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 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 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.

scholarly journals A Worpitzky boundary theorem for branched continued fractions of the special form

2016 ◽  
Vol 8 (2) ◽  
pp. 272-278 ◽  
Author(s):  
Kh.Yo. Kuchminska
Keyword(s):  
Continued Fractions ◽  
Special Form ◽  
Continued Fraction ◽  
Value Set ◽  
Limit Value ◽  
Branched Continued Fraction ◽  
Element Set

For a branched continued fraction of a special form we propose the limit value set for the Worpitzky-like theorem when the element set of the branched continued fraction is replaced by its boundary.

scholarly journals Research of absolute and figured absolute convergence of the branched continued fractions of the special form

2015 ◽  
Vol 6 (4(78)) ◽  
pp. 19 ◽  
Author(s):  
Тамара Миколаївна Антонова ◽  
Світлана Миколаївна Возна
Keyword(s):  
Continued Fractions ◽  
Special Form ◽  
Absolute Convergence

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.

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