Can Euler's generalization of factorial be done for double factorial?
Euler's generalization of factorial to non-integer values is
$$t! =\lim_{n \to \infty} \dfrac{n!n^t}{\prod_{k=1}^n(t+k)}.$$
I decided to see if I could come up with a similar generalization for double factorial, defined by $$n!! =\prod_{k \ge 0, 2k < n} (n-2k) =n(n-2)(n-4) \cdots. $$
My result is
$$(2t)!! =\lim_{n \to \infty}\dfrac{(2n)^t(2n)!!}{2^n\prod_{k=1}^{n} (t+k)}. $$
My questions:
Is this correct (there is a moderate amount of messy algebra involved)?
Is there a similar version for double factorial of Euler's formula $t! =\int_0^{\infty} x^t e^{-x} dx $?
I am quite confident that my formula generalizes to the m-factorial defined by $$n!_{(m)} =\prod_{k \ge 0, mk < n} (n-mk) =n(n-m)(n-2m) \cdots $$ just by replacing $2$ by $m$ everywhere.
I will post my proof in two days if there are no answers.