## Value distribution of the difference operator.(English)Zbl 1173.30018

This paper is devoted to considering value distribution of difference operators of transcendental entire functions. The key result is the following theorem: Let $$f$$ be a transcendental entire function of finite order, $$p$$ a non-vanishing polynomial, $$c$$ a non-zero complex number, and $$n\geq 2$$. If $$\Delta_{c}f:=f(z+c)-f(z)$$ is not vanishing identically, then $$f(z)^{n}\Delta_{c}f-p(z)$$ has infinitely many zeros. It will be shown by simple examples that the assertion fails if $$f$$ is of infinite order, resp. if $$n=1$$. This theorem is an extension to a previous result due to I. Laine and C.-C. Yang in [Proc. Japan Acad., Ser. A 83, 148–151 (2007; Zbl 1153.30030)] .

### MSC:

 30D35 Value distribution of meromorphic functions of one complex variable, Nevanlinna theory 39A05 General theory of difference equations

### Keywords:

entire functions; difference operator; finite order

Zbl 1153.30030
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