Some convergence regions of branched continued fractions of special form

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.

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“…parabolic regions of convergence, for BCF with independent variables. Such regions of convergence were studied in the works [1,3,5,7,8,11,14,15].…”

Section: Introductionmentioning

confidence: 99%

Parabolic convergence regions of branched continued fractions of the special form

Bodnar

Bilanyk

2021

Carpathian Math. Publ.

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“…Various problems of the convergence of other classes have been studied in [2,6,7,15] for multidimensional regular C-fractions with independent variables, in [20,21] for multidimensional g-fractions with independent variables, and in [3,4,8,11,12,18] for multidimensional A-and J-fractions with independent variables. Expansions of certain analytic functions of several variables in the above mentioned branched continued fractions may be found in [12,14,16,19,21].…”

Section: Introductionmentioning

confidence: 99%

On the convergence of multidimensional S-fractions with independent variables

Bodnar

Dmytryshyn

Sharyn

2020

Carpathian Math. Publ.

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The paper investigates the convergence problem of a special class of branched continued fractions, i.e. the multidimensional S-fractions with independent variables, consisting of \[\sum_{i_1=1}^N\frac{c_{i(1)}z_{i_1}}{1}{\atop+}\sum_{i_2=1}^{i_1}\frac{c_{i(2)}z_{i_2}}{1}{\atop+} \sum_{i_3=1}^{i_2}\frac{c_{i(3)}z_{i_3}}{1}{\atop+}\cdots,\] which are multidimensional generalizations of S-fractions (Stieltjes fractions). These branched continued fractions are used, in particular, for approximation of the analytic functions of several variables given by multiple power series. For multidimensional S-fractions with independent variables we have established a convergence criterion in the domain \[H=\left\{{\bf{z}}=(z_1,z_2,\ldots,z_N)\in\mathbb{C}^N:\;|\arg(z_k+1)|<\pi,\; 1\le k\le N\right\}\] as well as the estimates of the rate of convergence in the open polydisc \[Q=\left\{{\bf{z}}=(z_1,z_2,\ldots,z_N)\in\mathbb{C}^N:\;|z_k|<1,\;1\le k\le N\right\}\] and in a closure of the domain $Q.$

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“…A convergence criteria have been given for multidimensional regular C-fractions with independent variables by T. M. Antonova and D. I. Bodnar [1], O. E. Baran [2], R. I. Dmytryshyn [6].…”

Section: Introductionmentioning

confidence: 99%

On the convergence of multidimensional regular C-fractions with independent variables

Dmytryshyn

2018

Vìsn. Dnìpro unìv., Ser. Mat.

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Дослiджується збiжнiсть багатовимiрних регулярних C -дробiв з нерiвнозначни-ми змiнними, якi є багатовимiрним узагальненням регулярних C -дробiв. Цi гiллястi ланцюговi дроби є ефективним iнструментом для наближення функцiй, заданих формальними кратними степеневими рядами. Показано, що перетин параболiчної i кругової областей є областю збiжностi багатовимiрного регулярного C -дробу з нерiвнозначними змiнними, а параболiчна область є областю збiжностi гiллястого ланцюгового дробу, який є оберненим до багатовимiрного регулярного C -дробу з нерiвнозначними змiнними.Ключовi слова: збiжнiсть, рiвномiрна збiжнiсть, багатовимiрний регулярний С-дрiб з нерiвнозначними змiнними.Исследуется сходимость многомерных регулярных C -дробей с неравнозначны-ми переменными, которые являются многомерным обобщением регулярных C -дробей. Эти ветвящиеся цепные дроби являются эффективным инструментом для приближения функций, заданных формальными кратными степенными рядами. Показано, что сечение параболической и круговой областей является областью сходимости многомерной регулярной C -дроби с неравнозначными переменными, а параболическая область является областью сходимости ветвящейся цепной дроби, которая является обратной к многомерной регулярной C -дроби с неравнозначными переменным.Ключевые слова: сходимость, равномерная сходимость, многомерная регулярная С-дробь с неравнозначными переменными.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.

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Some convergence regions of branched continued fractions of special form

Cited by 11 publications

References 11 publications

Parabolic convergence regions of branched continued fractions of the special form

Parabolic convergence regions of branched continued fractions of the special form

On the convergence of multidimensional S-fractions with independent variables

On the convergence of multidimensional regular C-fractions with independent variables

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