How to represent factorial as a product notation?

Question

Original question: Is there any way to write $x!$ as a Pi notation? $$x!=\prod\text{?}$$

The answer should have been obvious $$x!=\prod_{r=1}^xr$$

Now I have an additional question:

Is it possible to write $\left(2(\ x+1 )\ \right)!$ as a product notation? $$(2(x+1))!=\prod?$$

Answer 1

You can write like this:
$$x!=\prod_{r=1}^x\text{r}$$

For additional question:

$$(2x+2)!=\prod_{r=1}^{(2x+2)}\text{r}$$

Answer 2

It is possible. We have definition of n! as product (r), running from 1 to n. I hope you asked for this pardon otherwise.

Answer 3

For a non-integer number,

$n!=\prod_{k=1}^{\infty}\left( \frac{k+1}{k}\right)^{n}\frac{k}{n+k}.$