Question

We consider the asymptotics of the coefficients of generating function $y(x)$, which is defined by the implicit function $y= F(x,y)$.

Let $F(x,y)$ be a rational function in $x$ and $y$, such that $F(x,y)$ has non-negative coefficients of development in $x$ and $y$. Further $F(x,y)$ is not a linear function of $y$. Can we conculde that $[x^n]y(x)\sim \rho^{-n} n^{-\frac{3}{2}}$?

If not, what kind of additional condition do we need to prove such asymptotics?

It seems that it can be derived from singularity analysis. One reference could be Flajolet and Sedgewick's Book "Analytic Combinatorics". However the proof in there is rather obscure to me.

I think that we still need to additional condition to make that conclusion, such as there exist $r>0$ and $s>0$ satisfying \begin{equation} F(r,s)=s, F_x(r,s)=1. \end{equation}

Any reference about this topic would be appreciated. Thank you in advance.

Answer 1

A set of conditions which imply this asymptotic is given by Theorem VII.6, which you mention in your remark. These additional condition is that the equation is "a-proper". The theorem is not true without this additional condition.