Zeros of sections of divergent power series
type: Item
instance of: scholarly article
title: Zeros of sections of divergent power series
zbMATH Open document ID: 0854.30001
author: Karl Dilcher
author: Lee A. Rubel
published in: Journal of Mathematical Analysis and Applications
publication date: 1996-07-14T00:00:00Z
review text: Let \(\sum^\infty_{n = 0} c_n z^n\) be a power series with zero radius of convergence. Put \(s_n : = \sum^\infty_{k = 0} c_k z^k\) and let \(Z_n (\alpha,a)\) denote the set of all roots of the equation \(s_n (z/ \alpha) = a\). The authors prove the following four theorems: (1) For any \(\varepsilon > 0\) and \(a \in \mathbb{C}\) the set \(Z_n ({\root n \of {|c_n |}}, a)\) is contained in \(\{z : |c_0 - a |/(1 + |c_0 - a |) \leq |z |\leq 2 + \varepsilon\}\) for infinitely many \(n\). (2) If \(c_n \geq 0\) for all \(n \geq 0\) and \(c_{n - 1} c_{n + 1} > c^2_n\) for all \(n \geq 1\), then for any \(a \neq c_0\) the points from \(Z_n ({\root n \of {|c_n |}}, a)\) converge to the unit circle as \(n \to + \infty\). (3) If \(c_n = n!\) for all \(n \geq 0\), then for any \(a \neq 1\) the set \(Z (n/e, a)\) is contained in \(\{z : 1 - 3 (|1 - a |\sqrt n) < |z |< 1 + (1 + |a |) \sqrt {2/ (\pi n)}\}\) for all \(n \geq \max \{(16/9) e^4 |1 - a |^4\), \((3/ |1 - a |)^2\}\). (4) If \(c_n = n!\) for all \(n \geq 0\), then for any \(a \neq 1\) and \(0 \leq \theta_1 < \theta_2 \leq 2 \pi\) the quotient \((\# \{z \in Z (1,a) : \theta_1 \leq \arg z \leq \theta_2\})/n\) tends to \((\theta_2 - \theta_1)/(2 \pi)\) as \(n \to + \infty\).
reviewed by: Marek Jarnicki
Mathematics Subject Classification ID: 30B10
zbMATH DE Number: 902489
Wikidata QID: Q56812629
MaRDI profile type: Publication
full work available at URL: jmaa.1996.0070
OpenAlex ID: W1981891006
DOI: 10.1006/JMAA.1996.0070
Recommended article: On the zeros of power series
Recommended article: Zeros of sections of exponential sums
Recommended article: Q4952792
Recommended article: Zeros of Dirichlet series
Recommended article: Zeros of sections of power series: deterministic and random
Recommended article: On zero distribution of sections and tails of power series
Recommended article: Q4227519
Recommended article: Zeros of certain Dirichlet series
description: scientific article; zbMATH DE number 902489