Fermat型复微分差分方程的整函数解

Entire Solutions of Fermat Type Complex Differential Difference Equations

文章利用复微分方程理论和复差分方程理论研究了形式为(μf(z)+λf′(z))^(2)+f(z)^(2)=P(z)的复微分方程和形式为(μf(z)+λf′(z))^(2)+f(z+c)^(2)=P(z)的复微分-差分方程的任意整函数解的存在形式。首先,用Weierstrass因式分解定理将两个方程进行分解,计算出f(z)和μf(z)+λf′(z)的具体形式;其次,对因式分解后产生的指数h(z)进行讨论,分为h(z)为常数和h(z)为非常数整函数两种情形;最后,研究每一种情形下整函数解中各个变量之间的关系。文章得到了两个关于Fermat型方程的整函数解的存在形式,在一定范围内推广和改进了前人的结论。

This paper investigates the existence of entire solutions of complex differential equations in the form of(μf(z)+λf′(z))^(2)+f(z)^(2)=P(z)and complex differential difference equations in the form of(μf(z)+λf′(z))^(2)+f(z+c)^(2)=P(z)through the complex differential equations theory and the complex difference equations theory.Firstly,the two equations were factored using Weierstrass factorization theorem to compute the specific forms of f(z)andμf(z)+λf′(z);Secondly,the exponent h(z)resulting from the factorization was discussed and divided into two cases,namely,h(z)as a constant and h(z)as a non-constant entire function;Lastly,the relationship between the individual variables in the entire solution was investigated in each of the cases.This article obtains two forms of the existence of entire solutions for Fermat type equations,generalising and improving the previous conclusions from a certain range.