Halburd, R. G.; Korhonen, R. J. Nevanlinna theory for the difference operator. (English) Zbl 1108.30022 Ann. Acad. Sci. Fenn., Math. 31, No. 2, 463-478 (2006). 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. Reviewer: Walter Bergweiler (Kiel) Cited in 4 ReviewsCited in 271 Documents 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 Keywords:difference equation; second main theorem; shared values; ramification PDFBibTeX XMLCite \textit{R. G. Halburd} and \textit{R. J. Korhonen}, Ann. Acad. Sci. Fenn., Math. 31, No. 2, 463--478 (2006; Zbl 1108.30022) Full Text: arXiv EuDML