## 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.

### MSC:

 30D35 Value distribution of meromorphic functions of one complex variable, Nevanlinna theory 39A70 Difference operators 39A10 Additive difference equations 39A12 Discrete version of topics in analysis
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