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Two-Dimensional Generalization of the Thron–Jones Theorem on the Parabolic Domains of Convergence of Continued Fractions

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Ukrainian Mathematical Journal Aims and scope

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

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Authors and Affiliations

  1. V. Hnatyuk Ternopil National Pedagogic University, Ternopil, Kyiv, Ukraine

    I. B. Bilanyk

  2. West-Ukrainian National University, Ternopil, Ukraine

    D. I. Bodnar

Authors
  1. I. B. Bilanyk
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  2. D. I. Bodnar
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Correspondence to I. B. Bilanyk.

Additional information

Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 74, No. 9, pp. 1155–1169, September, 2022. Ukrainian DOI: https://doi.org/10.37863/umzh.v74i9.7096.

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Bilanyk, I.B., Bodnar, D.I. Two-Dimensional Generalization of the Thron–Jones Theorem on the Parabolic Domains of Convergence of Continued Fractions. Ukr Math J 74, 1317–1333 (2023). https://doi.org/10.1007/s11253-023-02138-1

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  • DOI: https://doi.org/10.1007/s11253-023-02138-1

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