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Truncation error bounds for modified continued fractions with applications to special functions

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Summary

Much recent work has been done to investigate convergence of modified continued fractions (MCF's), following the proof by Thron and Waadeland [35] in 1980 that a limit-periodic MCFK(a n , 1;x 1), with\(\mathop {\lim }\limits_{n \to \infty } a_n = :a \in \mathbb{C} - ( - \infty , - \tfrac{1}{4}]\) andnth approximant

$$g_n = S_n (x_1 ) = \frac{{a_1 }}{1} + \frac{{a_2 }}{1} + \cdots + \frac{{a_{n - 1} }}{1} + \frac{{a_n }}{{1 + x_1 }},$$

converges more rapidly to its limitf than the ordinary reference continued fractionK(a n /1) withnth approximantf n =S n (0). Herex 1 denotes the smaller (in modulus) of the two fixed points ofT(w)=a/(1+w). The present paper gives truncation error bounds for bothf n andg n that exploit the limit-periodic property lima n =a. Certain a posteriori bounds given forg n are shown to be best possible, relative to the given (limited) information available. This is the first instance in which truncation error bounds for this problem have been shown to be best possible. Also included in this paper are results on speed of convergence, a practical method for constructing the bounds, and applications to a number of special functions. The given numerical examples indicate that the error bounds are indeed sharp. For the function arctanz, we give graphical contour maps of the number of significant digits in the approximationsf n (z),g n (z) andp n (z), thenth partial sum of the Maclaurin series, forz in a key region of the complex plane. These maps help us compare the various approximations with each other and add to our understanding of their convergence behavior.

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Authors and Affiliations

  1. Department of Mathematics, SUNY College at Oswego, 13126, Oswego, NY, USA

    Christopher Baltus

  2. Department of Mathematics, University of Colorado, 80309-0426, Boulder, CO, USA

    William B. Jones

Authors
  1. Christopher Baltus
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  2. William B. Jones
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This paper is dedicated to the memory of Peter Henrici

Research supported in part by the National Science Foundation under Grant DMS-8700498

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Baltus, C., Jones, W.B. Truncation error bounds for modified continued fractions with applications to special functions. Numer. Math. 55, 281–307 (1989). https://doi.org/10.1007/BF01390055

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