Nevanlinna theory of difference equations

Summary

Nevanlinna theory of difference equations applies the value distribution theory of meromorphic functions to equations such as $f(z + 1) = R(z, f(z))$ in the complex plane. Its central tool is a difference analogue of the lemma on the logarithmic derivative. Let $f$ be meromorphic of finite order and let $c \in \C$. Then the proximity function of $f(z + c)/f(z)$ is small compared with the Nevanlinna characteristic of $f$. With this, the classical methods of Nevanlinna theory for complex differential equations carry over to difference equations. It has been proposed as a test of integrability. Equations with enough meromorphic solutions of finite order are expected to be discrete analogues of the Painlevé equations.