Rate of convergence of Padé approximants for a particular Wynn series
type: Item
instance of: scholarly article
title: Rate of convergence of Padé approximants for a particular Wynn series
zbMATH Open document ID: 0840.65004
DOI: 10.1016/0168-9274(95)00074-5
published in: Applied Numerical Mathematics
publication date: 1996-06-30T00:00:00Z
review text: Given a sequence \(c(\omega)\) \((\omega \geq 0)\) of complex numbers and two integers \(i, j \geq 0\) it is, subject to suitable existence conditions, possible to derive two polynomials \(D(i,j)\), \(N(i,j)\) of degrees \(i\), \(j\) respectively, such that the first \(i+j+1\) coefficients of the series expansion in ascending powers of \(z\) of the single approximating fraction \(R(i,j \mid z)=N(i,j \mid z)/D(i,j \mid z)\) are \(c(\omega)\) \((0 \leq \omega \leq i+j)\). With \(m \geq 0\) fixed and assuming further conditions, each member of the sequence \(R[\Delta(m)] :\equiv R(\mu+1, \mu+m)\) \((0 \leq \mu<\infty)\) has its contact property as just described; the members may be expressed as successive partial sums of a series whose terms involve products of \(D\) of contiguous orders and of the successive partial numerators of the continued fraction associated with the series (\(*\)) \(\Sigma c(m+\omega)z^\omega\) \((0 \leq \omega<\infty)\). With \(k > 0\) fixed and other conditions assumed, the sequence \(R[\Xi (k)]:\equiv R(k,\mu)\) \((k - 1 \leq \mu<\infty)\) may be expressed as successive partial sums of a series whose terms involve products of \(D\) of contiguous orders and of simple functions of partial numerators featuring in continued fractions corresponding to various series of the form (\(*\)). [This theory is explained in detail in: \textit{P. Wynn}, Math. Comput. 14, 147-186 (1960; Zbl 0173.18803)]. In cases in which expressions for the denominators \(D\) and the relevant partial numerators are available in simple closed form, the convergence of the sequences \(R[\Delta(m)]\) and \(R[\Xi(k)]\) may be investigated by use of equivalent series. If \(c(\omega)=\Pi f(\mu)\) \((0 \leq \mu<\omega)\) where \(f(\mu)=(A - q^{\alpha+\mu})/(C - q^{\gamma+\mu})\) for \(\mu \geq 0\), the complex \(q\), \(A\), \(\alpha\), \(C\), \(\gamma\) being such that the bracketed terms involved are never zero, the \(D\) and the partial numerators may be expressed in simple close…
Mathematics Subject Classification ID: 65B05
Mathematics Subject Classification ID: 41A21
Mathematics Subject Classification ID: 40A15
zbMATH DE Number: 817336
zbMATH Keywords: Padé approximants
zbMATH Keywords: Wynn series
zbMATH Keywords: \(q\)-hypergeometric series
zbMATH Keywords: approximating fractions
zbMATH Keywords: convergence rates
zbMATH Keywords: continued fraction
author: M. Prevost
reviewed by: Peter Wynn
MaRDI profile type: Publication
full work available at URL: 0168-9274(95)00074-5
OpenAlex ID: W2032908703
cites work: Q4076249
cites work: Q3126793
cites work: Polynômes orthogonaux formels - applications
cites work: Convergence of Padé approximants for a \(q\)-hypergeometric series (Wynn's power series III)
cites work: Approximants de Padé et $U$-dérivation
cites work: Convergence of Padé approximants for a certain class of meromorphic functions
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cites work: Convergence of Padé approximants of partial theta functions and the Rogers-Szegö polynomials
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description: scientific article; zbMATH DE number 817336