Change of coordinate and Linear Ode

Question

Let $G(z)$ be a rational function. So if we have a series $$S(x):=\sum_{n}a_n x^n $$ where $$ a_n = \prod_{i=1}^{n}G((i-1)h) $$ We can conclude that the series satisfies a Linear differential equation with polynomial coefficients as the ratio of the $a_{n+1}/a_n $ is rational. Now we have a series of the form $$Sexp(x):=\sum_{n}b_n x^n \tag{*}$$ where $$ b_n = \exp\left(\sum_{i=1}^{n}G((i-1)h) \right)$$ can we conclude that there exist a change of coordinate say $X(x)$ such that it will also satisfy linear differential equation with polynomial coefficient? So what would be that change of coordinate?