Even and odd parts of limit periodic continued fractions
type: Item
instance of: scholarly article
title: Even and odd parts of limit periodic continued fractions
zbMATH Open document ID: 0604.30005
DOI: 10.1016/0377-0427(86)90029-4
author: Lisa Jacobsen
author: Haakon Waadeland
published in: Journal of Computational and Applied Mathematics
publication date: 1986-01-01T00:00:00Z
review text: The limit periodic continued fraction \(K(a_ n/1)\), \(a_ n\to a\in {\hat {\mathbb{C}}}={\mathbb{C}}\cup \{\infty \}\) converges if \(a\not\in [-\infty,- (1/4]\) in which case the even and odd parts of \(K(a_ n/1)\) also converge and to the same value as does \(K(a_ n/1)\). It is shown that if \(a_ n\to \infty\) so that \(a_{n+1}/a_ n\to 1\) and \(1+a_ 2\neq 0\neq 1+a_ n+a_{n+1}\), \(n=2,3,4,...\), then in the even and odd parts \(K(c_ n/1)\) and \(K(d_ n/1)\), one has \(c_ n\to -(1/4)\) and \(d_ n\to -(1/4)\). Applying convergence or divergence criteria for \(K(c_ n/1)\) or \(K(d_ n/1)\) one can conclude convergence or divergence of \(K(a_ n/1).\) Continued fractions of these types appear in connection with hypergeometric functions and are therefore of interest. A number of examples are considered.
Mathematics Subject Classification ID: 30B70
Mathematics Subject Classification ID: 40A15
zbMATH DE Number: 3975290
zbMATH Keywords: convergence of continued fractions
zbMATH Keywords: even parts
zbMATH Keywords: limit periodic continued fraction
zbMATH Keywords: odd parts
zbMATH Keywords: hypergeometric functions
MaRDI profile type: Publication
full work available at URL: 0377-0427(86)90029-4
OpenAlex ID: W2080627358
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description: scientific article; zbMATH DE number 3975290