Estimates of the Rate of Pointwise and Uniform Convergence for One-Periodic Branched Continued Fractions of a Special Form

2015 ◽  
Vol 208 (3) ◽  
pp. 289-300 ◽  
Author(s):  
D. I. Bodnar ◽  
M. M. Bubnyak
Keyword(s):  
Uniform Convergence ◽  
Continued Fractions ◽  
Special Form

scholarly journals Some properties of branched continued fractions of special form

2015 ◽  
Vol 7 (1) ◽  
pp. 72-77
Author(s):  
R.I. Dmytryshyn
Keyword(s):  
Uniform Convergence ◽  
Continued Fractions ◽  
Special Form ◽  
Continued Fraction ◽  
Positive Definite ◽  
Branched Continued Fraction

The fact that the values of the approximates of the positive definite branched continued fraction of special form are all in a certain circle is established for the certain conditions. The uniform convergence of branched continued fraction of special form, which is a particular case of the mentioned fraction, in the some limited parabolic region is investigated.

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 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 Representation of a one class function of two variables by bicontinued fractions

2020 ◽  
Vol 27 (2) ◽  
pp. 13
Author(s):  
M.M. Pahirya
Keyword(s):  
Uniform Convergence ◽  
Continued Fractions ◽  
Compact Set ◽  
Class Function ◽  
Function Of Two Variables

Let function $u (z, w) = f (z) h (w)$ be defined on the compact set  $\mathbf{K} \subset \mathbb{C}^2$. We study the problem of representation of functions of this class by the product of two continued fractions, which is called a bicontinued fraction. Some properties of Thiele reciprocal derivatives,  Thiele continued fractions and  regular C-fractions are proved. The possibility of representation of functions of this class by bicontinued fractions is shown. Examples are considered, domains of convergence and uniform convergence of obtained bicontinued fractions to the function are indicated.

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