Truncation error bounds for limit-periodic continued fractions \(K(a_ n/1)\) with \(\lim a_ n=0\)
type: Item
instance of: scholarly article
title: Truncation error bounds for limit-periodic continued fractions \(K(a_ n/1)\) with \(\lim a_ n=0\)
zbMATH Open document ID: 0564.40002
DOI: 10.1007/BF01389658
author: Christopher Baltus
author: W. B. Jones
published in: Numerische Mathematik
publication date: 1985-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/133013
review text: Truncation error bounds are developed for continued fractions \(K(a_ n/1)\) where \(| a_ n| \leq 1/4\) for all n sufficiently large. The bounds are particularly suited (some are shown to be best) for the limit- periodic case when lim \(a_ n=0\). Among the principal results is the following: If \(| a_ n| \leq \alpha /n^ p\) for all n sufficiently large (with constants \(\alpha >0\), \(p>0)\), then \(| f- f_ m| \leq C[D/(m+2)]^{p(m+2)}\) for all m sufficiently large (for some constants \(C>0\), \(D>0)\). Here f denotes the limit (assumed finite) of \(K(a_ n/1)\) and \(f_ m\) denotes its mth approximant. Applications are given for continued fraction expansions of ratios of Kummer functions \(_ 1F_ 1\) and of ratios of hypergeometric functions \({}_ 0F_ 1\). It is shown that \(p=1\) for \(_ 1F_ 1\) and \(p=2\) for \(_ 0F_ 1\), where p is the parameter determining the rate of convergence. Numerical examples indicate that the error bounds are indeed sharp.
Mathematics Subject Classification ID: 40A15
Mathematics Subject Classification ID: 65D15
zbMATH DE Number: 3899472
zbMATH Keywords: Truncation error bounds
zbMATH Keywords: Numerical examples
MaRDI profile type: Publication
cites work: Best Error Bounds for Padé Approximants to Convergent Series of Stieltjes
cites work: Q3336138
cites work: Padé Approximants and Bounds to Series of Stieltjes
cites work: Truncation Errors in Pade Approximations to Certain Functions: An Alternative Approach
cites work: Error Bounds for Elliptic Convergence Regions for Continued Fractions
cites work: A priori estimates for truncation error of continued fractions \(K(1/b_ n\))
cites work: Anomalous Convergence of a Continued Fraction for Ratios of Kummer Functions
cites work: Truncation error bounds for g-fractions
cites work: Truncation error bounds for $\pi$-fractions
cites work: Q4137395
cites work: Truncation error estimates for Stieltjes fractions
cites work: Truncation Error Estimates for T-Fractions
cites work: Truncation Error Bounds for Continued Fractions
cites work: Truncation error analysis by means of approximant systems and inclusion regions
cites work: Q3889331
cites work: Truncation Error Bounds for Continued Fractions $K({{a_n } / 1})$ with Parabolic Element Regions
cites work: A two-parameter family of best twin convergence regions for continued fractions
cites work: On value regions of continued fractions
cites work: Continued fractions with complex elements
cites work: The Padé Table and theτ-Method
cites work: On parabolic convergence regions for continued fractions
cites work: Truncation Error Bounds for Limit Periodic Continued Fractions
OpenAlex ID: W2008682508
description: scientific article; zbMATH DE number 3899472