Let $G(z)=\sum_{k\geq 0} a_kz^k$ be a generating function such that $z^aG(1-z)=P(z)$, where $P(z)$ is a polynomial and $a$ is a positive integer. I'm interested in $P(z)[z^n]$, the coefficient in front of $z^n$ in $P(z)$. A really simple example is $G(z)=z/(1-z)$ , $a=1$ and $P(z)=1-z$ with $P(z)[z^1]=-1$.
Here are some added difficulties. I know $G(z)$ converges for $|z|<1$. Unfortunately, $G(1)=\infty$, so we have no absolute convergence at $z=1$, hence we can't just expand $(1-z)^k$ and switch order of summation to extract $z^n$ via convolution or abstractly by contour integration. I'm interested in ways to approach this problem. I can assume $a_i\geq0$ for all $i$ and I can assume I know the exact minimum and maximum degree occurring in $P(z)$ (they are 0,1 respectively for $P(z)=1-z$). I presume that one way of going about this problem is to regularize $(1-z)^k$ as $(q-z)^k$ and then take the limit $q\rightarrow 1$. Is this a feasible approach? I feel like I'll still need some kind of generalized Abel theorem to justify the final result. Is there a good reference for this kind of a problem?