It is well-known that a Taylor series is an exponential generating function. $$ F = \sum_k a_k\frac{x^k}{k!} $$ where the coefficients are $k$th order derivatives of $F$.
It is also known that derivatives have a geometrical interpretation as elements of a tangent space above a point. I have been wondering if the coefficients of a generating function can be mapped to a set of $k$th order jet spaces above a point.
Thus, how can we construct the generating function given this picture? The coefficients would be the collection of all spaces above a point, how would the $x^k$ terms look? Some kind of product space?
Can an (ordinary) generating function be described in the language of differential geometry and does this introduce any interesting perspectives?
I have little background in maths.