I am trying to understand the last line in the excerpt below from my course notes. The discussion concerns an algorithm for turning a (compact, i.e. finite) expression for a generating function into a formula for the coefficients of the generating function.
The "we can express" part of the last line is giving me trouble. I cannot see why the two products are equal. Is there something obvious I'm missing? I see that they are both equal at $x = 0$, and that we can put $r_k$ in the denominator when we know none of the $r_k$ equal $0$, but why are the two expressions equal? I don't get it.
I appreciate any help.
The products are not the same, but $g(x)=\lambda\prod_k (x-r_k)^{m_k}$ where $\lambda$ is a constant. $g(0)=1$ means that $\lambda\prod_k(-r_k)^{m_k}=1$ and thus $\lambda=\prod_k\frac{1}{(-r_k)^{m_k}}$. Therefore $$ g(x)=\prod_k\frac{(x-r_k)^{m_k}}{(-r_k)^{m_k}}=\prod_{k}\left(1-\frac{x}{r_k}\right)^{m_k} $$