scholarly journals Chebyshev Polynomials and Continued Fractions Related

2019 ◽  
Vol 7 (4) ◽  
pp. 1-8
Author(s):  
Jerzy Szczepański
Keyword(s):  
Chebyshev Polynomials ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Complex Polynomials ◽  
The Family ◽  
Infinite Continued Fraction

Let $p$, $q$ be complex polynomials, $\deg p>\deg q\geq 0$. We consider the family of polynomials defined by the recurrence $P_{n+1}=2pP_n-qP_{n-1}$ for $n=1, 2, 3, ...$ with arbitrary $P_1$ and $P_0$ as well as the domain of the convergence of the infinite continued fraction $$f(z)=2p(z)-\cfrac{q(z)}{2p(z)-\cfrac{q(z)}{2p(z)-...}}$$ null

scholarly journals A generalized family of multidimensional continued fractions: TRIP Maps

2014 ◽  
Vol 10 (08) ◽  
pp. 2151-2186 ◽  
Author(s):  
Krishna Dasaratha ◽  
Laure Flapan ◽  
Thomas Garrity ◽  
Chansoo Lee ◽  
Cornelia Mihaila ◽  
...  
Keyword(s):  
Number Field ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Real Numbers ◽  
Multidimensional Continued Fractions ◽  
The Family ◽  
Before And After ◽  
Multidimensional Continued Fraction

Most well-known multidimensional continued fractions, including the Mönkemeyer map and the triangle map, are generated by repeatedly subdividing triangles. This paper constructs a family of multidimensional continued fractions by permuting the vertices of these triangles before and after each subdivision. We obtain an even larger class of multidimensional continued fractions by composing the maps in the family. These include the algorithms of Brun, Parry-Daniels and Güting. We give criteria for when multidimensional continued fractions associate sequences to unique points, which allows us to determine when periodicity of the corresponding multidimensional continued fraction corresponds to pairs of real numbers being cubic irrationals in the same number field.

scholarly journals Application of Wynn's epsilon algorithm to periodic continued fractions

1987 ◽  
Vol 30 (2) ◽  
pp. 295-299 ◽  
Author(s):  
M. J. Jamieson
Keyword(s):  
Continued Fractions ◽  
Continued Fraction ◽  
Rational Numbers ◽  
Quadratic Surd ◽  
Partial Quotients ◽  
Infinite Continued Fraction ◽  
Image Position ◽  
Epsilon Algorithm ◽  
Periodic Continued Fractions

The infinite continued fractionin whichis periodic with period l and is equal to a quadratic surd if and only if the partial quotients, ak, are integers or rational numbers [1]. We shall also assume that they are positive. The transformation discussed below applies only to pure periodic fractions where n is zero.

scholarly journals Restricted permutations, continued fractions, and Chebyshev polynomials

10.37236/1495 ◽  
2000 ◽  
Vol 7 (1) ◽  
Author(s):  
Toufik Mansour ◽  
Alek Vainshtein
Keyword(s):  
Explicit Expression ◽  
Chebyshev Polynomials ◽  
Generating Functions ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Restricted Permutations

Let $f_n^r(k)$ be the number of 132-avoiding permutations on $n$ letters that contain exactly $r$ occurrences of $12\dots k$, and let $F_r(x;k)$ and $F(x,y;k)$ be the generating functions defined by $F_r(x;k)=\sum_{n\ge 0} f_n^r(k)x^n$ and $F(x,y;k)=\sum_{r\ge 0}F_r(x;k)y^r$. We find an explicit expression for $F(x,y;k)$ in the form of a continued fraction. This allows us to express $F_r(x;k)$ for $1\le r\le k$ via Chebyshev polynomials of the second kind.

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 Extreme Value Theory for Hurwitz Complex Continued Fractions

Entropy ◽  
2021 ◽  
Vol 23 (7) ◽  
pp. 840
Author(s):  
Maxim Sølund Kirsebom
Keyword(s):  
Invariant Measure ◽  
Extreme Value Theory ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Value Theory ◽  
Extreme Value ◽  
Poisson Law ◽  
Complex Continued Fractions

The Hurwitz complex continued fraction is a generalization of the nearest integer continued fraction. In this paper, we prove various results concerning extremes of the modulus of Hurwitz complex continued fraction digits. This includes a Poisson law and an extreme value law. The results are based on cusp estimates of the invariant measure about which information is still limited. In the process, we obtained several results concerning the extremes of nearest integer continued fractions as well.

scholarly journals Calculating astronomical refraction by means of continued fractions

1979 ◽  
Vol 89 ◽  
pp. 95-101
Author(s):  
S. Mikkola
Keyword(s):  
Asymptotic Expansion ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Astronomical Refraction

A continued fraction was derived for the summation of the asymptotic expansion of astronomical refraction. Using simple approximations for the last denominator of the fraction, accurate formulae, useful down to the horizon, were obtained. The method is not restricted to any model of the atmosphere and can thus be used in calculations based on actual aerological measurements.

scholarly journals Continued Fractions Related to $(t,q)$-Tangents and Variants

10.37236/2014 ◽  
2011 ◽  
Vol 18 (2) ◽  
Author(s):  
Helmut Prodinger
Keyword(s):  
Continued Fractions ◽  
Continued Fraction ◽  
General Version ◽  
Continued Fraction Expansion ◽  
Special Cases ◽  
Tangent Function

For the $q$-tangent function introduced by Foata and Han (this volume) we provide the continued fraction expansion, by creative guessing and a routine verification. Then an even more recent $q$-tangent function due to Cieslinski is also expanded. Lastly, a general version is considered that contains both versions as special cases.

scholarly journals CONTINUED FRACTIONS, INTERMEDIATE FRACTIONS AND THEIR RELATION TO THE BEST APPROXIMATIONS

2020 ◽  
Vol 20 (3) ◽  
pp. 545-560
Author(s):  
LUKA MILINKOVIC ◽  
BRANKO MALESEVIC ◽  
BOJAN BANJAC
Keyword(s):  
Continued Fractions ◽  
Continued Fraction ◽  
Rational Approximations ◽  
Best Approximations ◽  
Current State ◽  
The Subject ◽  
State Of Art

The subject of this paper is the current state of art in theory of continued fractions, intermediate fractions and their relation to the best rational approximations of the first and second kind. The paper provides an overview of the some well known and even some new properties of continued fractions, and the various terms associated with them. In addition to intermediate fractions, paper considers the fine intermediate fractions and gave some statements to position these fractions in the continued fraction representation of numbers.

scholarly journals On the convergence of multidimensional regular C-fractions with independent variables

2018 ◽  
Vol 26 (1) ◽  
pp. 18 ◽  
Author(s):  
R.I. Dmytryshyn
Keyword(s):  
Power Series ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Open Disk ◽  
Efficient Tool ◽  
Independent Variables ◽  
Multidimensional Generalization ◽  
Multiple Power ◽  
Multiple Power Series ◽  
Branched Continued Fraction

In this paper, we investigate the convergence of multidimensional regular С-fractions with independent variables, which are a multidimensional generalization of regular С-fractions. These branched continued fractions are an efficient tool for the approximation of multivariable functions, which are represented by formal multiple power series. We have shown that the intersection of the interior of the parabola and the open disk is the domain of convergence of a multidimensional regular С-fraction with independent variables. And, in addition, we have shown that the interior of the parabola is the domain of convergence of a branched continued fraction, which is reciprocal to the multidimensional regular С-fraction with independent variables.

scholarly journals A characterization of rational numbers by p-adic Ruban continued fractions

1985 ◽  
Vol 39 (3) ◽  
pp. 300-305 ◽  
Author(s):  
Vichian Laohakosol
Keyword(s):  
Continued Fractions ◽  
Continued Fraction ◽  
Rational Numbers

AbstractA type of p–adic continued fraction first considered by A. Ruban is described, and is used to give a characterization of rational numbers.

Sign in / Sign up

Export Citation Format

Share Document