For Fibonacci numbers I usually see generating function $\frac{1}{1-x-x^2}$ or $\frac{x}{1-x-x^2}$ depending on initial terms.
But the denominator, $1-x-x^2$, seems different from the usual characteristic polynomial $x^2 - x - 1 = 0$. It doesn't appear to be a simple multiplication by $-1$ either since the $x$ term is negative in both cases.
I'm asking this simpler question because I am getting strange and unexpected results over here.
If you write down the recurrence relation for Fibonacci numbers and the consequence it has on the generating function, you will immediately notice that $1-x-x^2$ is just the characteristic polynomial of the Fibonacci sequence with its coefficients written in the opposite order, i.e. $x^2\,p\left(\frac{1}{x}\right)$ with $p(x)$ being $x^2-x-1$.
Addendum: such "reversal" process has an interesting side-effect: the radius of convergence of the generating function is exactly the reciprocal of the modulus of the largest root of the characteristic polynomial.