Some Circular Regions of Convergence for Branched Continued Fractions of a Special Form

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For branched continued fractions of a special form, we obtain circular regions of convergence. These regions are related to the multidimensional generalizations of some well-known theorems (W. Leighton, H. S. Wall, W. J. Thron, L. J. Lange, J. Mc Laughlin, and N. J. Wyshinski) on twin convergence regions for continued fractions. In the case where a branched continued fraction is transformed into a continued fraction ( N = 1 ), the obtained circular regions can of convergence be wider (under certain conditions imposed on the parameters) than some known twin convergence regions for continued fractions. An important place in the theory of continued fractions is occupied by twin convergence regions. Twin convergence regions are called pairs of regions E 1 ,E 2 of the complex plane such that the conditions a 2k−1 ∈E 1 and a 2k ∈E 2 , k ≥1 , ensure the convergence of the fraction k=1