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Series representations for the Apery constant \(\zeta (3)\) involving the values \(\zeta (2n)\)

  • Cezar LupuEmail author
  • Derek Orr
Article
  • 135 Downloads

Abstract

In this note, using the well-known series representation for the Clausen function, we also provide some new representations of Apery’s constant \(\zeta (3)\). In addition, by an idea from De Amo et al. (Proc Am Math Soc 139:1441–1444, 2011) we derive some new rational series representations involving even zeta values and central binomial coefficients. These formulas are expressed in terms of odd and even values of the Riemann zeta function and odd values of the Dirichlet beta function. In particular cases, we recover some well-known series representations of \(\pi \).

Keywords

Riemann zeta function Clausen integral Rational zeta series representations Apery’s constant 

Mathematics Subject Classification

Primary 41A58 41A60 Secondary 40B99 

Notes

Acknowledgements

We would like to thank Piotr Hajlasz, Bogdan Ion, George Sparling and William C. Troy for some fruitful conversations which led to an improvement of the present paper.

References

  1. 1.
    Adamchik, V.S.: Contributions to the theory of the Barnes function. Int. J. Math. Comput. Sci. 9, 11–30 (2014)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Adamchik, V.S., Strivastava, H.M.: Some series of the zeta and related functions. Analysis 18, 131–144 (1998)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Alzer, H., Karayannakis, D., Strivastava, H.M.: Series representations for some mathematical constants. J. Math. Anal. Appl. 320, 145–162 (2006)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Apostol, T.M.: Another elementary proof of Euler’s formula for \(\zeta (2n)\). Am. Math. Mon. 80, 425–431 (1973)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Boros, G., Moll, V.: Irresistible Integrals. Symbolics, Analysis and Experiments in the Evaluation of Integrals. Cambridge University Press, Cambridge (2004)CrossRefGoogle Scholar
  6. 6.
    Borwein, J.M., Bradley, D.M., Crandall, R.E.: Computational strategies for the Riemann zeta function. J. Comput. Appl. Math. 121, 247–296 (2000)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Borwein, J.M., Broadhurst, D.J., Kamnitzer, J.: Central binomial sums, multiple Clausen values, and zeta values. Exp. Math. 10, 25–34 (2001)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Bowman, F.: Note on the integral \(\int _0^{\frac{\pi }{2}}(\log \sin \theta )^nd\theta \). J. Lond. Math. Soc. 22, 172–173 (1947)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Choi, J.: Some integral representations of the Clausen function \(\operatorname{Cl}_{2}(x)\) and the Catalan constant. East Asian Math. J. 32, 43–46 (2016)CrossRefGoogle Scholar
  10. 10.
    Choi, J., Strivastava, H.: Certain classes of series involving the zeta function. J. Math. Anal. Appl. 231, 91–117 (1999)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Choi, J., Strivastava, H.M.: The Clausen function \(\operatorname{Cl}_{2}(x)\) and its related integrals. Thai J. Math. 12, 251–264 (2014)MathSciNetGoogle Scholar
  12. 12.
    Choi, J., Strivastava, H.M., Adamchik, V.S.: Multiple Gamma and related functions. Appl. Math. Comput. 134, 515–533 (2003)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Choi, J., Cho, Y.J., Strivastava, H.M.: Log-sine integrals involving series associated with the zeta function and polylogarithm function. Math. Scand. 105, 199–217 (2009)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Clausen, T.: Uber die function \(\sin \phi +\frac{1}{2^2}\sin 2\phi +\frac{1}{3^2}\sin 3\phi +etc.\). J. Reine Angew. Math. 8, 298–300 (1832)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Coffey, M.W.: On one-dimensional digamma and polygamma series related to the evaluation of Feynman diagrams. J. Comput. Appl. Math. 183, 84–100 (2005)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Cvijovic, D.: New integral representations of the polylogarithm function. Proc. R. Soc. A 643, 897–905 (2007)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Cvijovic, D.: Closed-form evaluation of some families of cotangent and cosecant integrals. Integral Transforms Spec. Func. 19, 147–155 (2008)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Cvijovic, D., Klinowski, J.: New rapidly convergent series representations for \(\zeta (2n+1)\). Proc. Am. Math. Soc. 125, 1263–1271 (1997)MathSciNetCrossRefGoogle Scholar
  19. 19.
    De Amo, E., Diaz Carrillo, M., Hernandez-Sanchez, J.: Another proof of Euler’s formula for \(\zeta (2k)\). Proc. Am. Math. Soc. 139, 1441–1444 (2011)MathSciNetCrossRefGoogle Scholar
  20. 20.
    de Doeder, P.J.: On the Clausen integral \(\operatorname{Cl}_{2}(\theta )\) and a related integral. J. Comput. Appl. Math. 11, 325–330 (1984)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Ewell, J.A.: A new series representation for \(\zeta (3)\). Am. Math. Mon. 97, 219–220 (1990)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Grosjean, C.C.: Formulae concerning the computation of the Clausen integral \(\operatorname{Cl}_{2}(\theta )\). J. Comput. Appl. Math. 11, 331–342 (1984)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Koblig, K.S.: Chebyshev coefficients for the Clausen function \(\operatorname{Cl}_{2}(x)\). J. Comput. Appl. Math. 64, 295–297 (1995)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Lewin, L.: Polylogarithms and Associated Functions. Elsevier North Holland Inc., New York (1981)zbMATHGoogle Scholar
  25. 25.
    Ogreid, O.M., Osland, P.: Some infinite series related to Feynman diagrams. J. Comput. Appl. Math. 140, 659–671 (2002)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Strivastava, H.M., Choi, J.: Series Associated with the Zeta and Related Functions. Kluwer Academic Publishers, Dordrecht (2001)CrossRefGoogle Scholar
  27. 27.
    Strivastava, H.M., Glasser, M.L., Adamchik, V.S.: Some definite integrals associated with the Riemann zeta function. Z. Anal. Anwend. 19, 831–846 (2000)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Tyler, D., Chernoff, P.R.: An old sum reappears-elementary problem 3103. Am. Math. Mon. 92, 507 (1985)Google Scholar
  29. 29.
    Wood, V.E.: Efficient calculation of Clausen’s integral. Math. Comput. 22, 883–884 (1968)MathSciNetzbMATHGoogle Scholar
  30. 30.
    Wu, J., Zhang, X., Liu, D.: An efficient calculation of the Clausen functions. BIT Numer. Math. 50, 193–206 (2010)CrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.Department of MathematicsUniversity of PittsburghPittsburghUSA

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