Linear Factorization of Hypercyclic Functions for Differential Operators

On the Fr\'{e}chet space of entire functions $H(\mathbb{C})$, we show that every nonscalar continuous linear operator $L:H(\mathbb{C})\to H(\mathbb{C})$ which commutes with differentiation has a hypercyclic vector $f(z)$ in the form of the infinite product of linear polynomials: $$f(z) = \prod_{j=1}^\infty \, \left( 1-\frac{z}{a_j}\right),$$ where each $a_j$ is a nonzero complex number.