Two-dimensional generalization of the Thron–Jones theorem on the parabolic domains of convergence of continued fractions
Abstract
UDC 517.5
For branched continued fractions of а special form (branched continued fractions with independent variables with fixed values of the variables), the notion of $ \mathcal{C} $-figure convergence was introduced and used to establish a two-dimensional generalization of the Thron–Jones theorem on the parabolic domains of convergence of continued fractions. A new method for the investigation of the domains of parabolic convergence of branched continued fractions of a special form is proposed. This method does not use the Stielties–Vitali theorem on convergence of the sequences of holomorphic functions. Hence, it enables us to extend the domain of parabolic convergence to a form similar to that established for the one-dimensional case. In proving this theorem, we essentially use a property of stability of continued fractions under perturbations established in the present work.
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