Abstract
In this paper, by using Nevanlinna value distribution theory, we consider a certain type of difference equation, which originates with the difference Painleve I equation, $f(z+1)+f(z-1)=\frac{A(z)}{f(z)}+C(z)$ , where $A(z)$ , $C(z)$ are small meromorphic functions relative to $f(z)$ , and we obtain the existence and the forms of rational solutions. We also discuss the properties of the Borel exceptional value, zeros, poles, and fixed points of finite order transcendental meromorphic solutions.
Highlights
In this paper, a meromorphic function means meromorphic in the whole complex plane C
We will discuss the existence and forms of rational solutions, and investigate the properties on finite order transcendental meromorphic solutions of a certain type of difference equation originating with the difference Painlevé I equation
What will happen if we consider a certain type of difference equation originating with the difference Painlevé I equation ( . )? Here, we obtain the following result
Summary
A meromorphic function means meromorphic in the whole complex plane C. A(z), C(z) are small meromorphic functions relative to f (z), and we obtain the existence and the forms of rational solutions. We discuss the properties of the Borel exceptional value, zeros, poles, and fixed points of finite order transcendental meromorphic solutions.
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