Research of absolute and figured absolute convergence of the branched continued fractions of the special form - Антонова, Тамара Миколаївна; Возна, Світлана Миколаївна (Eastern-European Journal of Enterprise Technologies, 2015)
The paper deals with the issue of convergence of one of the generalizations of continued fractions - branched continued fraction (BCF) of the special form with two branches, which has been proposed by the Polish mathematician W. Siemaszko in solving the problem of compliance between the formal double power series and a sequence of rational approximations of the function of two variables. Unlike continued fractions, approximations of which are constructed unambiguously, there are many ways to construct approximations of BCF of the general and special form. The paper examines the conventional approximations of one of the structures of figured approximations of the studied BCF, which is associated with the problem of compliance. The formulas of difference between two approximations (two conventional, two figured, figured and conventional) were given. Using the majorant method and known results of the theory of convergence of continued fractions, the theorem on some sufficient conditions for absolute and absolute figured convergence of BCF of the special form with two branches towards the same border was proved. It was shown that the values of the studied BCF and its approximations belong to a circle, the radius of which depends on the values of BCF elements on the first floor and the constants appearing in the formulation of the theorem. By choosing different values for these constants, it is possible to obtain various signs of absolute and figured absolute convergence of BCF of the special form.