Some Families of Series Representations for the Riemann ζ(3)

Results in Mathematics

Volumn 33, Issue 3-4, 1998, Pages 179-197

  Chen, Ming Po ^a   Srivastava, H M ^b  

^a INSTITUTE OF MATHEMATICS   (Taiwan)

^b UNIVERSITY OF VICTORIA   (Canada)

Author keywords

Bernoulli numbers; Catalan s constant; Clausen function; double hypergeometric series; Euler polynomials; Gauss nypergeometnc iunction; generalized Harmonic numbers; generating iunction; series expansion; series transformation; summation formulas; Zeta functions, l h pital s rule

Indexed keywords

EID: 0012662556     PISSN: 14226383     EISSN: 14209012     Source Type: Journal    
DOI: 10.1007/BF03322082     Document Type: Article

References (35)

1. R. Apéry, Irrationalité de ζ(2) et ζ(3), in Journées Arithmétiques de Luminy (Colloq. Internat. CNRS, Centre Univ. Luminy, Luminy, 1978), pp. 11–13, Astérisque, 61, Soc. Math. France, Paris, 1979.
2. T.M. Apostol, Some series involving the Riemann Zeta function, Proc. Amer. Math. Soc. 5(1954), 239–243.
3. R. Ayoub, Euler and the Zeta functions, Amer. Math. Monthly 81(1974), 1067–1086.
4. B.C. Berndt, Modular transformations and generalizations of several formulae of Ramanujan, Rocky Mountain J. Math. 7(1977), 147–189.
5. B.C. Berndt, Ramanujan’s Notebooks, Part II, Springer-Verlag, New York, 1989.
6. P.L. Butzer and M. Hauss, Riemann Zeta function: Rapidly converging series and integral representations, Appl. Math. Lett. 5(2) (1992), 83–88.
7. P.L. Butzer, C. Markett, and M. Schmidt, Stirling numbers, central factorial numbers, and representations of the Riemann Zeta function, Resultate Math. 19(1991), 257–274.
8. E. Catalan, Mémoire sur la transformation des séries et sur quelques intégrales défines, Mém. Acad. Roy. Belg. 33(1865), 1–50.
9. J. Choi and H.M. Srivastava, A reducible case of double hypergeometric series involving the Riemann ζ-function, Bull Korean Math. Soc. 33(1996), 107–110.
10. H. Cohen, Généralisation d’une construction de R. Apéry, Bull Soc. Math. France 109(1981), 269–281.
11. D. Cvijovic and J. Klinowski, New rapidly convergent series representations for ζ(2n + 1), Proc. Amer. Math. Soc. 125(1997), 1263–1271.
12. A. Dabrowski, A note on the values of the Riemann Zeta function at positive odd integers, Nieuw Arch. Wisk (4) 14(1996), 199-207.
13. J.A. Ewell, A new series representation for ζ(3), Amer. Math. Monthly 97(1990), 219–220.
14. J.A. Ewell, On values of the Riemann Zeta function at integral arguments, Canad. Math. Bull. 34(1991), 60–66.
15. J.A. Ewell, On the Zeta function values ζ(2k + 1), k = 1,2, …, Rocky Mountain J. Math. 25(1995), 1003–1012.
16. J.W.L. Glaisher, Summations of certain numerical series, Messenger Math. 42 (1912/ 1913), 19–34.
17. R.W. Gosper, Jr., A claculus of series rearrangements, in Algorithms and Complexity: New Directions and Recent Results (J. Traub, Editor), pp. 121–151, Academic Press, New York and London, 1976.
18. E. Grosswald, Die Werte der Riemannschen Zetafunktion an ungeraden Argumen-stellen, Nachr. Akad. Wiss. Göttingen Math.—Phys. Kl.II1970(1970), 9–13.
19. E.R. Hansen, A Table of Series and Products, Prentice-Hall, Englewood Cliffs, New Jersey, 1975.
20. N.S. Koshliakov, Application of the theory of sum-formulae to the investigation of a class of one-valued analytical functions in the theory of numbers, Messenger Math. 58(1928/1929), 1–23.
21. D. Leshchiner, Some new identities for ζi(k), J. Number Theory 13(1981), 355–362.
22. W. Magnus, F. Oberhettinger, and R.P. Soni, Formulas and Theorems for the Special Functions of Mathematical Physics, Die Grundlehren der Mathematischen Wissenschaften in Einzeldarstellungen, Bd. 52, Third Enlarged Edition, Springer-Verlag, New York, 1966.
23. Z.A. Melzak, Companion to Concrete Mathematics: Mathematical Techniques and Various Applications, John Wiley and Sons, New York, London, Sydney, and Toronto, 1973.
24. C. Nash and D. O’Connor, Ray-Singer torsion, topological field theories and the Riemann Zeta function at s = 3, in Low-Dimensional Topology and Quantum Field Theory(Proceedings of a NATO Advanced Research Workshop held at the Isaac Newton Institute at Cambridge, U.K.; September 6–12, 1992) (H. Osborn, Editor), pp. 279–288, Plenum Press, New York and London, 1993.
25. C. Nash and D. O’Connor, Determinants of Laplacians, the Ray-Singer torsion on lens spaces and the Riemann Zeta function, J. Math. Phys. 36(1995), 1462–1505.
26. S. Ramanujan, Notebooks of Srinivasa Ramanujan, Vols. I and II, Tata Institute of Fundamental Research, Bombay, 1957.
27. V. Ramaswami, Notes on Riemann’s ζ-function, J. London Math. Soc. 9(1934), 165–169.
28. H.M. Srivastava, A unified presentation of certain classes of series of the Riemann Zeta function, Riv. Mat. Univ. Parma (4) 14(1988), 1-23.
29. H.M. Srivastava and P.W. Karlsson, Multiple Gaussian Hypergeometric Series, Halsted Press (Ellis Horwood Limited, Chichester), John Wiley and Sons, New York, Chichester, Brisbane, and Toronto, 1985.
30. H.M. Srivastava and E.A. Miller, A simple reducible case of double hypergeometric series involving Catalan’s constant and Riemann’s ζ-function, Internat. J. Math. Ed. sci. Tech. 21(1990), 375–377.
31. A. Terras, Some formulas for the Riemann Zeta function at odd integer argument resulting from Fourier expansions of the Epstein Zeta function, Ada Arith. 29(1976), 181–189.
32. H. Tsumura, On evaluation of the Dirichlet series at positive integers by q- calculation, J. Number Theory48(1994), 383–391.
33. J.R. Wilton, A proof of Burnside’s formula for log Γ(x +1) and certain allied properties of Riemann’s ζ-function, Messenger Math. 52(1922/1923), 90–93.
34. N.-Y. Zhang and K.S. Williams, Some series representations of Γ(2n + 1), Rocky Mountain J. Math. 23(1993), 1581–1592.
35. N.-Y. Zhang and K.S. Williams, Some infinite series involving the Riemann Zeta function, in Analysis, Geometry and Groups: A Riemann Legacy Volume (H.M. Srivastava and Th. M. Rassias, Editors), Part II, pp. 691–712, Hadronic Press, Palm Harbor, Florida, 1993.