I found now the following theorem:
Let $f,g\in \mathbb{C}[e^{\lambda}x]$. Suppose that in the ring of analytic (over $\mathbb{C}$) functions $f \mid g$, i.e., there is an analytic (over $ \mathbb{C}$) function $h$ such that $g=fh$, or equivalently each root of $f$ (in $ \mathbb{C}$) with multiplicity $m$ is a root of $g$ with multiplicity at least $m$. Then $f \mid g$ in the ring $\mathbb{C}[e^{\lambda}x]$, i.e., there is a $h\in \mathbb{C}[e^{\lambda}x]$ such that $g=fh$.
