Good Linear Operators and Meromorphic Solutions of Functional Equations

Abstract

Nevanlinna theory provides us with many tools applicable to the study of value distribution of meromorphic solutions of differential equations. Analogues of some of these tools have been recently developed for difference, q-difference, and ultradiscrete equations. In many cases, the methodologies used in the study of meromorphic solutions of differential, difference, and q-difference equations are largely similar. The purpose of this paper is to collect some of these tools in a common toolbox for the study of general classes of functional equations by introducing notion of a good linear operator, which satisfies certain regularity conditions in terms of value distribution theory. As an example case, we apply our methods to study the growth of meromorphic solutions of the functional equation M(z,f)+P(z,f)=h(z), where M(z,f) is a linear polynomial in f and L(f), where L is good linear operator, P(z,f) is a polynomial in f with degree deg P≥2, both with small meromorphic coefficients, and h(z) is a meromorphic function.