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 Trudy Mat. Inst. Steklova, 2017, Volume 298, Pages 75–100 (Mi tm3821)

On the Van Vleck Theorem for Limit-Periodic Continued Fractions of General Form

V. I. Buslaev

Steklov Mathematical Institute of Russian Academy of Sciences, ul. Gubkina 8, Moscow, 119991 Russia

Abstract: The boundary properties of functions representable as limit-periodic continued fractions of the form $A_1(z)/(B_1(z)+A_2(z)/(B_2(z)+…))$ are studied; here the sequence of polynomials $\{A_n\}_{n=1}^\infty$ has periodic limits with zeros lying on a finite set $E$, and the sequence of polynomials $\{B_n\}_{n=1}^\infty$ has periodic limits with zeros lying outside $E$. It is shown that the transfinite diameter of the boundary of the convergence domain of such a continued fraction in the external field associated with the fraction coincides with the upper limit of the averaged generalized Hankel determinants of the function defined by the fraction. As a consequence of this result combined with the generalized Pólya theorem, it is shown that the functions defined by the continued fractions under consideration do not have a single-valued meromorphic continuation to any neighborhood of any nonisolated point of the boundary of the convergence set.

 Funding Agency Grant Number Russian Foundation for Basic Research 15-01-07531 Ministry of Education and Science of the Russian Federation ÍØ-9110.2016.1 The work was supported in part by the Russian Foundation for Basic Research (project no. 15-01-07531) and by a grant of the President of the Russian Federation (project no. NSh-9110.2016.1).

DOI: https://doi.org/10.1134/S0371968517030062

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English version:
Proceedings of the Steklov Institute of Mathematics, 2017, 298, 68–93

Bibliographic databases:

UDC: 517.53

Citation: V. I. Buslaev, “On the Van Vleck Theorem for Limit-Periodic Continued Fractions of General Form”, Complex analysis and its applications, Collected papers. On the occasion of the centenary of the birth of Boris Vladimirovich Shabat, 85th anniversary of the birth of Anatoliy Georgievich Vitushkin, and 85th anniversary of the birth of Andrei Aleksandrovich Gonchar, Trudy Mat. Inst. Steklova, 298, MAIK Nauka/Interperiodica, Moscow, 2017, 75–100; Proc. Steklov Inst. Math., 298 (2017), 68–93

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/tm3821
• https://doi.org/10.1134/S0371968517030062
• http://mi.mathnet.ru/eng/tm/v298/p75

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This publication is cited in the following articles:
1. V. I. Buslaev, “On Singular points of Meromorphic Functions Determined by Continued Fractions”, Math. Notes, 103:4 (2018), 527–536
2. V. I. Buslaev, “Schur's criterion for formal power series”, Sb. Math., 210:11 (2019), 1563–1580
3. V. I. Buslaev, “Necessary and sufficient conditions for extending a function to a Schur function”, Sb. Math., 211:12 (2020), 1660–1703
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