Two-dimensional generalization of the Thron–Jones theorem on the parabolic domains of convergence of continued fractions

• Date: 2022
• Main Authors: Bilanyk, I. B., Bodnar, D. I., Біланик, І. Б., Боднар, Д. І., Боднар, Дмитрий
• Format: Article
• Language: Ukrainian
• Published: Institute of Mathematics, NAS of Ukraine 2022
• Online Access: https://umj.imath.kiev.ua/index.php/umj/article/view/7096
• Journal Title: Ukrains’kyi Matematychnyi Zhurnal

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.