Is there any mathematic meaning of generating function at some value?

Question

Out of curiosity I am trying to learn some material about generating functions. Now I understand that if I will expand Fibonacci generating function, $f(x) = \frac{1}{1-x-x^2}$ I will get a series where coefficients are Fibonacci numbers.

What I can't understand is whether there is any mathematical meaning $f(a)$ where $a$ is some value? From what I learnt so far (I hope that I understood this correctly) it looks like there is some meaning for any x that is in the convergence radius ($\frac{2}{1+ \sqrt{5}}$).

So what is the mathematical meaning of f(1.5) or f(0) or f(0.5) for example?

Answer 1

I'm quite, quite far from an expert but I do have a specific literature reference to contribute: the whole section 2.4 Power series, analytic theory in the book generatingfunctionology by Herbert S. Wilf. The second edition (1994) is downloadable (there is a third edition but not on the web) and this is how the author introduces the topic in the section:

If they do converge though, and they represent functions, that's a big advantage, for then we may be in a position to find analytic information about the recurrence relation that might not otherwise be easily obtainable.

I think the relevant message is: If you have the skills, it's possible to learn something about the coefficients of the series from the function represented by the power series. If the power series represents a function. But see Wilf for the actual wisdom.