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Packing measure and dimension of the limit sets of IFSs of generalized complex continued fractions
Journal of difference equations and applications (Print), 2022We consider a family of conformal iterated function systems (for short, CIFSs) of generalized complex continued fractions. Note that in our previous paper we showed that the proper-dimensional Hausdorff measure of the limit set of each CIFS is zero and ...
K. Inui, Hiroki Sumi
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Generalization of continued fractions. I
Journal of Mathematical Sciences, 2012We constructed a new algebraic object, namely, recursion fractions of the n th order that are n -dimensional generalizations of continued fractions. For the representation and the study of such fractions, we used paradeterminants and triangular matrices.
D. I. Bodnar, R. A. Zators’kyi
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Convergence acceleration for generalized continued fractions
Transactions of the American Mathematical Society, 1988The main result in this paper is the proof of convergence acceleration for a suitable modification (as defined by de Bruin and Jacobsen) in the case of an n n
Levrie, Paul, Jacobsen, Lisa
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Generalized continued fractions
Discrete Mathematics and Applications, 1998The author considers a generalized continued fraction whose expression has the form \[ a_0+\frac{(-1)^{u_1}}{a_1+{\displaystyle \frac{(-1)^{u_2}}{a_2+{\displaystyle \frac{(-1)^{u_3}}{a_3+\dots}}}}}, \] where \(a_i\in\mathbb R\) (\(i=0,1,2,\dots\)) and \(u_i\in\{0,1\}\) (\(i=1,2,\dots\)). In this paper the concept how to represent a real number \(\alpha\
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Generalizing the continued fraction algorithm to arbitrary dimensions
30th Annual Symposium on Foundations of Computer Science, 1989A new \(N\)-dimensional continued fraction algorithm is presented. Its most remarkable property is that it produces infinitely many solutions of the diophantine inequality \(\max_{1\leq i\leq N}| x_ i q-p_ i|\ll q^{-w(N)}\) with \(w(N)=1/2N(N+1)\). It also detects linear dependence.
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On some generalizations of Ramanujan’s continued fraction identities
Proceedings of the Indian Academy of Sciences - Section A, 1987Many of the continued fraction expansions of Ramanujan can be viewed as special cases of the expansion of the ratio of basic hypergeometric series: \(_ 2\phi_ 1(a,b;c;xq)/_ 2\phi_ 1(a,b;c;x).\) The authors find the related expansions for the ratios in which the numerator has been replaced by one of a number of similar basic hypergeometric series.
Bhargava, S. +2 more
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A generalization of the Jarník–Besicovitch theorem by continued fractions
Ergodic Theory and Dynamical Systems, 2015We apply the tools of continued fractions to tackle the Diophantine approximation, including the classic Jarník–Besicovitch theorem, localized Jarník–Besicovitch theorem and its several generalizations. As is well known, the classic Jarník–Besicovitch sets, expressed in terms of continued fractions, can be written as $$\begin{eqnarray}\{x\in [0,1):a_{n+
Wang, Bao-Wei, Wu, Jun, Xu, Jian
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Generalized continued fractions associated with the Gauss transform
Russian Mathematical Surveys, 2002In this short note the author describes some properties of the so-called \(\omega\)-continued fractions \(x=[a_0,a_1,\dots]_\omega\), which are a one-dimensional variant of the multidimensional \((A,\omega)\)-continued fractions, which was introduced by the author in [Math. Notes 56, No. 6, 1315--1317 (1994); translation from Mat. Zametki 56, No.
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Other Generalizations of Continued Fractions
2013In this chapter we present some other generalizations of regular continued fractions to the multidimensional case. The main goal for us here is to give different geometric constructions related to such continued fractions (whenever possible). We say a few words about Minkowski–Voronoi continued fractions, triangle sequences related to Farey addition, O’
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