Nevanlinna theory for the difference operator.

*(English)*Zbl 1108.30022The 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)