关于齐次线性微分方程的复振荡

On the Complex Oscillation of Homogeneous Linear Differential Equations

考虑二阶方程 f″ +(R1(z)eP1(z) +R2 (z)eP2 (z) +Q(z) ) f =0 ,其中P1(z) =ζ1zn+… ,P2 (z) =ζ2 zn+…为非常数多项式 ,R1(z) 0 ,R2 (z) 0 ,Q(z)为级小于n的整函数 ,ζ2 / ζ1是实数 ,ρ =ζ2 / ζ1,得到下列结果 :(i)若 0 <ρ <1/ 2 ,则上述微分方程的任一非平凡解的零点收敛指数大于或等于n ;(ii)若Q(z)≡ 0 ,3/ 4 <ρ <1,则上述微分方程任一非平凡解的零点收敛指数大于或等于n .

In this paper,we consider the second order equation f″+(R 1e P 1 +R 2e P 2 +Q)f=0,where P 1(z)=ζz n+…,P 2(z)=ζ 2z n+…(ζ 1ζ 2≠0,ρ=ζ 1/ζ 2 is real),are non constant polynomials ,R 1(z)0,R 2(z)0,Q(z)are entire functions and their orders are less than n.We obtain the following results:(i) if 0<ρ<1/2,then any non trivial solution of the equation satisfies λ(f)≥n;(ii)if Q(z)≡0,3/4<ρ<1,then any non trivial solution of the equation satisfies λ(f)≥n.