The thesis is fulfilled within the analytic theory of continued and branched continued fractions and is devoted to the study of convergence of branched continued fractions of the special form (branched continued fractions with independent variables at fixed values of variables), to establishing truncation error bounds.
Continued and branched continued fractions are an effective apparatus for constructing rational approximations of analytic functions. The founder of the scientific school of the theory of branched continued fractions is V. Ya. Skorobogatko. During the process of formation of analytic theory, three types of branched continued fractions were distinguished: branched continued fractions of general form with a fixed number of branches of branching, two-dimensional continued fractions, and branched continued fractions with independent variables. They were the object of research of domestic and foreign scientists, in particular, D. I. Bodnar, Ch. Yo. Kuchminska, M. O. Nedashkovsky, R. I. Dmytryshyn, T. M. Antonova, S. V. Sharyn, V. R. Hladun, O. M. Sus, M. M. Pahirya, N. P. Hoyenko, O. S. Manziy, O. E. Baran, M. M. Bubnyak, S. M. Vozna as well as V. Semashko, M. R. O’Donohoe and J. Murphy, A. Cuyt and B. Verdonk, H. Waadeland, T. Komatsu and others.
The most general algorithms for the expansion of analytic functions into continued fractions use the principle of correspondence between power series and different types of functional continued fractions. The study of the correspondence between multiple power series and multidimensional C-fractions contributed to the appearance of branched continued fractions with independent variables, which were first introduced by D. I. Bodnar. Such fractions are effectively used to construct multidimensional rational approximations of analytic functions with several variables.
The main task of the dissertation research is to establish unbounded conditional convergence regions of branched continued fractions of the special form. It is known that unbounded regions cannot be regions of convergence of continued fractions whose partial denominators are equal to one. Therefore, certain additional conditions are imposed that guarantee their convergence. As a rule, this is the divergence of a series composed of elements of a continued fraction. The question of establishing the criterion and effective sufficient conditions for convergence of such fractions with positive elements is considered in order to obtain the formulas of general terms of similar series in the study of unbounded conditional convergence regions of branched continued fractions of the special form. These results are used to construct unbounded conditional convergence regions: angular, parabolic, and others. The thesis consists of the abstract, introduction, four chapters, conclusions for each chapter and general conclusions, bibliography, and appendix that contains the list of author’s publications.
