The Ramanujan Journal

, Volume 38, Issue 1, pp 123–128 | Cite as

Ramanujan’s asymptotic expansion for the harmonic numbers



In this paper, we provide a recurrence relation for determining the coefficients of Ramanujan’s asymptotic expansion for the \(n\)th harmonic number.


Harmonic numbers Euler–Mascheroni constant Asymptotic expansion 

Mathematics Subject Classification

Primary 41A60 40A25 



The authors thank the referees for their careful reading of the manuscript and insightful comments.


  1. 1.
    Berndt, B.C.: Ramanujan’s Notebooks, Part V. Springer, Berlin (1998)MATHCrossRefGoogle Scholar
  2. 2.
    Chen, C.P., Elezović, N., Vukšić, L.: Asymptotic formulae associated with the Wallis power function and digamma function. J. Class. Anal. 2, 151–166 (2013)MathSciNetGoogle Scholar
  3. 3.
    Gould, H.W.: Coefficient identities for powers of Taylor and Dirichlet series. Am. Math. Mon. 81, 3–14 (1974)MATHCrossRefGoogle Scholar
  4. 4.
    Hirschhorn, M.D.: Ramanujan’s enigmatic formula for the harmonic series. Ramanujan J. 27, 343–347 (2012)MATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    Nemes, G.: Asymptotic expansion for \(\log n!\) in terms of the reciprocal of a triangular number. Acta Math. Hung. 129, 254–262 (2010)MATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    Ramanujan, S.: Notebook II. Narosa, New Delhi (1988)Google Scholar
  7. 7.
    Srivastava, H.M., Choi, J.: Zeta and \(q\)-Zeta functions and associated series and integrals. Elsevier Science Publishers, Amsterdam, London and New York (2012)Google Scholar
  8. 8.
    Villarino, M.B.: Ramanujan’s harmonic number expansion into negative powers of a triangular number. J. Inequal. Pure Appl. Math. 9(3), Article 89 (2008).

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.School of Mathematics and InformaticsHenan Polytechnic UniversityJiaozuoChina

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