On singular points of meromorphic functions determined by continued fractions
type: Item
instance of: scholarly article
title: On singular points of meromorphic functions determined by continued fractions
zbMATH Open document ID: 1400.30004
DOI: 10.1134/S0001434618030203
published in: Mathematical Notes
publication date: 2018-07-23T00:00:00Z
review text: Let a general continued-fraction be given, \[ \frac{a_1z^{\alpha_1}}{1+\frac{a_2z^{\alpha_2}}{1+\cdots}}=\mathcal K_{n=1}^\infty(\alpha_nz^{\alpha_n}/1) \] where \(a_n \in \mathbb C\backslash \{0\}\), \(\alpha_n \in \mathbb N\) for all \(n=1,2,\dots\) and \( \lim\limits_{n \to \infty} \alpha_n=\infty, \; \lim\limits_{n \to \infty} |a_n|^{1/\alpha_n}=1\). Leighton's Conjecture. The continued-fraction converges uniformly in the spherical metric on compact subsets of the disk \(\mathbb D=\{|z|<1\}\) to a meromorphic (in \(\mathbb D\)) function \(f\), and the disk \(\mathbb D\) is the natural domain of existence of the meromorphic function \(f\). In [Sb. Math. 197, No. 10, 1405--1416 (2006; Zbl 1151.30022); translation from Mat. Sb. 197, No. 10, 3--14 (2006)], \textit{A. A. Gonchar} proved a theorem, which implies, in particular, that Leighton conjecture holds for a nondecreasing sequence of exponents without any additional assumptions. In this paper it is shown that the Leighton conjecture holds also for meromorphic functions represented by continued fractions \(\mathcal K_{n=1}^{\infty}(a_n A_n(z)/1)\), where \(A_1, A_2,\dots\) is a sequence of polynomials with limit distribution of zeros whose degrees tend to infinity.
reviewed by: Mykhaylo Pahirya
Mathematics Subject Classification ID: 30B70
Mathematics Subject Classification ID: 30D30
Mathematics Subject Classification ID: 32A20
Mathematics Subject Classification ID: 40A15
zbMATH DE Number: 6909362
zbMATH Keywords: continued fraction
zbMATH Keywords: Hankel determinant
zbMATH Keywords: transfinite diameter
zbMATH Keywords: meromorphic function
author: Viktor I. Buslaev
MaRDI profile type: Publication
cites work: Q3611747
cites work: Q5792555
cites work: Continued fraction expansions for arbitrary power series
cites work: Singular Points of Functions Defined by C-Fractions
cites work: Q3508679
cites work: Q3788266
cites work: Convergence of m-point Padé approximants of a tuple of multivalued analytic functions
cites work: Distribution of the zeros of Hermite-Padé polynomials
cites work: Distribution of the zeros of Padé polynomials and analytic continuation
cites work: The capacity of the rational preimage of a compact set
cites work: On the existence of compacta of minimal capacity in the theory of rational approximation of multi-valued analytic functions
cites work: Zero distribution of Hermite-Padé polynomials and localization of branch points of multivalued analytic functions
cites work: Convergence of Shafer quadratic approximants
cites work: An analog of Gonchar's theorem for the \(m\)-point version of Leighton's conjecture
cites work: Q4360405
cites work: Capacity of a compact set in a logarithmic potential field
cites work: On the Van Vleck theorem for limit-periodic continued fractions of general form
cites work: An analogue of Polya's theorem for piecewise holomorphic functions
full work available at URL: s0001434618030203
OpenAlex ID: W2801404054
Wikidata QID: Q129943004
Recommended article: An analog of Gonchar's theorem for the \(m\)-point version of Leighton's conjecture
Recommended article: On the Van Vleck theorem for limit-periodic continued fractions of general form
Recommended article: On the domain of meromorphy of limit k-periodic continued fractions
Recommended article: On the growth of meromorphic functions defined by quasi-regular \(c\)-fractions
Recommended article: Analytic continuation of functions given by continued fractions, revisited
Recommended article: Q3508679
Recommended article: A nearness result for PC-fractions
Recommended article: Q4552352
Recommended article: On Hankel determinants of functions given by their expansions in P-fractions
Recommended article: Order and type of entire functions arising from separately convergent continued fractions
description: scientific article; zbMATH DE number 6909362