Meromorphic solutions of complex differential-difference equations
type: Item
instance of: scholarly article
title: Meromorphic solutions of complex differential-difference equations
zbMATH Open document ID: 1387.34121
author: Juan-Miguel Gracia
published in: Results in Mathematics
publication date: 2018-01-30T00:00:00Z
review text: In this paper, the authors study the differential-difference equations of the form \[ f(z+c)=e^{P(z)}f^{(k)}(z), \] where \(P(z)\) is a polynomial with \(\deg P=n\geq1\). In fact, they prove that if \(f(z)\) is a finite order entire solution of the above differential-difference equation with a Borel exceptional value b, then \(b = 0\) and the order \(\sigma(f)\) of \(f\) is \(n + 1.\) They also prove that if \(f\) is a transcendental entire solution with finite order of the following differential-difference equation \[ h_n(z)f^{(n)}(z+\eta_n)+\cdots+h_1(z)f'(z+\eta_1)+h_0(z)f(z+\eta_0)=0, \] where \(h_0(z),h_1(z),\cdots, h_n(z)\not\equiv0\) are polynomials, \(\eta_0,\eta_1,\cdots,\eta_n\) are distinct complex constants, then \(\lambda(f-a)\geq \sigma(f)-1\) for every finite value \(a\), where \(\lambda(f-a)\) is the exponent of convergence of zeros of \(f-a\). They also study the following differential-difference equation \[ h_n(z)f^{(n)}(z+\eta_n)+\cdots+h_1(z)f'(z+\eta_1)+h_0(z)f(z+\eta_0)=q(z), \] where \(h_0(z),h_1(z),\cdots, h_n(z)\not\equiv0\) are polynomials, \(\eta_0,\eta_1,\cdots,\eta_n\) are distinct complex constants and \(q(z)\) is a non-zero polynomial.
Mathematics Subject Classification ID: 34M05
Mathematics Subject Classification ID: 30D35
Mathematics Subject Classification ID: 34M10
zbMATH DE Number: 6832339
zbMATH Keywords: differential-difference equations
zbMATH Keywords: meromorphic solutions
zbMATH Keywords: exponent of convergence of zeros
zbMATH Keywords: finite order
reviewed by: Liangwen Liao
MaRDI profile type: Publication
full work available at URL: s00025-017-0736-y
OpenAlex ID: W2743850697
cites work: Zeros of entire solutions to complex linear difference equations
cites work: Growth and zeros of meromorphic solution of some linear difference equations
cites work: On entire solutions of certain type of differential-difference equations
cites work: On the Nevanlinna characteristic of \(f(z+\eta)\) and difference equations in the complex plane
cites work: Difference analogue of the lemma on the logarithmic derivative with applications to difference equations
cites work: Value Distribution and Linear Operators
cites work: Growth of meromorphic solutions of delay differential equations
cites work: Q5723337
cites work: Q4754498
cites work: Nevanlinna theory and complex differential equations
cites work: Some results related to complex differential-difference equations of certain types
cites work: On entire solutions of some differential-difference equations
cites work: On a differential-difference equation
cites work: On meromorphic solutions of a linear differential-difference equation with constant coefficients
cites work: Q3282117
cites work: Q4280901
cites work: On entire solutions of nonlinear difference-differential equations
cites work: Q5257252
cites work: Q4827955
DOI: 10.1007/S00025-017-0736-Y
description: scientific article; zbMATH DE number 6832339