Twin circular domains of convergence of branched continued fractions with inequivalent variables

Baran, Ondrej

doi:10.1007/s10958-011-0291-0

Cited by 6 publications

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References 3 publications

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“…The numerator of ratio (3) A n (z) is called nth numerator and denominator B n (z) -nth denominator of the approximant (2).…”

Section: Definition Of a Positive Definite Bcf Of Special Formmentioning

confidence: 99%

“…Using the methods of majorants, fundamental inequalities, theorems about compact family of holomorphic functions, the convergence of some numerical and functional BCF of special form are investigated in [1,2,3,5,6,7]. Taking into account the formulas for the numerators and denominators of approximants as determinants, the properties of positive definite BCF are defined and considered in the monograph [4, pp.…”

Section: Introductionmentioning

confidence: 99%

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Positive definite branched continued fractions of special form

Dmytryshyn

2013

Carpathian Math. Publ.

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“…The numerator of ratio (3) A n (z) is called nth numerator and denominator B n (z) -nth denominator of the approximant (2).…”

Section: Definition Of a Positive Definite Bcf Of Special Formmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Positive definite branched continued fractions of special form

Dmytryshyn

2013

Carpathian Math. Publ.

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“…Perhaps the most important subclass of such fractions is the branched continued fractions with independent variables. These fractions are an efficient tool for the approximation of multivariable functions, which are represented by multiple power series [3,5,8,9]. By structure the branched continued fractions with independent variables are their analogues.…”

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confidence: 99%

“…A convergence criteria have been given in [1,3] for multidimensional C -fractions with independent variables…”

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confidence: 99%

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Convergence of some branched continued fractions with independent variables

Dmytryshyn

2017

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In this paper, we investigate a convergence of associated multidimensional fractions and multidimensional J -fractions with independent variables that are closely related to each other; the coefficients of its partial numerators are positive constants or are non-zero complex constants from parabolic regions. We have established the uniform convergence of the sequences of odd and even approximants of the above mentioned fractions to holomorphic functions on compact subsets of certain domains of C N . And also, we have proved that a condition of convergence for the considered branched continued fractions in certain subsets of C N is the divergence of the series composed of its coefficients. Moreover, we have established that the convergence is uniform to a holomorphic function on all compact subsets of domains of C N , which are interior of the above mentioned subsets.

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Some Circular Regions of Convergence for Branched Continued Fractions of a Special Form

Baran

2015

J Math Sci

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517.524For branched continued fractions of a special form, we obtain circular regions of convergence. These regions are related to the multidimensional generalizations of some well-known theorems ) on twin convergence regions for continued fractions. In the case where a branched continued fraction is transformed into a continued fraction ( N = 1 ), the obtained circular regions can of convergence be wider (under certain conditions imposed on the parameters) than some known twin convergence regions for continued fractions.An important place in the theory of continued fractions is occupied by twin convergence regions. Twin convergence regions are called pairs of regions E 1 , E 2 of the complex plane such that the conditions a 2 k−1 ∈E 1 and a 2 k ∈E 2 , k ≥ 1 , ensure the convergence of the fraction k=1 ∞ D a k 1 . The first twin convergence regions were obtained by W. Leighton and H. S. Wall [11]: the fraction k=1 ∞ D a k 1 converges if a 2 k−1 ≤ 1 4 and a 2 k ≥ 25 4 , k ≥ 1 . Setting a k = c k 2 , k ≥ 1 , W. J. Thron [7] established the convergence of the continued fraction k=1 ∞ D c k 2 1 in the case where c 2 k−1 ≤ ρ and c 2 k ± i ≥ ρ , 0 < ρ < 1, k ≥ 1 . In [7], it was shown that, for ρ = 1 , the fraction k=1 ∞ D c k 2 1 converges if c 2 k−1 ≤ 1, c 2 k ± i ≥ 1 , k ≥ 1 , and c 2 k > ε , where ε is any positive number. L. J. Lange [9] proved the convergence of the fraction k=1 ∞ D c k 2 1 for c 2 k−1 ± ia ≤ ρ and c 2 k ± i(a + 1) ≥ ρ , k ≥ 1 , where a ∈C , and a and ρ satisfy the inequality a < ρ < a + 1 . J. Mc Laughlin and N. J. Wyshinski [10] showed that the fraction k=1 ∞ D a k 1 converges if a 2 k−1 ≤ c and a 2 k ≥ 1 + 3c + 2 c 2c + 1 , k ≥ 1 , where c > 0 . The twin convergence regions for continued fractions were also studied, e.g., in the work by L. Lorentzen [12]. The branched continued fractions are multidimensional generalizations of continued fractions. For the branched continued fractions of various structures, the twin convergence regions were studied by T. M. Antonova [1], E. A. Boltarovich [5], V. R. Gladun [6]. The twin convergence regions for two-dimensional continued fractions were described in the works by T. M. Antonova and O. M. Sus' [2] and Kh. Yo. Kuchmins'ka [8].

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Twin circular domains of convergence of branched continued fractions with inequivalent variables

Cited by 6 publications

References 3 publications

Positive definite branched continued fractions of special form

Positive definite branched continued fractions of special form

Convergence of some branched continued fractions with independent variables

Some Circular Regions of Convergence for Branched Continued Fractions of a Special Form

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