A family of continued fractions
type: Item
instance of: scholarly article
title: A family of continued fractions
zbMATH Open document ID: 1196.11013
author: David Angell
published in: Journal of Number Theory
publication date: 2010-04-08T00:00:00Z
review text: The author investigates a one-parameter family of infinite generalized continued fractions \[ \frac{1+s}{1+}\frac{2+s}{2+}\frac{3+s}{3+}\cdots \] He proves that for any positive integer \(s\) this continued fraction converges to a rational number \[ \frac{w(0,s)}{w(1,s)}, \] where \(w(z,s)\) is a sequence of polynomials defined by formulae \[ w(z,1)=1,\qquad w(z,2)=z+4, \] \[ w(z,s)=(z+2s)w(z,s-1)-(s-2)(z+s)w(z,s-2)\quad\text{for}\quad s\geq 3. \] In the second part of the paper the author describes interesting combinatorial properties of the numerators \(u(s)=w(0,s)\) and denominators \(v(s)=w(-1,s)\). Other properties of these numbers can be found in the Online Encyclopedia of Integer Sequences (see sequences A052852 and A000262) [see Zbl 0845.11001].
Mathematics Subject Classification ID: 11A55
Mathematics Subject Classification ID: 11J70
Mathematics Subject Classification ID: 11Y65
zbMATH DE Number: 5690726
zbMATH Keywords: generalized continued fractions
zbMATH Keywords: convergence
Wikidata QID: Q60692229
reviewed by: Alexey Ustinov
describes a project that uses: OEIS
MaRDI profile type: Publication
full work available at URL: j.jnt.2009.12.003
OpenAlex ID: W2049301013
cites work: Irrationality and Transcendence in Number Theory
cites work: A remarkable continued fraction
cites work: Q4040824
DOI: 10.1016/J.JNT.2009.12.003
description: scientific article; zbMATH DE number 5690726