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 \[∑i_1=1^N{cᵢ₍₁₎zi_1}{1}{+}∑i_2=1i_1{cᵢ₍₂₎zi_2}{1}{+} ∑i_3=1i_2{cᵢ₍₃₎zi_3}{1}{+}⋯,\] 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 \[H=\{{z}=(z_1,z_2,…,z_N)^N:\;|(z_k+1)|<π,\; 1≤ k≤ N\}\] as well as the estimates of the rate of convergence in the open polydisc \[Q=\{{z}=(z_1,z_2,…,z_N)^N:\;|z_k|<1,\;1≤ k≤ N\}\] and in a closure of the domain $Q.$