A multidimensional generalization of Lagrange’s theorem on continued fractions

2012 ◽  
Vol 92 (3-4) ◽  
pp. 312-326 ◽  
Author(s):  
A. V. Bykovskaya
Keyword(s):  
Continued Fractions ◽  
Multidimensional Generalization ◽  
Lagrange's Theorem

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.

Cyclotomic approximation lattices

2015 ◽  
Vol 11 (02) ◽  
pp. 557-567
Author(s):  
Antonino Leonardis
Keyword(s):  
Continued Fractions ◽  
The Third ◽  
Lagrange's Theorem ◽  
Cyclotomic Integers

In this paper, we will consider the Approximation Lattices for a p-adic number, as defined in a work of de Weger, and construct a generalization called the Cyclotomic Approximation Lattices. In the latter case, we consider approximation by a pair of cyclotomic integers instead of rational ones. This can be useful for studying p-adic continued fractions with cyclotomic integral part. The first section will introduce this work and provide motivations. The second one will give some background theorems on number rings. In the third section, we will recall the work of de Weger with a new proof for Theorem 3.6, the analogue of classical Lagrange's theorem for continued fractions. In the fourth one, we will then see the cyclotomic variant and its analogous properties.

scholarly journals On some of convergence domains of multidimensional S-fractions with independent variables

2019 ◽  
Vol 11 (1) ◽  
pp. 54-58 ◽  
Author(s):  
R.I. Dmytryshyn
Keyword(s):  
Continued Fractions ◽  
Continued Fraction ◽  
Continuation Theorem ◽  
Independent Variables ◽  
Multidimensional Generalization ◽  
Connected Set ◽  
Small Domain ◽  
Partial Quotients ◽  
Circular Domains ◽  
Branched Continued Fraction

The convergence of multidimensional S-fractions with independent variables is investigated using the multidimensional generalization of the classical Worpitzky's criterion of convergence, the criterion of convergence of the branched continued fractions with independent variables, whose partial quotients are of the form $\frac{q_{i(k)}^{i_k}q_{i(k-1)}^{i_k-1}(1-q_{i(k-1)})z_{i(k)}}{1}$, and the convergence continuation theorem to extend the convergence, already known for a small domain (open connected set), to a larger domain. It is shown that the union of the intersections of the parabolic and circular domains is the domain of convergence of the multidimensional S-fraction with independent variables, and that the union of parabolic domains is the domain of convergence of the branched continued fraction with independent variables, reciprocal to it.

On the Continued Fractions of Conjugate Quadratic Irrationalities

1980 ◽  
Vol 23 (2) ◽  
pp. 199-206
Author(s):  
Fritz Herzog
Keyword(s):  
Continued Fractions ◽  
Continued Fraction ◽  
Irrational Number ◽  
Quadratic Irrationality ◽  
Partial Quotients ◽  
Image Position ◽  
The Right ◽  
Lagrange's Theorem

Let1be the simple continued fraction (SCF) of an irrational number x. The partial quotients ai which we shall sometimes refer to as the "terms" of the SCF are integers and, for i ≥ 2, they are positive. If x is a quadratic irrationality then, by Lagrange's Theorem, the right side of (1) becomes periodic from some point on.

scholarly journals On the Eventually Periodic Continued β-Fractions and Their Lévy Constants

Mathematics ◽  
2022 ◽  
Vol 10 (1) ◽  
pp. 127
Author(s):  
Qian Xiao ◽  
Chao Ma ◽  
Shuailing Wang
Keyword(s):  
Real Number ◽  
Continued Fractions ◽  
Golden Ratio ◽  
Irreducible Polynomial ◽  
Negative Real Number ◽  
Lagrange's Theorem

In this paper, we consider continued β-fractions with golden ratio base β. We show that if the continued β-fraction expansion of a non-negative real number is eventually periodic, then it is the root of a quadratic irreducible polynomial with the coefficients in Z[β] and we conjecture the converse is false, which is different from Lagrange’s theorem for the regular continued fractions. We prove that the set of Lévy constants of the points with eventually periodic continued β-fraction expansion is dense in [c, +∞), where c=12logβ+2−5β+12.

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