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The Ramanujan Journal

, 26:35 | Cite as

On the continued fraction \(\frac{1 |}{| z } +\frac{1 |}{| 1 } + \frac{2 |}{| z } +\frac{3 |}{| 1 } + \frac{4 |}{| z} + \cdots\)

  • Harald Riede
Article

Abstract

The article on hand deals with the continued fraction
$$\frac{1 |}{| z } +\frac{1 |}{| 1 } + \frac{2 |}{| z } +\frac{3 |}{| 1 } + \frac{4 |}{| z} + \cdots.$$
The famous Indian mathematician Srinivasa Ramanujan has given a pre-presentation by a power series, but he however concealed a proof. Subsequently a proof has been established, but a direct verification is intricate. Here we give a quick and direct approach with comparitively little effort.

Keywords

Continued fraction Ramanujan Continued fraction of Stieltjes Power series Exponential function Ordinary differential equation Gaussian error function 

Mathematics Subject Classification (2000)

30B70 30B10 11A155 

References

  1. 1.
    Berndt, B.C.: Ramanujan’s Notebooks, Part II. Springer, Berlin (1999) Google Scholar
  2. 2.
    Perron, O.: Die Lehre von den Kettenbrüchen, vol. I (Elementare Kettenbrüche), Darmstadt (1977) Google Scholar
  3. 3.
    Perron, O.: Die Lehre von den Kettenbrüchen, vol. II (Analytisch-funktionentheoretische Kettenbrüche), Darmstadt (1977) Google Scholar
  4. 4.
    Ramanujan, S.: Notebooks. Tata Institute of Fundamental Research, Bombay (1957) (2 volumes) MATHGoogle Scholar
  5. 5.
    Riede, H.: Über eine Formel von S. Ramanujan, Bad Honnef 2007, unpublished manuscript Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Mathematisches Institut der Universität Koblenz-LandauKoblenzGermany

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