Convergence of some branched continued fractions with independent variables

Abstract

In this paper, we investigate a convergence of associated multidimensional fractions and multidimensional J-fractions with independent variables that are closely related to each other; the coefficients of its partial numerators are positive constants or are non-zero complex constants from parabolic regions. We have established the uniform convergence of the sequences of odd and even approximants of the above mentioned fractions to holomorphic functions on compact subsets of certain domains of $\mathbb{C}^N.$ And also, we have proved that a condition of convergence for the considered branched continued fractions in certain subsets of $\mathbb{C}^N$ is the divergence of the series composed of its coefficients. Moreover, we have established that the convergence is uniform to a holomorphic function on all compact subsets of domains of $\mathbb{C}^N,$ which are interior of the above mentioned subsets