Best to ask by example. Given the recurrence relation $a_{n}=a_{n-1}+a_{n-2}$, and some given initial conditions, we can find a similar relation for the generating function for the sequence, $f(x)=\sum_{n=0}^\infty a_nx^n$: $f(x)=xf(x)+x^2f(x)+c(x)$ Where $c(x)$ is a polynomial encoding the initial conditions. My main question is how can this polynomial be computed as painlessly as possible from the initial conditions? This is interesting not only for Fibonacci but in general, of course.
A similar, probably equivalent questions is this: If for some sequence we have a rational generating function $\frac{p(x)}{q(x)}$ then the coefficients of $q(x)$ are exactly the coefficients of the recurrence relation for the sequence. Also, $p(x)$ is dependent on the initial conditions - but again, it's not clear to me how to compute those initial conditions from $p(x)$.