Arithmetical properties of the leaping convergents of \(e^{1/s}\)
type: Item
instance of: scholarly article
title: Arithmetical properties of the leaping convergents of \(e^{1/s}\)
zbMATH Open document ID: 1075.11004
author: Takao Komatsu
published in: Tokyo Journal of Mathematics
publication date: 2005-01-04T00:00:00Z
review text: The regular continued fraction expansion of \(e^{1/s}\) for \(s\geq 2\) is \([1;\,\overline{s(2k-1)-1,\,1,\,1}]_{k=1}^\infty\). It is natural to take the 3-contraction of this expansion. In theorem 1 it is proved that this contraction is \[ e^{1/s}=1+\frac{2}{2s-1}\,{\quad\atop +}\,\frac 1{6s}\,{\quad\atop +}\,\frac 1{10 s}\,{\quad\atop +}\,\frac 1{14 s}\,{\quad\atop +\dots} \] and that its canonical approximants \(P_n/Q_n\) satisfy \[ \sum_{t=0}^r\frac{P_tQ_{r-t}}{t!(r-t)!}=(4s)^r. \] Inspired by this the author also derives the \((2\ell+1)\)-contraction of the regular continued fraction \([a_0;\,\overline{c_0+dk,\,c_1,\,\dots,\,c_{2\ell}}]_{k=0}^\infty\).
Mathematics Subject Classification ID: 11A55
zbMATH DE Number: 2124813
zbMATH Keywords: continued fractions
zbMATH Keywords: contractions
zbMATH Keywords: exponential function
reviewed by: Lisa Lorentzen
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.3836/tjm/1244208469
OpenAlex ID: W2086690724
DOI: 10.3836/TJM/1244208469
description: scientific article; zbMATH DE number 2124813