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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.

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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