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Ramanujan and the regular continued fraction expansion of real numbers

Published online by Cambridge University Press:  26 April 2005

J. MC LAUGHLIN
Affiliation:
Mathematics Department, Trinity College, 300 Summit Street, Hartford, CT 06106-3100. e-mail: [email protected]
NANCY J. WYSHINSKI
Affiliation:
Mathematics Department, Trinity College, 300 Summit Street, Hartford, CT 06106-3100. e-mail: [email protected]

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
Copyright
2005 Cambridge Philosophical Society

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