On the convergence of multidimensional S-fractions with independent variables
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 \[_i_1=1^Nc_i(1)z_i_1 1+_i_2=1^i_1c_i(2)z_i_2 1+ _i_3=1^i_2c_i(3)z_i_3 1+…,\] 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 \[H={z =(z_1,z_2,…,z_N)C^N: |(z_k+1)|<π, 1kN}\] as well as the estimates of the rate of convergence in the open polydisc \[Q={z =(z_1,z_2,…,z_N)C^N: |z_k|<1, 1kN}\] and in a closure of the domain Q. , 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