一类高阶线性微分方程解的超级估计  

Estimate of the Hyper-Order of Solutions for Certain Higher Order Linear Differential Equations

作　　者：范水平[1] 陈宗煊[1] 

机构地区：[1]华南师范大学数学科学学院,广州510631

出　　处：《华南师范大学学报（自然科学版）》2015年第4期146-149,共4页Journal of South China Normal University(Natural Science Edition)

基　　金：国家自然科学基金项目(11171119);广东省自然科学基金项目(2014A030313422)

摘　　要：利用Nevanlinna的值分布理论和分类讨论的思想方法,研究了一类高阶齐次线性微分方程f(k)+Hk-1f(k-1)+…+H1f'+H0f=0解的增长性,得到了一些有意义的结果:当Hj(z)(j=0,1,…,k-1)是整函数时,根据线性微分方程的一般理论,上述方程的每个解都是整函数.当方程系数满足:Hj(z)=hj(z)ePj(z)(j=0,1,…,k-1),Pj(z)是首项系数为aj的n(n≥1)次多项式,hj(z)为整函数,σ(hj(z))<n,aj是复数,存在as和al,使得l>s,as=dseiφ,al=-dleiφ,ds>0,dl>0.对j≠s,l,aj=djeiφ(dj≥0)或aj=-djeiφ,max{dj;j≠s,l}=d<min{ds,dl},hshl0,给出了该微分方程的每个超越解的超级的精确估计.结果可以推广到亚纯函数系数的微分方程.By utilizing Nevanlinna＇s value distribution theory of meromorphic functions and categorized discussion method, the growth of solutions of higher order differential equations is investigated and some important results are obtained. When Hj（z）（j=0,1,…,k-1） are entire functions, according to the general theory of linear differential equations, every solution of the above equations with entire coefficients is entire function. When the coefficients of the above equations satisfy：Hj（z）=hj（z）e^Pj（z）（j=0,1,…,k-1）,Pj（z）are polynomials.with degree n and leading coefficients aj, hj（z） are entire functions,σ（hj（z））〈n,aj are complex number,l〉s,as=dse^iφ,al=-dle^iφ,ds〉0,dl〉0.For j≠s,l,aj=dje^iφ（dj≥0） or aj=-dje^iφ,max{dj;j≠s,l}=d〈min{ds,dl},hshl≠0,and the precise estimation of the hyper-order of their transcendental solutions of the class of linear differential equations is given. The results obtained in this paper can be extended to differential equations with meromorphic coefficients.

关 键 词：微分方程 整函数 超级 

分 类 号：O174.5[理学—数学]