Ramanujan-like series for \(\frac{1}{\pi }\) involving harmonic numbers



We introduce new classes of Ramanujan-like series for \(\frac{1}{\pi }\), by devising methods for evaluating harmonic sums involving squared central binomial coefficients, such as the Ramanujan-type series
$$\begin{aligned} \sum _{n=1}^{\infty } \frac{\left( {\begin{array}{c}2 n\\ n\end{array}}\right) ^2 \left( H_n^2+H_n^{(2)}\right) }{16^n (2 n-1)} = \frac{4 \pi }{3}-\frac{32 \ln ^2(2) - 32 \ln (2) + 16 }{\pi } \end{aligned}$$
introduced in this article. While the main technique used in this article is based on the evaluation of a parameter derivative of a beta-type integral, we also show how new integration results involving complete elliptic integrals may be used to evaluate Ramanujan-like series for \(\frac{1}{\pi }\) containing harmonic numbers.


Ramanujan-like series Harmonic number Pi formula Complete elliptic integral 

Mathematics Subject Classification

Primary 33C75 33C20 Secondary 65B10 



The author would like to thank Dr. Jonathan Sondow for a useful discussion concerning Ramanujan-like formulas for \(\frac{1}{\pi }\). The author would also like to thank two anonymous reviewers for many useful comments.


  1. 1.
    Boyadzhiev, K.N.: Series with central binomial coefficients, Catalan numbers, and harmonic numbers. J. Integer Seq. 15 (2012). Article 12.1.7Google Scholar
  2. 2.
    Chen, H.: Interesting series associated with central binomial coefficients, Catalan numbers and harmonic numbers. J. Integer Seq. 19 (2016). Article 16.1.5Google Scholar
  3. 3.
    Guillera, J.: More hypergeometric identities related to Ramanujan-type series. Ramanujan J. 32, 5–22 (2013)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Kaplan, E.L.: Multiple elliptic integrals. Stud. Appl. Math. 29, 69–75 (1950)MathSciNetMATHGoogle Scholar
  5. 5.
    Sofo, A.: Integrals of logarithmic and hypergeometric functions. Commun. Math. 24, 7–22 (2016)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.York UniversityTorontoCanada

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