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Ramanujan and the regular continued fraction expansion of real numbers
Published online by Cambridge University Press: 26 April 2005
Abstract
In some recent papers, the authors considered regular continued fractions of the form \[ \bigg[a_{0};\underbrace{a,\ldots, a}_{m}, \underbrace{a^{2},\ldots, a^{2}}_{m}, \underbrace{a^{3},\ldots, a^{3}}_{m}, \ldots \bigg], \] where $a_{0} \,{\geq}\, 0, a \,{\geq}\, 2$ and $m \,{\geq}\, 1$ are integers. The limits of such continued fractions, for general $a$ and in the cases $m\,{=}\,1$ and $m\,{=}\,2$, were given as ratios of certain infinite series.
However, these formulae can be derived from known facts about two continued fractions of Ramanujan. Motivated by these observations, we give alternative proofs of the results of the previous authors for the cases $m\,{=}\,1$ and $m\,{=}\,2$ and also use known results about other $q$-continued fractions investigated by Ramanujan to derive the limits of other infinite families of regular continued fractions.
- Type
- Research Article
- Information
- Mathematical Proceedings of the Cambridge Philosophical Society , Volume 138 , Issue 3 , May 2005 , pp. 367 - 381
- Copyright
- 2005 Cambridge Philosophical Society
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