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

Дослiджується збiжнiсть багатовимiрних регулярних C -дробiв з нерiвнозначними 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 неравнозначными 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.

In this paper, we investigate the convergence of multidimensional regular С -fractions with independent variables, which are a multidimensional generalization of regular Сfractions. 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.

Introduction
Let N be a fixed natural number and be the sets of multiindices. Our research is devoted to the convergence of multidimensional regular C-fraction with independent variables where the a i(k) , i(k) ∈ I k , k ≥ 1, are complex constants such that a i(k) = 0, i(k) ∈ I k , k ≥ 1, and where z = (z 1 , z 2 , . . . , z N ) ∈ C N , and the multidimensional regular Cfraction with independent variables which are reciprocal to it We note that these branched continued fractions with independent variables are the expansions of multiple power series [5,7]. 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].
In the present paper we derive some new convergence criteria for the mentioned above fractions. For establishing the convergence criteria, we use the convergence continuation theorem (Theorem 24.2 [9, pp. 108-109] (see also Theorem 2.17 [4, p. 66])) to extend the convergence, already known for a small region, to a larger region. The Theorems 1 and 2 give us the intersection of the interior of the parabola and the open disk for the domains of convergence of (1). In Theorems 3 and 4 the interior of the parabola are obtained for the domains of convergence of (2).

Convergence
We give two convergence criteria for multidimensional regular C-fraction with independent variables (1). For use in the following theorems we introduce the notation for the tails of (1): Then it is clear that the following recurrence relations hold Let be the nth approximant of (1), n ≥ 1.
We shall prove the following result.
converges to a function holomorphic in the domain for every constant M > 0. The convergence is uniform on every compact subset of D l 1 ,l 2 ,...,l N ,M .
Proof. We set Let n be an arbitrary natural number. By induction on k we show that, for arbitrary of multiindex i(k) ∈ I k , the following inequalities are valid where 1 ≤ k ≤ n. From (3) it is clear that for k = n the inequalities (11) hold. By induction hypothesis that (11) hold for k = p + 1, p + 1 ≤ n, i(p + 1) ∈ I p+1 , we prove (11) for k = p, i(p) ∈ I p . Indeed, use of (5) and (10), for the arbitrary of multiindex i(p) ∈ I p , leads to In the proof of Lemma 4.41 [8] it is shown that if x ≥ c > 0 and v 2 ≤ 4u + 4, We From this inequality it is easily shown that v 2 ≤ 4u + 4.

CONVERGENCE OF BRANCHED CONTINUED FRACTIONS
Theorem 3. A multidimensional regular C-fraction with independent variables (2), where the a i(k) , i(k) ∈ I k , k ≥ 1, satisfy the conditions arg(a i(k) ) = ϕ, −π < ϕ < π, i(k) ∈ I k , k ≥ 1, converges to a function holomorphic in the domain The convergence is uniform on every compact subset of D l 1 ,l 2 ,...,l N .
Finally, the following theorem can be proved in much the same way as Theorem 3 using Theorem 5 [3].