We call a function $f(z)\in \mathbb{C}[[z]]$ holonomic if there exists a differential operator of the from
$$ a_{i}(x)(D_x)^i + \ldots+a_{0}(x) $$ which annihilated $f(x)$. $D_{x}=\frac{\partial}{\partial x}$ and $a_i (x)\in \mathbb{C}(x)$ are polynomials.
For example $e^x=\sum_{d}\frac{1}{d!}x^d$ is holonomic.
The following series $$\sum_{n=0}^{\infty} \left(\prod_{i=1}^{n}G( iz)\right)z^n \tag{*}$$ is a holonomic function in $z$ if $G(z)$ is rational. My question is when $G(z) =e^z$ what kind of differential we obtain? We know that $e^z$ can be approximated by rational function. In that case, we get a sequence of differential equations how does it relate to help conclude existence of differential eq that annihilate $(*)$. There are many closure properties share by holonomic function. When those properties can be carried to the function like (*)? Any literature, reference will be helpful.