Estimating the \(k\)th coefficient of \((f(z))^{n}\) when \(k\) is not too large
type: Item
instance of: scholarly article
title: Estimating the \(k\)th coefficient of \((f(z))^{n}\) when \(k\) is not too large
zbMATH Open document ID: 1124.30005
published in: Journal of Mathematical Analysis and Applications
publication date: 2007-07-19T00:00:00Z
review text: Assume that (i) the radius of covergence of the series \(f(z)=1+ \sum^\infty_{k=1} a_k(f)z^k\) is positive, (ii) all the coefficients \(a_k(f)\) are real, (ii) \(f\) is strongly positive, that is, for all sufficiently small \(r>0\), \(|f(re^{i\theta})| <f(r)\), \(0<\theta<2\pi\). Let \([f(z)]^n=1+ \sum^\infty_{k=1}a_k^{(n)}(f)z^k\). This article is a continuation of previous papers [Ergodic Theory Dyn. Syst. 14, No. 1, 23--51 (1994; Zbl 0840.26011), Proc. Am. Math. Soc. 123, No. 10, 2999--3007 (1995; Zbl 0933.26004)] by the same author. Theorem 1 extends results concerning the asymptotic expansion of \(a_k^{(n)}(f)\) while Theorem 2 provides asymptotic estimations for \(a_k^{(n)}(f)\), provided \(k\) belongs to a specified range. Some examples are given.
reviewed by: Eligiusz Złotkiewicz
Mathematics Subject Classification ID: 30C45
Mathematics Subject Classification ID: 30B10
Mathematics Subject Classification ID: 30E15
zbMATH DE Number: 5173024
zbMATH Keywords: large powers of polynomials
zbMATH Keywords: multinomial coefficients
zbMATH Keywords: asymptotic for coefficients
author: Valerio de Angelis
MaRDI profile type: Publication
full work available at URL: j.jmaa.2007.01.017
OpenAlex ID: W2064312800
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DOI: 10.1016/J.JMAA.2007.01.017
description: scientific article; zbMATH DE number 5173024