Development of an N-Multiple Power Series Into an N-Dimensional Regular C-Fraction

2020 ◽  
Vol 246 (2) ◽  
pp. 201-208
Author(s):  
Kh. Yo. Kuchminska ◽  
S. M. Vozna
Keyword(s):  
Power Series ◽  
Multiple Power ◽  
Multiple Power Series

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 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 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.

On the convergence of multiple power series

1966 ◽  
Vol 91 (4) ◽  
pp. 277-279
Author(s):  
Togo Nishiura ◽  
Daniel Waterman
Keyword(s):  
Power Series ◽  
Multiple Power ◽  
Multiple Power Series

A Tauberian theorem for multiple power series

2016 ◽  
Vol 207 (2) ◽  
pp. 286-313 ◽  
Author(s):  
A L Yakymiv
Keyword(s):  
Power Series ◽  
Tauberian Theorem ◽  
Multiple Power ◽  
Multiple Power Series

scholarly journals Local Limit Theorem for the Multiple Power Series Distributions

Mathematics ◽  
2020 ◽  
Vol 8 (11) ◽  
pp. 2067
Author(s):  
Arsen L. Yakymiv
Keyword(s):  
Limit Theorem ◽  
Power Series ◽  
Local Limit Theorem ◽  
Sufficient Conditions ◽  
Local Limit ◽  
Necessary And Sufficient Conditions ◽  
Multiple Power ◽  
Power Series Distributions ◽  
Necessary And Sufficient ◽  
Multiple Power Series

We study the behavior of multiple power series distributions at the boundary points of their existence. In previous papers, the necessary and sufficient conditions for the integral limit theorem were obtained. Here, the necessary and sufficient conditions for the corresponding local limit theorem are established. This article is dedicated to the memory of my teacher, professor V.M. Zolotarev.

On the distribution of multiple power series regularly varying at the boundary point

2019 ◽  
Vol 29 (6) ◽  
pp. 409-421 ◽  
Author(s):  
Arsen L. Yakymiv
Keyword(s):  
Limit Theorem ◽  
Power Series ◽  
Boundary Point ◽  
Local Version ◽  
Random Vectors ◽  
Integral Limit ◽  
Nonnegative Coefficients ◽  
Multiple Power ◽  
Regularly Varying ◽  
Multiple Power Series

Abstract Let B(x) be a multiple power series with nonnegative coefficients which is convergent for all x ∈ (0, 1)n and diverges at the point 1 = (1, …, 1). Random vectors (r.v.)ξx such that ξx has distribution of the power series B(x) type is studied. The integral limit theorem for r.v. ξx as x ↑ 1 is proved under the assumption that B(x) is regularly varying at this point. Also local version of this theorem is obtained under the condition that the coefficients of the series B(x) are one-sided weakly oscillating at infinity.

scholarly journals On the convergence of a multiple power series

1952 ◽  
Vol 4 (1) ◽  
pp. 111-114
Author(s):  
Sin Hitotumatu
Keyword(s):  
Power Series ◽  
Multiple Power ◽  
Multiple Power Series
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