关于一类高阶齐次线性微分方程解的增长性

摘要

研究了高阶线性齐次微分方程f(k)+Ak-1(z)Rk-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 equationrnf(k)+Ak-1(z)Pk-1(ez)f'+…+A1(z)P1(ez)f'+A0(z)P0(ez)f=0rnis investigated, whereAj(z)≠0(j=0,1,…,k-1) are entire functions, Pj(ez)(j=0,1,…,k-1) are nonconstant polynomials of ez 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.