@article{Bodnar_Dmytryshyn_Sharyn_2020, title={On the convergence of multidimensional S-fractions with independent variables: Array}, volume={12}, url={https://journals.pnu.edu.ua/index.php/cmp/article/view/4320}, DOI={10.15330/cmp.12.2.353-359}, abstractNote={<p>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 $Q.$</p>}, number={2}, journal={Carpathian Mathematical Publications}, author={Bodnar, O.S. and Dmytryshyn, R.I. and Sharyn, S.V.}, year={2020}, month={Dec.}, pages={353–359} }