I've come across a type of differential equation and I'm wondering if it can be solved, or if there are special conditions on the equation in order that it is solvable.
The equation can be written rather concisely as:
$c_{jk} \in \mathbb{C}$
$$\sum_{j,k=0} c_{jk} (x)_k\frac{d^j}{dx^j} f(x+k)=0$$
where this is a finite sum and $(x)_k = x \cdot (x+1) \cdots (x+k-1) = \frac{\Gamma(x+k)}{\Gamma(x)}$.
I want to know if I specify initial conditions whether or not there is some unique function satisfying this differential equation. I'm unsure because the $f(x+k)$ as opposed to $f(x)$ throws me off a bit. I've never seen differential equations with the transfer operator before.