On the primeness of \(z+\cos \alpha(z)\) (Q1260841)

An entire function \(F\) is called prime if it cannot be written in the form \(F(z)=f(g(z))\) with nonlinear entire functions \(f\) and \(g\). The main result of this paper is that if \(\alpha\) is a nonconstant entire function, then \(z+\cos\alpha(z)\) is prime. The proof that there do not exist transcendental entire functions \(f\) and \(g\) such that \(z+\cos\alpha(z)=f(g(z))\) uses a method which is based on Wiman-Valiron theory and was introduced by the reviewer [Math. Z. 204, 381-390 (1990; Zbl 0681.30014)] to prove that \(z+p(z)e^{\alpha(z)}\) is prime for any polynomial \(p\), that is, compositions of entire functions have infinitely many fixpoints. The proof that there does not exist a factorization \(z+\cos\alpha(z)=f(g(z))\) where \(f\) or \(g\) is a nonlinear polynomial requires different arguments. To rule out that \(g\) is a nonlinear polynomial two different branches of \(g^{-1}\) in the equation \(g^{- 1}+\cos\alpha(g^{-1})=f\) are considered while Nevanlinna theory is used to obtain a contradiction in the case that \(f\) is a nonlinear polynomial. Finally, it is shown in this paper that if \(p\) is a polynomial and \(z+p(\cos\alpha(z))=f(g(z))\) where \(f\) or \(g\) are entire functions, then \(g\) must be linear if \(f\) is transcendental.