Rate of convergence of Padé approximants for a particular Wynn series
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 closed form; this is also true when \(f\) assumes the special forms \(f(\mu)=A - q^{\alpha+\mu}, \dots, 1/(\gamma+\mu)\) and when \((**)\) \(c(\omega)=1/(C - q^{\gamma+\omega})\) [see: \textit{P. Wynn}, A general system of orthogonal polynomials, Math. Res. Center (Univ. of Wisconsin) Report (1965); Q. J. Math., Oxf. II. Ser. 18, 81-96 (1967; Zbl 0185.30001)]. Use of these closed forms in the investigation of convergence was made by \textit{K. A. Driver} and \textit{D. S. Lubinsky} [Aequationes Math. 45, No. 1, 1-23 (1993; Zbl 0767.41019)]. The paper under review is concerned with the coefficients \((**)\); in this case the series \((*)\) with \(m=0\) defines a sum \(G(C,\gamma;q|z)\) for sufficiently small \(z\). It is shown that, with \(m \geq 0\), \(q\) such that \(|q|\neq 1\), \(z\) sufficiently small and \(\phi(\mu) = 1/\mu^2\), \(\lim \sup \mid G(C, \gamma; q\mid z) - R(\mu + 1, \mu + m\mid z)^{\phi(\mu)} = \min (|q|\), \(1/|q|)\) as \(\mu\) increases (the quotients \(R\) concerned are those of \(R[\Delta (m)]\)). Use is made of the fact that the coefficients of the special series considered may be expressed in the form \(c(\omega) = 1/t(\omega)\) where the \(t\) satisfy a difference equation of the form \(t(\omega + 1) = st(\omega) - pt(\omega -1)\); considerable simplification of the analysis in the general case is possible.
scientific article; zbMATH DE number 817336
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