Double step Pochhammer symbol?

Question

The usual Pochhammer symbol is defined as

$$(x)_n=x(x+1)(x+2)...(x+n)=\frac{\Gamma(x+n)}{\Gamma(x)}$$

I am interested in a generalized Pochhammer-like symbol that produces the following output

$$x(x+2)(x+4)...(x+2n)$$

Is there a convenient definition for the above in terms of $\Gamma$ or other continuous functions?

Or perhaps there is a function like $(x)_n$ defined in its own right, which directly produces this output?

EDIT:

Just found that this is called the Pochhammer k-symbol, which is essentially exactly the solution provided in the answer by Mostafa Ayaz.

Answer 1

$$x(x+2)(x+4)...(x+2n)=2^{n+1}\dfrac{x}{2}(\dfrac{x}{2}+1)(\dfrac{x}{2}+2)...(\dfrac{x}{2}+n)=2^{n+1}\dfrac{\Gamma(\dfrac{x}{2}+n)}{\Gamma(\dfrac{x}{2})}$$

Answer 2

The Wikipedia article on falling and rising factorials have some notation on this

https://en.wikipedia.org/wiki/Falling_and_rising_factorialsWikipedia

In general

$$[f(x)]^{k/h}=f(x)f(x-kh)\cdots f(x-(k-1)h).$$

((I like to use the notation

$x^{(k,h)}=x(x-h)\cdots (x-(k-1)h)$ ))