Product Pi Notation

Question

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.^

Answer 1

Yes, the definition is $$ \prod_{a\leq k \leq b} f(k) = f(a) f(a+1) \cdots f(b), $$ with the empty product (which occurs e.g., if $b<a$) being defined as $1$ (the multiplicative identity).

Answer 2

The formula $\beta^n \prod_{k=0}^{n-1}(\alpha+k) $ does generate the right hand sides of those equations.

I hope the superscript on the left refers to indices and not power.