Growth and zeros of meromorphic solution of some linear difference equations
type: Item
instance of: scholarly article
title: Growth and zeros of meromorphic solution of some linear difference equations
zbMATH Open document ID: 1208.39028
published in: Journal of Mathematical Analysis and Applications
publication date: 2010-10-22T00:00:00Z
review text: The article deals with the growth of meromorphic solutions and the growth of their zeros for linear difference equation \[ P_n(z)f(z + n) + \dots + P_1(z)f(z + 1) + P_0(z)f(z) = F(z)\tag{1} \] where \(F(z),P_0(z),P_1(z),\dots,P_n(z)\) are polynomials with \(FP_0P_n \not\equiv 0\) and \[ \deg (P_n + \dots + P_1 + P_0) = \max \;\{\deg P_j:\;j = 0,1,\dots,n\} \geq 1.\tag{2} \] The following three theorems are proved: (i) If (2) holds then every finite order transcendental meromorphic solution \(f(z)\) to (1) satisfies \(\sigma(f) \geq 1\) and \(\sigma(f) = \lambda(f)\); (ii) If (2) holds and \(F(z) \equiv 0\) then every finite order transcendental meromorphic solution \(f(z)\) to (1) satisfies \(\sigma(f) \geq 1\), takes every non-zero value \(a \in {\mathbb C}\) infinitely often and \(\sigma(f) = \lambda(f - a)\); (iii) Without assumption (2) every meromorphic solution \(f(z)\) to (1) with infinitely many poles satisfies \(\sigma(f) \geq 1\). In these results \(\sigma(f)\) is the order of growth of meromorphic function \(f(z)\) and \(\lambda(f)\) the exponent of the growth of zeros of \(f(z)\). In the article there are some simple illustrative examples.
Mathematics Subject Classification ID: 39A45
Mathematics Subject Classification ID: 39A06
Mathematics Subject Classification ID: 30D05
zbMATH DE Number: 5804455
zbMATH Keywords: linear difference equation
zbMATH Keywords: meromorphic solution
zbMATH Keywords: growth of functions
zbMATH Keywords: zeroes of functions
author: Zongxuan Chen
reviewed by: Peter P. Zabreĭko
MaRDI profile type: Publication
full work available at URL: j.jmaa.2010.06.049
OpenAlex ID: W2083518652
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DOI: 10.1016/J.JMAA.2010.06.049
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description: scientific article; zbMATH DE number 5804455