Understanding the conditions for the Lagrange Inversion Formula

Question

Lagrange Inversion Formula: Let $A(u) = \sum_{k \ge 0} a_k z^k$ be a power series in $\mathbb{C}[[z]]$ with $a_0 \ne 0$. Then the equation

$$B(z) = zA(B(Z)) \qquad (1)$$

has a unique solution $B(z) \in \mathbb{C}[[z]]$ such that $b_n = \frac{1}{n}[z^{n-1}]A(z)^n$.

Consider what happens if $a_0 = 0$. What value does $[z^{n-1}]A(z)^n$ take? Is $B(z)$ still a solution to $(1)$?

I understand that the condition $a_0 \ne 0$ is equivalent to $A(z)$ having a (multiplicative) inverse (see Wikipedia). However, I do not see what, in general, can be now said about $[z^{n-1}]A(z)^n$. Could you please give me a hint?

Answer 1

Hint: If $a_0=0$ we have \begin{align*} \left(A(z)\right)^n&=\left(a_1z+a_2z^2+\cdots\right)^n=a_1^nz^n+\cdots \end{align*} so that the $n$-th power of $A$ has terms in $z$ with powers greater or equal $n$ only. It follows that for $n\geq 1$ \begin{align*} \color{blue}{[z^{n-1}]\left(A(z)\right)^n=0} \end{align*}