Does $prod_k (x - r_k)^{m_k} = prod_k (1 - frac{x}{r_k})^{m_k}$?

Question

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.

Tuvasbien · Accepted Answer

The products are not the same, but $g(x)=lambdaprod_k (x-r_k)^{m_k}$ where $lambda$ is a constant. $g(0)=1$ means that $lambdaprod_k(-r_k)^{m_k}=1$ and thus $lambda=prod_kfrac{1}{(-r_k)^{m_k}}$. Therefore $$ g(x)=prod_kfrac{(x-r_k)^{m_k}}{(-r_k)^{m_k}}=prod_{k}left(1-frac{x}{r_k}right)^{m_k} $$

Correct answer by Tuvasbien on January 23, 2021

Does $prod_k (x - r_k)^{m_k} = prod_k (1 - frac{x}{r_k})^{m_k}$?

One Answer

Add your own answers!

Ask a Question