$\frac{g(x)}{f^*(x)}=\sum_{n\geq 0}s_nx^n$ generates recursion

Question

I need help proving that in $\mathbb{F}_q$, if $g$ is a polynomial of degree less than $k$ and $f^*(x)=1-(a_{k-1}x+\dots+a_0x^k)$ a polynomial of degree $k$, then $$\frac{g(x)}{f^*(x)}=\sum_{n\geq 0}s_nx^n$$ fulfills the recursion $s_{n+k}=a_{k-1}s_{n+k-1}+\dots+a_1s_{n+1}+a_0s_n$.

I know that this is all related to LFSRs, but cannot see how to prove the result.