Solving a Pi product.

Question

How the value of this $P_k$ is calculated from the first equation. Thank you.

$$k \geq m$$ $$P_k=P_0\prod_{i=0}^{m-1}\frac{\alpha}{(i+1)\mu}\prod_{j=m}^{k-1}\frac{\alpha}{m\mu}$$ $$P_k=\frac{P_0\left(\frac{\alpha}{\mu}\right)^k}{m!m^{k-m}}$$

Answer 1

Assuming that the correct expression is $$ P_k = P_0\prod_{i=0}^{m-1}\frac{\alpha}{(i+1)\mu}\prod_{j=m}^{k-1}\frac{\alpha}{m\mu} $$ we just have to count. The total number of factors of $\alpha/\mu$ are $k$, hence we should have $(\alpha/\mu)^k$. The total number of factors of $1/m$ are $k-m$, so we should have $1/m^{k-m}$. The product with $1/(i+1)$, when $i$ runs from $0$ to $m-1$ gives you $1/m!$. All in all, the result is $$ P_0\frac{(\alpha/\mu)^k}{m^{k-m}m!}, $$ as stated.