Links between difference and differential equations? Does there exist any correspondence between difference equations and differential equations?
In particular, can one cast some classes of ODEs into difference equations or vice versa?

(Adding to user26872's answer as this was not obvious to me so it might help someone else going through his derivation)
The identity
$$x_{n+1} = Ex_n = \sum\limits_{k=0}^\infty \frac{h^k}{k!}D^k x_n=e^{hD}x_n$$
is true if we consider the following. Let $x_n = x(t_n)$. Let's assume that $x(t)$ is differentiable around some point $t_n$, then it's Taylor series can be defined as
$$x(t) = \sum\limits_{k=0}^\infty \frac{x^{(k)}(t_n)}{k!}(t-t_n)^k.$$
If we now perform the same expansion at $t' = t_n+h$ we have
$$x(t_n+h) = \sum\limits_{k=0}^\infty \frac{x^{(k)}(t_n)}{k!}h^k = \sum\limits_{k=0}^\infty \frac{h^k D^k}{k!}x(t) = e^{hD} x(t_n)$$
and so
$$x_{n+1} = Ex_n \leftrightarrow x(t_n + h) = E x(t_n)$$
thus giving
$$E = e^{hD}.$$
