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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 ${f}^{\text{'}}\left(z\right)$ occurring in many estimates, in particular in the ramification term ${N}_{1}\left(r,f\right),$ is replaced by the difference ${▵}_{c}f\left(z\right)=f\left(z+c\right)-f\left(z\right)·$ 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 $N\left(r,1/\left(f-a\right)\right)$ one can ignore here those $a$-points of $f$ which occur in $c$-separated pairs, that is, points $z$ for which $f\left(z+c\right)=f\left(z\right)=a,$ provided ${▵}_{c}f¬\equiv 0·$ 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.

##### MSC:
 30D35 Distribution of values (one complex variable); Nevanlinna theory 39A70 Difference operators 39A10 Additive difference equations 39A12 Discrete version of topics in analysis