On certain entire functions which together with their derivatives are prime

Abstract

Introduction. In studying the factorization of meromorphic functions, we may ask the relationship between the factors of a function and those of its derivatives. A meromorphic function F(z)=f(g(z)) is said to have / and g as left and right factors, respectively, provided that f is meromorphic and g is entire (g may be meromorphic if /is rational). F(z) is said to be prime (pseudoprime, left-prime, right-prime) if every factorization of the above form into factors implies either / is linear or g is linear (either / is rational or g is a polynomial, / is linear whenever g is transcendental, g is linear whenever / is transcendental). When factors are restricted to entire functions, it is called to be a factorization in entire sense. In this paper only entire factors will be considered. We note here it is known ([7]) that, when F is not periodic, then F is prime if F is prime in entire sense. Because of this observation, in this note entire factors only need to be considered. Suppose that a transcendental entire function F(z) is prime. Does it follow that its tt-th derivative F\z) is also prime? In general, there is not much that we can really say. For example, take F(z)=e*+z, then F is known to be prime (cf. [5] or [10] etc.), but F'(z)'=e+1 is not prime (F'(z) is pseudo-prime). Further take F(z)=exp [e]+z, then F(z) is prime (cf. [6] or [10]), but F'(z) =e-exp [>]+l is composite (both factors are transcendental). While if we take F(z)=z-e, then F(z) is prime for n=0,1, (F(z)=F(z)). (Note that F(z)=z-exp [>] is prime but F'(z) is not prime, since F'(z) is an even function.) Another interesting example is given by