scholarly journals Parabolic convergence regions of branched continued fractions of the special form

2021 ◽  
Vol 13 (3) ◽  
pp. 619-630
Author(s):  
D.I. Bodnar ◽  
I.B. Bilanyk
Keyword(s):  
Continued Fractions ◽  
Special Form ◽  
Multidimensional Analogue ◽  
Independent Variables ◽  
Positive Elements

Using the criterion of convergence of branched continued fractions of the special form with positive elements, effective sufficient criteria of convergence for these fractions are established. To study the parabolic regions of convergence, the element regions and value regions technique was used. In particular, half-planes are considered as value regions. A multidimensional analogue of Tron's twin convergence regions for branched continued fractions of the special form is established. The obtained results made it possible to establish the conditions for the convergence of the multidimensional $S$-fractions with independent variables.

scholarly journals Convergence criterion for branched contіnued fractions of the special form with positive elements

2017 ◽  
Vol 9 (1) ◽  
pp. 13-21 ◽  
Author(s):  
D.I. Bodnar ◽  
I.B. Bilanyk
Keyword(s):  
Continued Fractions ◽  
Special Form ◽  
Sufficient Conditions ◽  
Convergence Criterion ◽  
Sufficient Condition ◽  
Independent Variables ◽  
Positive Elements ◽  
Multiple Power ◽  
Necessary And Sufficient ◽  
Multiple Power Series

In this paper the problem of convergence of the important type of a multidimensional generalization of continued fractions, the branched continued fractions with independent variables, is considered. This fractions are an efficient apparatus for the approximation of multivariable functions, which are represented by multiple power series. When variables are fixed these fractions are called the branched continued fractions of the special form. Their structure is much simpler then the structure of general branched continued fractions. It has given a possibility to establish the necessary and sufficient conditions of convergence of branched continued fractions of the special form with the positive elements. The received result is the multidimensional analog of Seidel's criterion for the continued fractions. The condition of convergence of investigated fractions is the divergence of series, whose elements are continued fractions. Therefore, the sufficient condition of the convergence of this fraction which has been formulated by the divergence of series composed of partial denominators of this fraction, is established. Using the established criterion and Stieltjes-Vitali Theorem the parabolic theorems of branched continued fractions of the special form with complex elements convergence, is investigated. The sufficient conditions gave a possibility to make the condition of convergence of the branched continued fractions of the special form, whose elements lie in parabolic domains.

scholarly journals On the convergence criterion for branched continued fractions with independent variables

2018 ◽  
Vol 9 (2) ◽  
pp. 120-127 ◽  
Author(s):  
R.I. Dmytryshyn
Keyword(s):  
Power Series ◽  
Continued Fractions ◽  
Special Form ◽  
Absolute Convergence ◽  
Convergence Criterion ◽  
Complex Numbers ◽  
Independent Variables ◽  
Nonnegative Coefficients ◽  
Multiple Power ◽  
Multiple Power Series

In this paper, we consider the problem of convergence of an important type of multidimensional generalization of continued fractions, the branched continued fractions with independent variables. These fractions are an efficient apparatus for the approximation of multivariable functions, which are represented by multiple power series. We have established the effective criterion of absolute convergence of branched continued fractions of the special form in the case when the partial numerators are complex numbers and partial denominators are equal to one. This result is a multidimensional analog of the Worpitzky's criterion for continued fractions. We have investigated the polycircular domain of uniform convergence for multidimensional C-fractions with independent variables in the case of nonnegative coefficients of this fraction.

scholarly journals Some properties of approximants for branched continued fractions of the special form with positive and alternating-sign partial numerators

2018 ◽  
Vol 10 (1) ◽  
pp. 3-13 ◽  
Author(s):  
T.M. Antonova ◽  
M.V. Dmytryshyn ◽  
S.M. Vozna
Keyword(s):  
Continued Fractions ◽  
Special Form ◽  
Sufficient Conditions ◽  
Two Dimensional ◽  
Rational Approximants ◽  
Independent Variables ◽  
Function Of Two Variables ◽  
Necessary And Sufficient ◽  
Double Power Series ◽  
Two Independent Variables

The paper deals with research of convergence for one of the generalizations of continued fractions -- branched continued fractions of the special form with two branches. Such branched continued fractions, similarly as the two-dimensional continued fractions and the branched continued fractions with two independent variables are connected with the problem of  the correspondence between a formal double power series and a sequence of the rational approximants of a function of two variables. Unlike continued fractions, approximants of which are constructed unambiguously, there are many ways to construct approximants of branched continued fractions of the general and the special form. The paper examines the ordinary approximants and one of the structures of figured approximants of the studied branched continued fractions, which is connected with the problem of correspondence. We consider some properties of approximants of such fractions, whose partial numerators are positive and alternating-sign  and partial denominators are equal to one. Some necessary and sufficient conditions for figured convergence are established. It is proved that under these conditions from the convergence of the sequence of figured approximants it follows the convergence of the sequence of ordinary approximants  to the same limit.

Multidimensional regular C-fraction with independent variables corresponding to formal multiple power series

2019 ◽  
Vol 150 (4) ◽  
pp. 1853-1870 ◽  
Author(s):  
R. I. Dmytryshyn
Keyword(s):  
Power Series ◽  
Continued Fractions ◽  
Numerical Experiments ◽  
The Other ◽  
Independent Variables ◽  
Classical Algorithm ◽  
Multiple Power ◽  
The One ◽  
The Given ◽  
Multiple Power Series

AbstractIn the paper the correspondence between a formal multiple power series and a special type of branched continued fractions, the so-called ‘multidimensional regular C-fractions with independent variables’ is analysed providing with an algorithm based upon the classical algorithm and that enables us to compute from the coefficients of the given formal multiple power series, the coefficients of the corresponding multidimensional regular C-fraction with independent variables. A few numerical experiments show, on the one hand, the efficiency of the proposed algorithm and, on the other, the power and feasibility of the method in order to numerically approximate certain multivariable functions from their formal multiple power series.

scholarly journals Some convergence regions of branched continued fractions of special form

2013 ◽  
Vol 5 (1) ◽  
pp. 4-13 ◽  
Author(s):  
O.E. Baran
Keyword(s):  
Continued Fractions ◽  
Special Form

Some circular and parabolic convergence regions for branched continued fractions of special form are established.

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 On the differential equations of H. Lewy

1961 ◽  
Vol 2 (1) ◽  
pp. 11-16
Author(s):  
W. B. Smith-White
Keyword(s):  
Differential Equations ◽  
Special Form ◽  
Linear Equations ◽  
Systems Of Equations ◽  
Independent Variables ◽  
Image Position ◽  
Two Independent Variables

It is known that the theory of Cauchy's problem for differential equations with two independent variables is réducible to the corresponding problem for systems of quasi-linear equations. The reduction is carried further, by means of the theory of characteristics, to the case of systems of equations of the special form first considered by H. Lewy [1]. The simplest case is that of the pair of equationswhere the aii depend on z1 and z2. The problem to be considered is that of finding functions z1(x, y), z2(x, y) which satisfy (1) and which take prescribed values on x + y = 0.

scholarly journals Positive definite branched continued fractions of special form

2013 ◽  
Vol 5 (2) ◽  
pp. 225-230
Author(s):  
R.I. Dmytryshyn
Keyword(s):  
Quadratic Form ◽  
Continued Fractions ◽  
Special Form ◽  
Positive Definite

Research of the class of branched continued fractions of special form, whose denominators do not equal to zero, is proposed and the connection of such fraction with a certain quadratic form is established. It furnishes new opportunities for the investigation of convergence of branching continued fractions of special form.

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