一类高阶线性微分方程解的增长性

The Growth of Solutions of Some Higher Order Linear Differential Equation

本文讨论一类一般的齐次和非齐次高阶线性微分方程解的增长性,证明了当整函数F,A_j,D_j和s≥1次多项式P_j(z)(j=0,1,…,k-1)满足某些条件时,方程(其中k≥2),f^(k)+ (A_(k-1)(z)e^(P_(k-1)(z))+D_(k-1)(z))f^((k-1))+…+(A_0(z)e^(P_0(z))+D_0(z))f=F当F≡0时,所有非零解具无穷级；当F≠0时,至多除去一个有限级解f_0外,其余所有解均满足■(f)=λ(f)=σ(f)=∞且σ_2(f)≤max{s,σ(F)},从而推广了M．Frei,M．Ozawa,G．Gundersen,J．K．Langley,陈宗煊,李纯红等人的结果。

In this paper, the growth of solutions of some general homogeneous and nonhomogeneous higher order linear differential equation is investigated. It is proved that under some conditions for entire functions F, Aj, Dj and polynomials Pj（z） with degree s ≥ 1 （j = 0, 1,… ,k - 1）, the equation （where k ≥ 2） f^（k） ＋ （Ak-1（z）e^Pk-1（z） ＋ Dk-1（z））f^（k-1） ＋ … ＋ （Ao（z）e^P0（z） ＋ Do（z））f = F satisfy the properties： when F = 0, all non-zero solutions are of infinite order; when F ≠ 0, at most one exceptional solution f0 with finite order, all other solutions satisfy λ^-（f） = λ（f） = δ（f） = ∞ and δ2（f） ≤ max{s,δ（F）}. These results generalize those of M. Frei, M. Ozawa, G. Gundersen, J. K. Langley, Chen Z. X., Li C. H..