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Nevanlinna theory for the difference operator. (English) Zbl 1108.30022
The authors consider to what extent the results of Nevanlinna theory remain valid if the derivative $f'(z)$ occurring in many estimates, in particular in the ramification term $N_1(r,f),$ is replaced by the difference $\triangle_c f(z)=f(z+c)-f(z).$ In particular, the authors obtain, for functions of finite order, an analogue of Nevanlinna’s second main theorem in this context. In the counting function $N(r, 1/(f-a))$ one can ignore here those $a$-points of $f$ which occur in $c$-separated pairs, that is, points $z$ for which $f(z+c)=f(z)=a,$ provided $\triangle_cf\not\equiv 0.$ A corollary is a version of Picard’s theorem for functions of finite order, where three values occur only in $c$-separated pairs. The authors also obtain a version of Nevanlinna’s famous five value theorem where points in $c$-separated pairs are ignored. Applications of the results to difference equations are also given. The results are illustrated by a number of examples. The paper concludes with a discussion of the results and some open questions.

30D35Distribution of values (one complex variable); Nevanlinna theory
39A70Difference operators
39A10Additive difference equations
39A12Discrete version of topics in analysis
Full Text: EuDML