Nevanlinna theory for the difference operator

TL;DR

This paper extends classical Nevanlinna theory to the setting of difference operators, analyzing the distribution of paired points of meromorphic functions and establishing analogues of key theorems with applications to difference equations.

Contribution

It introduces a Nevanlinna theory framework for difference operators, replacing ramification with paired point counts, and proves analogues of fundamental theorems.

Findings

Derived a second main theorem analogue with paired points

Established Nevanlinna defect relations for difference setting

Applied results to difference equations and provided illustrative examples

Abstract

Certain estimates involving the derivative $f\mapsto f'$ of a meromorphic function play key roles in the construction and applications of classical Nevanlinna theory. The purpose of this study is to extend the usual Nevanlinna theory to a theory for the exact difference $f\mapsto \Delta f=f(z+c)-f(z)$. An $a$-point of a meromorphic function $f$ is said to be $c$-paired at $z\in\C$ if $f(z)=a=f(z+c)$ for a fixed constant $c\in\C$. In this paper the distribution of paired points of finite-order meromorphic functions is studied. One of the main results is an analogue of the second main theorem of Nevanlinna theory, where the usual ramification term is replaced by a quantity expressed in terms of the number of paired points of $f$. Corollaries of the theorem include analogues of the Nevanlinna defect relation, Picard's theorem and Nevanlinna's five value theorem. Applications to difference equations are discussed, and a number of examples illustrating the use and sharpness of the results are given.