关于一类高阶齐次线性微分方程解的增长性

On the Growth of Solutions of Higher Order Homogeneous Linear Differential Equations

研究了高阶线性齐次微分方程f(k)+Ak-1(z)Pk-1(ez)f'+…+A1(z)P1(ez)f'+A0(z)P0(ez)f=0解的增长性,其中Aj(z)■0(j=0,1,…,k-1)是整函数,Pj(ez)(j=0,1,…,k-1)是ez的非常数多项式,它们的常数项都为零,且次数不相等。证明了该微分方程的每一个非零解有无穷级。

The growth of solutions of higher order homogeneous linear differential equationf（1）＋A k-1（e2）f1＋……＋A1（z）P1（e2）f＋A0（z）P0（ez）f=0 is investigated, whereA,（z）≠O（j=0,1,…,k-1）are entire functions,P1（ez）（J=0,1,…,k-1）are nonconstant polynomials of e~ without constant term, and deg P（z） is not equal to deg Q（z） . It is showed that the order of growth of each nonzero solution of the above equations is infinite.