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The Ramanujan Journal

, Volume 32, Issue 1, pp 109–124 | Cite as

Generalized Brouncker’s continued fractions and their logarithmic derivatives

  • Olga Kushel
Article
  • 118 Downloads

Abstract

In this paper, we study the continued fraction y(s,r) which satisfies the equation y(s,r)y(s+2r,r)=(s+1)(s+2r−1) for \(r > \frac{1}{2}\). This continued fraction is a generalization of the Brouncker’s continued fraction b(s). We extend the formulas for the first and the second logarithmic derivatives of b(s) to the case of y(s,r). The asymptotic series for y(s,r) at ∞ are also studied. The generalizations of some Ramanujan’s formulas are presented.

Keywords

Brouncker’s continued fraction Ramanujan’s formula Asymptotic series Functional equations 

Mathematics Subject Classification (2010)

11A55 11J70 30B70 

Notes

Acknowledgements

The author thanks Prof. S. Khrushchev for helpful suggestions and valuable comments.

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Copyright information

© Springer Science+Business Media, LLC 2013

Authors and Affiliations

  1. 1.Institut für Mathematik, MA 3-6Technische Universität BerlinBerlinGermany

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