整函数导数的幂次分担的唯一性

Uniqueness of entire function for sharing condition of the power of derivative

证明了一个关于整函数导数幂次分担条件的唯一性结论,如果f(z)和g(z)是两个非常数整函数,c 1,c 2是两个有穷复数,n,k是两个正整数,且n≥3,若[f(k)(z)]n-c 1和[g(k)(z)]n-c 2在C上IM分担4个互不相同的有穷复数,那么,当c 1≠c 2时,f(z)和g(z)均为次数不超过k的多项式;当c 1=c 2时,f(z)=t ng(z)+p(z),其中t n=1,p(z)为次数不超过k-1的多项式。

In this paper,we prove the uniqueness of sharing condition for the power of the derivative of entire functions.If f(z)and g(z)are two nonconstant entire functions,c1,c2 are two finite complex numbers,n,k are two positive integers and n≥3,suppose[f(k)(z)]n-c1and[g(k)(z)]n-c2 share four different finite complex numbers on by IM,then when c1≠c2,f(z)and g(z)are polynomials with degree no more than k;when c1=c2f(z)=tng(z)+p(z),where tn=1,p(z)is a polynomial with degree no more than k-1.