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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 {\it I. Laine} and {\it C.-C. Yang} in [Proc. Japan Acad., Ser. A 83, 148--151 (2007; Zbl 1153.30030)] .

30D35Distribution of values (one complex variable); Nevanlinna theory
39A05General theory of difference equations
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