Convergence criterion for branched continued fractions of the special form with positive elements

Abstract

In this paper the problem of convergence of the important type of a multidimensional gener- alization of continued fractions, the branched continued fractions with independent variables, is considered. This fractions are an efficient apparatus for the approximation of multivariable func- tions, which are represented by multiple power series. When variables are fixed these fractions are called the branched continued fractions of the special form. Their structure is much simpler then the structure of general branched continued fractions. It has given a possibility to establish the necessary and sufficient conditions of convergence of branched continued fractions of the spe- cial form with the positive elements. The received result is the multidimensional analog of Seidel’s criterion for the continued fractions. The condition of convergence of investigated fractions is the di- vergence of series, whose elements are continued fractions. Therefore, the sufficient condition of the convergence of this fraction which has been formulated by the divergence of series composed of par- tial denominators of this fraction, is established. Using the established criterion and Stieltjes-Vitali Theorem the parabolic theorems of branched continued fractions of the special form with complex elements convergence, is investigated. The sufficient conditions gave a possibility to make the con- dition of convergence of the branched continued fractions of the special form, whose elements lie in parabolic domains.