# On the convergence of multidimensional S-fractions with independent variables

## Authors

• O.S. Bodnar Volodymyr Gnatiuk Ternopil National Pedagogical University, 2 Kryvonosa str., 46027, Ternopil, Ukraine
• R.I. Dmytryshyn Vasyl Stefanyk Precarpathian National University, 57 Shevchenka str., 76018, Ivano-Frankivsk, Ukraine https://orcid.org/0000-0003-2845-0137
• S.V. Sharyn Vasyl Stefanyk Precarpathian National University, 57 Shevchenka str., 76018, Ivano-Frankivsk, Ukraine https://orcid.org/0000-0003-2547-1442

## Keywords:

branched continued fraction, convergence criterion, uniform convergence, estimates of the rate of convergence, continued fraction
Published online: 2020-12-13

### Abstract

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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How to Cite
(1)
Bodnar, O.; Dmytryshyn, R.; Sharyn, S. On the Convergence of Multidimensional S-Fractions With Independent Variables. Carpathian Math. Publ. 2020, 12, 353-359.

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