Let $P$ be an analytic linear differential operator defined on some open interval $X=(a,b)$ and $\mathcal{M}=\mathcal{D}_X / \mathcal{D}_X \bullet P$ the corresponding $\mathcal{D}$-module. I'm trying to understand **what the correct definition of a Green's function is for $P$ in this context.**

For starters I'm not sure what kind of algebraic object it is. Should it be a...

- A function on $X \times X - \triangle$?
- A $1$-form on $X \times X - \triangle$?
- A class in local cohomology $H^1_{\triangle}(X \times X, \Omega^1_{X \times X})$?
- A $\mathcal{D}_{X \times X}$- module?
- None of the above... :( ?

Obviously I'm phrasing everything in the most simple terms but ideally I'm looking for the answer that will generalize easily and immediately to general case.

If there is a standard reliable reference for these kind of questions I'd be happy to know about it.