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Some observations on the distribution of values of continued fractions. (English) Zbl 0704.40001

If all elements \(a_ n\) of a continued fraction \(K\frac{a_ n}{1}\) lie in an element region \(E\) then the corresponding values of \(K\frac{a_ n}{1}\) lie in a limit region \(V(E)\), for example \(E=\{z:| z| \leq 1/4\}\) the Worpitzky circle and \(V(E)=\{w:| w| \leq 1/2\}\) the limit region. The authors consider the probability distribution of \(w\) in \(V(E)\) on the assumption that \(a_ n\) is distributed evenly in \(E\). Theorems are stated and proven for several classical choices of \(E\) and are supported by numerical evidence. The paper is very readable with only a couple of misprints.
Reviewer: A.Magnus

MSC:

40A15 Convergence and divergence of continued fractions
30B70 Continued fractions; complex-analytic aspects
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References:

[1] Jacobsen, L., Thron, W.J.: Oval convergence regions and circular, limit regions for continued fractionsK(a n /1), Analytic Theory of Continued Fractions II. In: Thron, W.J. (ed.), Proceedings, Pitlochry and Aviemore 1985, pp. 90-126. Lect. Notes Math. No. 1199, Heidelberg Berlin New York: Springer 1986
[2] Jones, W.B., Thron, W.J.: Continued fractions: Analytic theory and applications. Encydopedia of Mathematics and its Applications., 11, Addison-Wesley, Reading, Mass. (1980). Now available from Cambridge University Press · Zbl 0445.30003
[3] Jones, W.B., Thron, W.J., Waadeland, H.: Truncation error bounds for continued fractionsK(a n /1) with parabolic element regions. SIAM J. Numer. Anal20, 1219-1230 (1983) · Zbl 0567.40002 · doi:10.1137/0720092
[4] Rye, E., Waadeland, H.: Reflections on value regions, limit regions and truncation errors for continued fractions. Numer. Math.47, 191-219 (1985) · Zbl 0545.30002 · doi:10.1007/BF01389709
[5] Scott, W.T., Wall, H.S.: A convergence theorem for continued fractions. Trans. Amer. Math. Soc.47, 155-172 (1940) · Zbl 0022.32603 · doi:10.1090/S0002-9947-1940-0001320-1
[6] Worpitzky, J.: Untersuchung über die Entwickelung der monodromen und monogenen Funktionen durch Kettenbrüche Friedrichs-Gymnasium und Realschule Jahresbericht, Berlin pp. 3-39 (1865)
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