Given a polynomial function $f$ on $1$ variable, the following Taylor series: $$\sum_{n=0}^\infty \frac{h^n}{n!} \frac{d^n f}{dx^n}(x)$$ can be written in closed form as $f(x+h)$. (A cool fact to observe, which is relevant later, is that although the above expression seems to make sense only in a field of characteristic zero, a priori, it is true in characteristic $p$ as well, as any $p$ in the denominator cancels out.)
Now, suppose we instead have a polynomial $f$ in $2$ variables $x$ and $y$, and I consider a similar Taylor series-like expression: $$\sum_{n=0}^\infty \frac{h^n}{n!} \frac{d^{2n}f}{dx^ndy^n}(x,y). $$ Is there a nice, concise, closed form expression for this sum? I know that this question is slightly vague, but hopefully the $1$ variable base indicates the direction I am looking at. (Really though, I’d be interested in any way of writing this expression more simply.)
Is the problem easier if I assume that I am working in positive characteristic? (This is the case I am chiefly interested in.) In this case, as can be checked easily, all the terms $n=p$ onwards die, but I don’t know if that helps.