Abstract
Abstract This paper focuses on finite-order meromorphic solutions of nonlinear difference equation f n ( z ) + q ( z ) e Q ( z ) Δ c f ( z ) = p ( z ) {f}^{n}(z)+q(z){e}^{Q(z)}{\text{Δ}}_{c}f(z)=p(z) , where p , q , Q p,q,Q are polynomials, n ≥ 2 n\ge 2 is an integer, and Δ c f {\text{Δ}}_{c}f is the forward difference of f. A relationship between the growth and zero distribution of these solutions is obtained. Using this relationship, we obtain the form of these solutions of the aforementioned equation. Some examples are given to illustrate our results.
Highlights
Introduction and main resultsIn 2010, Yang and Laine [1] studied the existence of finite-order entire solutions of the differentialdifference equation f n(z) + L(z, f ) = h(z), (1.1)where L(z, f ) is a linear differential-difference polynomial in f with small meromorphic coefficients, h(z) is a meromorphic function of finite order, and n ≥ 2 is an integer
This paper results in an extensive focus on the integrability of nonlinear difference equations
Assuming that h(z) = p1 eα1z + p2 eα2z, the authors in [2,3,4,5] gave some conditions how to judge the existence of admissible finite-order entire solutions of (1.1), where p1, p2, α1, α2 are constants
Summary
In 2010, Yang and Laine [1] studied the existence of finite-order entire solutions of the differentialdifference equation f n(z) + L(z, f ) = h(z),. It is proved that if n ≥ 4, equation (1.1) has at most one admissible entire solution of finite order. Assuming that h(z) = p1 eα1z + p2 eα2z, the authors in [2,3,4,5] gave some conditions how to judge the existence of admissible finite-order entire solutions of (1.1), where p1 , p2 , α1, α2 are constants. If the condition admissible solution is omitted, that is, the coefficients of L(z, f ) are not small functions of f, Wen, Heittokangas, and Laine [7] proved that every finite-order meromorphic solution of the difference equation f n(z) + q(z)eQ(z)f (z + c) = p(z). For the convenience of our statement, we recall some notions and definitions
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