Asymptotic coefficients of generating function $\frac{F(z)}{G(z)}$

Question

I have the OGF in form of $\frac{F(z)}{G(z)}$, both $F(z), G(z)$ are polynomial expression and relatively prime.

Do we have the systematic way to estimate $[z^n]$?

I know that we can estimate $[z^n]$ base on the smallest modulus $1/\beta$ of $G(z) = 0$. i.e $[z^n] = C\beta^nn^{v-1}$, $C=v\frac{(-\beta)^vF(1/\beta)}{G^{(v)}(1/\beta)}$, $v$ is the multiplicity of $\beta$. However, how to estimate when $G(z)$ does have complex roots, for example, $G(z) = 2z^2 - 2z + 1$

Thanks,

Answer 1

Expand $F(z)/G(z)$ in partial fractions in terms of the roots of $G$. The Maclaurin series for $1/(r - z)^k = r^{-k} (1-z/r)^{-k} $ is $ \sum_{n=0}^\infty {{k+n-1} \choose k} z^n/r^{n+k}$, so the largest coefficients (asymptotically as $n \to \infty$) correspond to the root(s) closest in absolute value to $0$, with ties broken by the highest power $k$.