【摘要】 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.更多还原

【Abstract】 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.更多还原