks B. C. Berndt
  5. Proc. Amer. Math. Soc. v.123 On an intriguing integral and some series related to ${\zeta}$ (4) D. Borwein;J. M. Borwein https://doi.org/10.2307/2160718
  6. Proc. Edinburgh Math. Soc. (Ser. 2) v.38 Explicit evalution of Eulersums D. Borwein;J. M. Borwein;R. Girgensohn https://doi.org/10.1017/S0013091500019088
  7. Electron. J. Combin. Research Paper 23 v.3 no.1 Evaluation of triple Euler sums J. M. Borwein;R. Girgensohn
  8. Amer. Math. Monthly v.108 Euler's formula for Zeta function convolutions P. Bracken
  9. Experiment. Math. v.3 On the evaluation of Euler sums R. E. Crandall;J. P. Buhler https://doi.org/10.1080/10586458.1994.10504297
  10. J. Comput. Appl. Math. v.37 On some series containing ${\psi}$(x)-${\psi}$(y) and (${\psi}$(x)-${\psi}$(y))$^2$ for certain values of x and y P. J. de Doelder https://doi.org/10.1016/0377-0427(91)90112-W
  11. Experiment. Math. v.7 Euler sums and contour integral representations P. Flajolet;B. Salvy https://doi.org/10.1080/10586458.1998.10504356
  12. Pacific J. Math. v.152 Multiple harmonic series M. E. Hoffman https://doi.org/10.2140/pjm.1992.152.275
  13. Polylogarithms and Associated Functions L. Lewin
  14. J. Number Theory v.48 Triple sums and the Riemann Zeta function C. Markett https://doi.org/10.1006/jnth.1994.1058
  15. Die Gammafunktion N. Nielsen
  16. Problems and Theorems in Analysis v.I G. Polya;G. Szego
  17. Appl. Math. Comput. v.131 Some classes of infinite series associated with the Riemann Zeta and Polygamma functions and generalized harmonic numbers Th. M. Rassias;H. M. Srivastava https://doi.org/10.1016/S0096-3003(01)00172-2
  18. Trans. Amer. Math. Soc. v.347 Remarks on some integrals and series involving the Stirling numbers and ${\zeta}$ (n) L.-C. Shen https://doi.org/10.2307/2154819
  19. J. Number Theory v.25 A formula of S. Ramanujan R. Sitaramachandrarao https://doi.org/10.1016/0022-314X(87)90012-6
  20. Indian J. Pure Appl. Math. v.10 Some identities involving the Riemann Zeta function R. Sitaramachandrarao;A. Sivaramsarma
  21. Indian J. Pure Appl. Math. v.11 Two identities due to Ramanujan R. Sitaramachandrarao;A. Sivaramsarma
  22. Pacific J. Math. v.113 Transformation formulae for multiple series R. Sitaramachandrarao;M. V. Subbarao https://doi.org/10.2140/pjm.1984.113.471
  23. Series Associated with the Zeta and Related Functions H. M. Srivastava;J. Choi
  24. A Course of Modern Analysis: An Introduction to the General Theory of Infinite Processes and of Analytic Functions; With an Account of the Principal Transcendental Functions E. T. Whittaker;G. N. Watson
  25. Amer. Math. Monthly v.60 A new method of evaluating ${\zeta}$(2n) G. T. Williams https://doi.org/10.2307/2306473
  26. First European Congress of Mathematics (Paris, 1992);Progr. Math. v.II;120 Values of Zeta functions and their applications D. Zagier;A. Joseph(ed.);F. Mignot(ed.);F. Murat(ed.);B. Prum(ed.);R. Rentschler(ed.)

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