I wonder what is the properties of Product Pi Notation? I can't found anywhere about the properties.

First of all, i have:

$X=\beta\alpha\\
X^2=\beta^2\alpha(\alpha + 1) \\
X^3=\beta^3\alpha(\alpha + 1)(\alpha + 2) \\
. \\
. \\
. \\
\text{And so forth.}
$

My question is. I want to write this form into Pi Notation (If it's possible):

We know that the pattern for $X^n$ is the following:

$X^n=\beta^n\alpha(\alpha + 1)(\alpha + 2)\cdots (\alpha+(n-1))$

And with the Product Pi, is the following true?

$X^n=\beta^{n}\displaystyle\prod\_{k=0}^{n-1}\left(\alpha + k\right)\quad,n=1, 2, 3,\ldots$

If it's wrong, please to tell me what is the right one? Please help, Thanks.^