一类高阶周期微分方程的解和小函数的关系

The Relation Between Solutions of a class of HigherOrder Differential Equations With Periodic Coefficients andFunctions of Small Growth

摘要: 研究了微分方程~f^(k)+P_k-1(\mathrme^z)+Q_k-1(\mathrme^-z)f^(k-1)+\cdots+P_0(\mathrme^z)+Q_0(\mathrme^-z)f=0和 ~f^(k)+P_k-1(\mathrme^z)+Q_k-1(\mathrme^-z)f^(k-1)+\cdots+P_0(\mathrme^z)+Q_0(\mathrme^-z)f=R_1(\mathrme^z)+R_2(\mathrme^-z)~的解以及它们的一阶导数与小函数的关系, 其中~P_j(z)~,~Q_j(z)~(j=0，1,2，\cdots，k-1)~和~R_i(z)(i=1，2)~是关于~z~的多项式.

Abstract: Linear differential equations ~f^(k)+P_k-1(\mathrme^z)+Q_k-1(\mathrme^-z)f^(k-1)+\cdots+P_0(\mathrme^z)+Q_0(\mathrme^-z)f=0~and ~f^(k)+P_k-1(\mathrme^z)+Q_k-1(\mathrme^-z)f^(k-1)+\cdots+P_0(\mathrme^z)+Q_0(\mathrme^-z)f=R_1(\mathrme^z)+R_2(\mathrme^-z)~ where ~P_j(z)~,~Q_j(z)~(j=0，1,2，\cdots，k-1)~and~R_i(z)(i=1，2)~are polynomials in z are investigated. The relationship between solutions and their 1th derivatives and small functions are studied.