摘要
本文研究复合意义下整函数及其导函数的分解问题.文[1,P^213]中曾提出下述臆测设F(z)为一有穷级的超越整函数,若F(z)为E-拟素的,所有其导函数F^(n)(z)(n=1.2,…)亦必为拟素的(或试构造一个有穷级超越拟素整函数F(z),对某个k,F^(k)(z)非为拟素的).问题1 当F(z)为无穷级时如何?问题2 是否可找到一个整函数F(z),使得在函数簇{F^(n)(z);n=0,1,2,…}中有无限多个元素为复合的及无限多个元素为素的?
In this paper, we deal with the factorization of entire functions in the sense of composition, and prove the following.Theorem. For any real number ρ≥1, there exists a transcendental entire function F(z) of order p , such that there are infinitely many prime elements and infinitely many composite elements in the class {F(n) n = 0.1, 2, ···}.
