一类高阶微分方程亚纯解的增长性

ON THE GROWTH OF MEROMORPHIC SOLUTIONS OF A CLASS OF HIGHER ORDER DIFFERENTIAL EQUATIONS

研究了高阶微分方程f(k)+Ak-1f(k-1)+…+A1f′+A0f=0亚纯解的增长性,得到了:假设b≠0是复常数,定义指标集Λ={a|a=cab,-1<ca<1},有限集ΛjΛ(j=0,…,k-1),A0=Hbebz+∑a∈Λ0H0aeaz,Aj=∑a∈ΛjHjaaze(j=1,…,k-1),其中Hb和Hja(a∈Λj,j=0,…,k-1)都是级小于1的亚纯函数,且Hb■0,若微分方程有非零亚纯函数解f,则每个非零亚纯解的级为无穷.特别地,如果f的极点重数一致有界,则它的超级为1.

The growth of meromorphic solutions of the differential equation f^（k）＋Ak-1f^（k-1）＋…＋A1f′＋A0f=0 investigated. Suppose that b ≠0 is a complex constant. Define a setA={a｜a=cab,-1〈ca〈1},A（j=0,…,k-1）,A0=Hbe^bc＋∑∈A0H0de^az,Aj=∑a∈Aj Hja e （j=1,…,k-1）, where Hb≠0,and Hja（a∈Aj,j=0,…,k-1） are meromorphic functions with their order of growth smaller than 1. The following result is obtained： σ（f） = w if f（≠0） is the meromorphic solution of the equation above. Furthermore, if the order of pole of the solution f is consistent bounded, then σ2 （f） = 1.