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] Laohakosol , V. and Ubolsri , P. , p -adic continued fractions of Liouville type , Proc. Amer. Math. Soc. , 101 ( 3 ) ( 1987 ), 403 – 410 . [4

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Periodica Mathematica Hungarica
Authors: Pakwan Riyapan, Vichian Laohakosol, and Tuangrat Chaichana

Summary  

Two types of explicit continued fractions are presented. The continued fractions of the first type include those discovered by Shallit in 1979 and 1982, which were later generalized by Pethő. They are further extended here using Peth\H o's method. The continued fractions of the second type include those whose partial denominators form an arithmetic progression as expounded by Lehmer in 1973. We give here another derivation based on a modification of Komatsu's method and derive its generalization. Similar results are also established for continued fractions in the field of formal series over a finite base field.

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Abstract  

Criteria for independence, both algebraic and linear, are derived for continued fraction expansions of elements in the field of Laurent series. These criteria are then applied to examples involving elements recently discovered to have explicit series and continued fraction expansions.

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Summary Let F q be a finite field with q elements. We consider formal Laurent series of F q -coefficients with their continued fraction expansions by F q -polynomials. We prove some arithmetic properties for almost every formal Laurent series with respect to the Haar measure. We construct a group extension of the non-archimedean continued fraction transformation and show its ergodicity. Then we get some results as an application of the individual ergodic theorem. We also discuss the convergence rate for limit behaviors.

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Abstract  

This paper is concerned with the divergence points with fast growth orders of the partial quotients in continued fractions. Let S be a nonempty interval. We are interested in the size of the set of divergence points

\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$E_\varphi (S) = \left\{ {x \in [0,1):{\rm A}\left( {\frac{1} {{\varphi (n)}}\sum\limits_{k = 1}^n {\log a_k (x)} } \right)_{n = 1}^\infty = S} \right\},$$ \end{document}
where A denotes the collection of accumulation points of a sequence and φ: ℕ → ℕ with φ(n)/n → ∞ as n → ∞. Mainly, it is shown, in the case φ being polynomial or exponential function, that the Hausdorff dimension of E φ(S) is a constant. Examples are also given to indicate that the above results cannot be expected for the general case.

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93 KRAAIKAMP, C. and LIARDET, P., Good approximations and continued fractions, Proc. Amer. Math. Soc. 112 (1991), 303-309. MR 91i :11079 Good approximations and

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a timer and a randomly-timed gated mechanism , J. Math. Res. and Exposition , 29 ( 4 ) ( 2009 ), 721 – 729 . [7] Parthasarathy , P. R ., Vijayashree , K. V . and Lenin , R. B ., An M/M/1 driven fluid queue–continued fraction approach

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Abstract  

The aim of this article is to give some property of continued fraction with matrices arguments, about their convergence and others applications. At the end of this work, we present a resolution of the Algebraic Riccati Equation by giving an explicit continued fraction development of its solution.

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