Solutions of complex difference and q-difference equations

Yue Wang

doi:10.1186/s13662-016-0790-x

Abstract

Using Nevanlinna theory of the value distribution of meromorphic functions, the growth of entire solutions and the form of transcendental meromorphic solutions of some types of systems of higher-order complex difference equations are investigated. Some new results are obtained. We also investigate the problem of the existence of solutions of complex q-difference equations, and we obtain some new results, which are different from analogue differential equations. Improvements and extensions of some results in the literature are presented. Some examples show that our results are, in a sense, the best possible.

Highlights

1 Introduction and notation Throughout the paper, we use the standard notations of the Nevanlinna theory of meromorphic functions
In Section, we study the existence of admissible meromorphic solutions of systems of complex difference equation ( . ), the problem of the order of entire solutions of systems of complex difference equation ( . ), and the form of transcendental meromorphic solutions of systems of complex difference equation ( . ), and obtain three theorems
In Section, we study the problem of the existence of solutions of complex q-difference equations ( . ), ( . ) and ( . ), and we obtain three theorems, and we give some remarks and some examples, which show that the results obtained in Section are, in a sense, the best possible

Summary

Introduction

Introduction and notationThroughout the paper, we use the standard notations of the Nevanlinna theory of meromorphic functions (see [ ]).Many authors, such as Weissenborn [ ], Toda [ ], Gao [ , ] and Malmquist [ ] etc. have investigated complex differential equation theory, they obtained many results, such as the following.Theorem A (Malmquist theorem) (see [ ]) Let a (z), . . . , ap(z), b (z), . . . , bq(z) be rational functions. First of all, we will investigate solutions of systems of the higher-order complex difference equations In Section , we study the existence of admissible meromorphic solutions of systems of complex difference equation

Results

Conclusion