Dmytryshyn R. Some classes of functional branched continued fractions with independent variables and multiple power series

In the thesis the criterions of convergence are established for branched continued fractions with independent variables, whose partial denominators are equal to units. We describe the general theory of correspondence for the sequences of multivariable functions meromorphic at the origin. A classification of functional branched continued fractions with independent variables is carried out. The correspondence properties are investigated for these branched continued fractions. The algorithms for the expansion of given formal multiple power series into a multidimensional regular C-fraction with independent variables and a multidimensional associated fraction with independent variables have been constructed. And also a multidimensional Rutishauser qd-algorithm and a multidimensional Bauer g-algorithm are constructed. The convergence of functional branched continued fractions with independent variables are investigated. The criterions of convergence for S-fractions are established. The truncation error bounds for the multidimensional S-fractions with independent variables and for the multidimensional g-fractions with independent variables are obtained in some constrained domains of complex N-space. The expansions of some analytic multivariable functions, represented by formal multiple power series, into corresponding functional branched continued fractions with independent variables are constructed. The numerical examples show the efficiency of constructed approximations.