Explicit evaluation of euler and related sums

Ramanujan Journal

Volumn 10, Issue 1, 2005, Pages 51-70

  Choi, Junesang ^a   Srivastava, H M ^b  

^a Dongguk University   (South Korea)

^b UNIVERSITY OF VICTORIA   (Canada)

Author keywords

(Generalized) harmonic numbers; Euler sums; Gamma function; Hurwitz Zeta function; Hypergeometric series; Polygamma functions; Polylogarithm functions; Psi function; Recurrence relations and recursion formulas; Riemann Zeta function

Indexed keywords

EID: 29344458033     PISSN: 13824090     EISSN: None     Source Type: Journal    
DOI: 10.1007/s11139-005-3505-6     Document Type: Article

References (36)

1. V.S. Adamchik and H.M. Srivastava, "Some series of the zeta and related functions," Analysis 18 (1998), 131-144.
2. D.H. Bailey, J.M. Borwein, and R. Girgensohn, "Experimental evaluation of euler sums," Experimental Math. 3(1994), 17-30.
3. A. Basu and T.M. Apostol, "A new method for investigating euler sums," Ramanujan J. 4 (2000), 397-419
4. B.C. Berndt, Ramanujan's Notebooks, Part I, Springer-Verlag, New York, Berlin, Heidelberg, and Tokyo, 1985.
5. D. Borwein and J.M. Borwein, "On an intriguing integral and some series related to ζ (4)," Proc. Amer. Math. Soc. 123 (1995), 1191-1198
6. D. Borwein, J.M. Borwein, and R. Girgensohn, "Explicit evaluation of euler sums," Proc. Edinburgh Math. Soc. (Ser. 2) 38 (1995), 277-294
7. J.M. Borwein and R. Girgensohn, "Evaluation of triple euler sums," Electron. J. Combin. 3(1) (1996), Research Paper 23, 27 pp. (electronic).
8. D. Bowman and D.M. Bradley, "Multiple Polylogarithms: A brief survey," in q-Series with Applications to Combinatorics, Number Theory, and Physics (Papers from the Conference held at the University of Illinois, Urbana, Illinois; October 26-28, 2000)
9. (B.C. Berndt and K. Ono, eds), Contemporary Mathematics 291, American Mathematical Society, Providence, Rhode Island, (2001), pp. 71-92.
10. P. Bracken, "Euler's formula for zeta function convolutions," Amer. Math. Monthly 108 (2001), 771-778.
11. J. Choi andT.Y. Seo "Evaluation of some infinite series," Indian J. Pure Appl. Math. 28 (1997), 791-796.
12. J. Choi and H.M. Srivastava, "Certain classes of infinite series," Monatsh. Math. 127 ( 1999), 15-25.
13. J. Choi, H.M. Srivastava, and Y. Kim, "Applications of a certain family of hypergeometric summation formulas associated with Psi and zeta functions," Comm. Korean Math. Soc. 16 (2001), 319-332.
14. J. Choi, H.M. Srivastava, T.Y. Seo, and A.K. Rathie, "Some families of infinite series," Soochow J. Math. 25 (1999), 209-219.
15. L. Comtet, "Advanced Combinatorics: The Art of Finite and Infinite Expansions" Translated from the French by J. W. Nienhuys (Reidel, Dordrecht and Boston 1974).
16. R.E. Crandall and J.P. Buhler, "On the evaluation of euler sums," Experimental Math. 3 (1994), 275-285.
17. 2 for certain values of x and y," J. Comput. Appl. Math. 37 (1991), 125-141.
18. P. Flajolet and B. Salvy, "Euler sums and contour integral representations," Experimental Math. 7 (1998), 15-35.
19. M.E. Huffman, "Multiple harmonic series," Pacific J. Math. 152 (1992), 275-290.
20. Y. Kim, "Infinite series associated with psi and zeta functions," Honam Math. J. 22 (2000), 53-60.
21. L. Lewin, Polylogarithms and associated functions, Elsevier, North-Holland, New York, London, and Amsterdam 1981.
22. C. Markett, "Triple sums and the Riemann Zeta function," J. Number Theory 48 (1994), 113-132.
23. N. Nielsen, Die Gammafunktion, Chelsea Publishing Company, (Bronx, New York 1965).
24. G. Polya and G. Szegö, "Problems and Theorems in Analysis," (Translated from the German by D. Aeppli), Vol. I (Springer-Verlag, New York, Heidelberg, and Berlin, 1972).
25. E.D. Rainville, Special Functions, Macmillan Company, New York, (1960).
26. Th. M. Rassias and H.M. Srivastava, "Some classes of infinite series associated with the riemann zeta and polygamma functions and generalized harmonic numbers," Appl. Math. Comput. 131 593-605, 2002
27. L.-C. Shen, "Remarks on some integrals and series involving the Stirling numbers and ζ (n)," Trans. Amer. Math. Soc. 347 (1995), 1391-1399.
28. R. Sitaramachandrarao, "A formula of S. Ramanujan," J. Number Theory 25 (1987), 1-19.
29. R. Sitaramachandrarao and A. Sivaramsarma, Some identities involving the riemann zeta function, Indian J. Pure Appl. Math. 10 (1979), 602-607.
30. R. Sitaramachandrarao and A. Sivaramsarma, Two identities due to Ramanujan, Indian J. Pure Appl. Math. 11(1980), 1139-1140.
31. R. Sitaramachandrarao and M.V. Subbarao, "Transformation formulae for multiple series," Pacific J. Math. 113 (1984) 471-479.
32. H.M. Srivastava and, J. Choi, Series Associated with the Zeta and Related Functions, Kluwer Academic Publishers, Dordrecht, Boston, and London, (2001).
33. E.T. Whittaker and G.N. Watson, 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, Fourth ed. Cambridge University Press, Cambridge, London, and New York, (1963).
34. G.T. Williams "A New method of Evaluating ζ (2n)," Amer. Math. Monthly, 60 (1953), 19-25.
35. D. Zagier, "Values of zeta functions and their applications," First European Congress of Mathematics, Vol. II (Paris, 1992)
36. (A. Joseph, F. Mignot, F. Murat, B. Prum, and R. Rentschler, eds.), Progress in Mathematics 120 (1994) 497-512. Birkhäuser, Basel.