一类高阶周期线性微分方程解的性质及复振荡

A PROPERTY OF SOLUTIONS AND THE COMPLEX OSCILLATION FOR A CLASS OF HIGHER ORDER PERIODIC LINEAR DIFFERENTIAL EQUATIONS

本文对一类超越型高阶周期线性微分方程解的性质及复振荡证明了:设B(ξ)=g1(1/ξ)+g2(ξ),其中g1(t)和g2(t)是整函数,以及g1(t)(或gz(t))是超越的且级小于1/2.令A(z)=B(e^z).(i)如果方程ω^(k)+A(z)ω=0(k≥3)有解f(z)=0满足log^+N(r,手1/f)=O(r),则,f(z)和f(z+2πi)线性相关.(ii)如果B(ξ)在ξ=∞(或相应地在ξ=0)有一p阶极点,p不被k整除,则前方程的任一解,f(z)≠o的零点收敛指数都是无穷,且更强的结论log^+N(r,1/f)≠0(r)成立.

On the investigations of a property of solutions and the complex oscillation for the transcendental-type periodic seond order linear differential equations, a breakthrough was made in 1990. But the method is not valid for the higher order differential equations. In this paper we prove using a new method: Let B(ξ)=g1(l/ξ) + g2(ξ), where g1(t),g2(t) are entire, and g1(t) (or g2(t) is transcendental with order less than 1/2. Set A(z) = B(ez). (i) If the equation w(k) + A(z)w = 0 (k ≥ 3) has a solution f(z)(?) 0 satisfying log+ N(r, 1/f) = O (r), then f(z) and f(z + 2πi) are linearly dependent. (ii) If B(ξ) at ξ = ∞ (or resp. at ξ = 0) has a pole of order p, where p is not divisible by k, then for every solution f(z) (?) 0 of above equation, the exponent of convergence of its zero-sequence is infinite, and, in fact, the stronger conclusion log+ N(r, 1/f) ≠ o (r) holds.