An example concerning infinite factorizations of transcendental entire functions (Q1567437)

A nonlinear entire function is called prime if it cannot be expressed as the composition of nonlinear entire functions. By a classical result of Ritt, a polynomial has an essentially unique factorization into prime polynomials. In particular, the number of prime factors in two factorizations is equal. Here, it is shown that there is no bound on the number of prime factors in a factorization of an entire function. More specifically, there exists a sequence \((c_n)\) such that \(F_n(z)= (c_ne^z+ z)\circ\cdots\circ(c_1e^z+ z)\) converges locally uniformly to an entire function \(F\) and \(F(z)= H_n(z)\circ (c_ne^z+ z)\circ\cdots\circ (c_1 e^z+ z)\) for some entire function \(H_n\).