On Some Continued Fractions and Series

2021 ◽  
Vol 58 (2) ◽  
pp. 230-245
Author(s):  
Khalil Ayadi ◽  
Chiheb Ben Bechir ◽  
Iheb Elouaer
Keyword(s):  
Continued Fractions ◽  
Continued Fraction ◽  
Continued Fraction Expansions

We exhibit some explicit continued fraction expansions and their representation series in different fields. Some of these continued fractions have a type of symmetry, known as folding symmetry. We will extracted those whose are specialized.

scholarly journals A Lochs-Type Approach via Entropy in Comparing the Efficiency of Different Continued Fraction Algorithms

Mathematics ◽  
2021 ◽  
Vol 9 (3) ◽  
pp. 255
Author(s):  
Dan Lascu ◽  
Gabriela Ileana Sebe
Keyword(s):  
Dynamical Systems ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Unit Interval ◽  
Probability Measures ◽  
Absolutely Continuous ◽  
Continued Fraction Expansions ◽  
Absolutely Continuous Invariant

We investigate the efficiency of several types of continued fraction expansions of a number in the unit interval using a generalization of Lochs theorem from 1964. Thus, we aim to compare the efficiency by describing the rate at which the digits of one number-theoretic expansion determine those of another. We study Chan’s continued fractions, θ-expansions, N-continued fractions, and Rényi-type continued fractions. A central role in fulfilling our goal is played by the entropy of the absolutely continuous invariant probability measures of the associated dynamical systems.

scholarly journals Renewal-type limit theorem for continued fractions with even partial quotients

2009 ◽  
Vol 29 (5) ◽  
pp. 1451-1478 ◽  
Author(s):  
FRANCESCO CELLAROSI
Keyword(s):  
Limit Theorem ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Type Theorem ◽  
Natural Extension ◽  
Gauss Map ◽  
Continued Fraction Expansion ◽  
Special Flow ◽  
Partial Quotients ◽  
Continued Fraction Expansions

AbstractWe prove the existence of the limiting distribution for the sequence of denominators generated by continued fraction expansions with even partial quotients, which were introduced by Schweiger [Continued fractions with odd and even partial quotients. Arbeitsberichte Math. Institut Universtät Salzburg4 (1982), 59–70; On the approximation by continues fractions with odd and even partial quotients. Arbeitsberichte Math. Institut Universtät Salzburg1–2 (1984), 105–114] and studied also by Kraaikamp and Lopes [The theta group and the continued fraction expansion with even partial quotients. Geom. Dedicata59(3) (1996), 293–333]. Our main result is proven following the strategy used by Sinai and Ulcigrai [Renewal-type limit theorem for the Gauss map and continued fractions. Ergod. Th. & Dynam. Sys.28 (2008), 643–655] in their proof of a similar renewal-type theorem for Euclidean continued fraction expansions and the Gauss map. The main steps in our proof are the construction of a natural extension of a Gauss-like map and the proof of mixing of a related special flow.

scholarly journals On convergence of branched continued fraction expansions of Horn's hypergeometric function $H_3$ ratios

2021 ◽  
Vol 13 (3) ◽  
pp. 642-650
Author(s):  
T.M. Antonova
Keyword(s):  
Uniform Convergence ◽  
Hypergeometric Function ◽  
Compact Subset ◽  
Complex Variable ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Sufficient Conditions ◽  
Two Dimensional ◽  
Continued Fraction Expansions ◽  
Branched Continued Fraction

The paper deals with the problem of convergence of the branched continued fractions with two branches of branching which are used to approximate the ratios of Horn's hypergeometric function $H_3(a,b;c;{\bf z})$. The case of real parameters $c\geq a\geq 0,$ $c\geq b\geq 0,$ $c\neq 0,$ and complex variable ${\bf z}=(z_1,z_2)$ is considered. First, it is proved the convergence of the branched continued fraction for ${\bf z}\in G_{\bf h}$, where $G_{\bf h}$ is two-dimensional disk. Using this result, sufficient conditions for the uniform convergence of the above mentioned branched continued fraction on every compact subset of the domain $\displaystyle H=\bigcup_{\varphi\in(-\pi/2,\pi/2)}G_\varphi,$ where \[\begin{split} G_{\varphi}=\big\{{\bf z}\in\mathbb{C}^{2}:&\;{\rm Re}(z_1e^{-i\varphi})<\lambda_1 \cos\varphi,\; |{\rm Re}(z_2e^{-i\varphi})|<\lambda_2 \cos\varphi, \\ &\;|z_k|+{\rm Re}(z_ke^{-2i\varphi})<\nu_k\cos^2\varphi,\;k=1,2;\; \\ &\; |z_1z_2|-{\rm Re}(z_1z_2e^{-2\varphi})<\nu_3\cos^{2}\varphi\big\}, \end{split}\] are established.

scholarly journals Closed Form Continued Fraction Expansions of Special Quadratic Irrationals

2013 ◽  
Vol 2013 ◽  
pp. 1-5
Author(s):  
Daniel Fishman ◽  
Steven J. Miller
Keyword(s):  
Positive Integer ◽  
Closed Form ◽  
Recurrence Relation ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Fibonacci Numbers ◽  
Continued Fraction Expansions

We derive closed form expressions for the continued fractions of powers of certain quadratic surds. Specifically, consider the recurrence relation with , , a positive integer, and (note that gives the Fibonacci numbers). Let . We find simple closed form continued fraction expansions for for any integer by exploiting elementary properties of the recurrence relation and continued fractions.

Reversibility, Continued Fractions, and Infinite Meander Permutations of Planar Homoclinic Orbits in Linear Hyperbolic Anosov Maps

2014 ◽  
Vol 24 (08) ◽  
pp. 1440008
Author(s):  
Bernold Fiedler
Keyword(s):  
Continued Fractions ◽  
Continued Fraction ◽  
Singularity Theory ◽  
Global Attractors ◽  
Homoclinic Orbits ◽  
Stable And Unstable Manifolds ◽  
Unstable Manifolds ◽  
The Difference ◽  
Continued Fraction Expansions

Meander permutations have been encountered in the context of Gauss words, singularity theory, Sturm global attractors, plane Cartesian billiards, and Temperley–Lieb algebras, among others. In this spirit, we attempt to investigate the difference of orderings of homoclinic orbits on the stable and unstable manifolds of a planar saddle. As an example, we consider reversible linear Anosov maps on the 2-torus, and their relation to continued fraction expansions.

scholarly journals Alternative derivation of some regular continued fractions

1968 ◽  
Vol 8 (2) ◽  
pp. 205-212 ◽  
Author(s):  
R. F. C. Walters
Keyword(s):  
Positive Integer ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Matrix Approach ◽  
Alternative Derivation ◽  
The Matrix ◽  
Continued Fraction Expansions

In this paper we find an expression for ex as the limit of quotients associated with a sequence of matrices, and thence, by using the matrix approach to continued fractions ([5] 12–13, [2] and [4]), we derive the regular continued fraction expansions of e2/k and tan 1/k (where k is a positive integer).

ON TRANSCENDENTAL CONTINUED FRACTIONS IN FIELDS OF FORMAL POWER SERIES OVER FINITE FIELDS

2021 ◽  
pp. 1-12
Author(s):  
BÜŞRA CAN ◽  
GÜLCAN KEKEÇ
Keyword(s):  
Power Series ◽  
Finite Field ◽  
Formal Power Series ◽  
Finite Fields ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Continued Fraction Expansions ◽  
Formal Power

Abstract In the field of formal power series over a finite field, we prove a result which enables us to construct explicit examples of $U_{m}$ -numbers by using continued fraction expansions of algebraic formal power series of degree $m>1$ .

scholarly journals On the frequency of partial quotients of regular continued fractions

2009 ◽  
Vol 148 (1) ◽  
pp. 179-192 ◽  
Author(s):  
AI-HUA FAN ◽  
LINGMIN LIAO ◽  
JI-HUA MA
Keyword(s):  
Variational Principle ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Real Numbers ◽  
Hausdorff Dimensions ◽  
Partial Quotients ◽  
Continued Fraction Expansions

AbstractWe consider sets of real numbers in [0, 1) with prescribed frequencies of partial quotients in their regular continued fraction expansions. It is shown that the Hausdorff dimensions of these sets, always bounded from below by 1/2, are given by a modified variational principle.

On normal numbers for continued fractions

2000 ◽  
Vol 20 (5) ◽  
pp. 1405-1421 ◽  
Author(s):  
COR KRAAIKAMP ◽  
HITOSHI NAKADA
Keyword(s):  
Continued Fractions ◽  
Continued Fraction ◽  
Normal Numbers ◽  
Continued Fraction Expansions

In this paper we will show that any number normal with respect to the regular continued fraction (RCF) is also normal with respect to the continued fraction from below (BCF) and the nearest integer continued fraction (NICF). Furthermore, we will show that in case of the NICF a number is normal with respect to the RCF-expansion if and only if it is NICF-normal. This can be generalized to a larger class of semi-regular continued fraction expansions.

Irrationality measures for almost periodic continued fractions

2016 ◽  
Vol 23 (1) ◽  
Author(s):  
Silvia Dodulíková ◽  
Jaroslav Hančl ◽  
Ondřej Kolouch ◽  
Marko Leinonen ◽  
Kalle Leppälä
Keyword(s):  
Continued Fractions ◽  
Continued Fraction ◽  
Almost Periodic ◽  
Continued Fraction Expansions ◽  
Periodic Continued Fractions

AbstractWe give general results which provide asymptotic irrationality measures and estimations for the denominators of the convergents for certain almost periodic simple continued fraction expansions. As an application we obtain new irrationality measures, for example, to Napier's constant

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