Hurwitz sequences, the Farey process, and general continued fractions
type: Item
instance of: scholarly article
title: Hurwitz sequences, the Farey process, and general continued fractions
zbMATH Open document ID: 0667.10007
DOI: 10.1016/0001-8708(88)90029-1
author: Jay R. Goldman
published in: Advances in Mathematics
publication date: 1988-01-01T00:00:00Z
review text: The author studies Hurwitz sequences, obtained by the Farey process, and continued fractions, i.e. fractions \[ [b_ 0;a_ 1/b_ 1,a_ 2/b_ 2,...]=b_ 0+\frac{a_ 1}{b_ 1+}\frac{a_ 2}{b_ 2+}... \] for which \(a_ i\) and \(b_ i\) are integers; \(b_ i>0\) and \(| a_ i| =1\) for \(i=1,2,..\). They have convergents \(P_ n=p_ n/q_ n=[b_ 0;a_ 1/b_ 1,...,a_ n/b_ n].\) Such a unitary continued fraction is a continued fraction for a real number \(\alpha\), if \(P_ n\to \alpha\), as \(n\to \infty.\) The main results of this paper are in connection with the so-called full continued fraction - this is a unitary continued fraction for any irrational \(\alpha\), the sequence of convergents of which contains the sequence of convergents of each other unitary continued fraction of the same \(\alpha\). He proves, for instance, that if \(\alpha\) is a positive irrational number, then there exists a full continued fraction \([b_ 0;a_ 1/b_ 1,...]\) for \(\alpha\), for which \(b_ 0=1\) and \(a_ i/b_ i=1/1\) or - 1/2 for all i and the inverse proposition is also true. \(\{\) Misprint: On page 241 in Definition (2) \((a+c)/(c+d)\) must be read as \((a+c)/(b+d).\}\)
Mathematics Subject Classification ID: 11A55
Mathematics Subject Classification ID: 11J70
zbMATH DE Number: 4091588
zbMATH Keywords: Farey interval
zbMATH Keywords: Hurwitz sequences
zbMATH Keywords: continued fractions
zbMATH Keywords: full continued fraction
reviewed by: Tonko Tonkov
MaRDI profile type: Publication
full work available at URL: 0001-8708(88)90029-1
OpenAlex ID: W2091977715
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description: scientific article; zbMATH DE number 4091588