Maybe this is a stupid question but I'm lost.
The double factorial is defined as:
$$n!!=\prod_{k=1}^\frac n2 2k=n\times(n-2)\times(n-4)\times\dots\times2$$
For $n$ even. By definition $0!!=1$ as an empty product (pretty similar to how $0!=1$) but then Wikipedia says that for an even integer the double factorial can be expressed as (https://en.wikipedia.org/wiki/Double_factorial under the complex argument section):
$$(2k)!!=2^kk!\sqrt \frac 2\pi$$
But how can this be ? The two definitions are different and if I put $k=0$, as an example, I get that:
$$0!!=\sqrt \frac 2\pi\neq1$$
Is that an error of Wikipedia ? If not how can I arrive to a definition different from my original one just manipulating terms ?
The referred formula is an analytic continuation of the formula for odd integers, so it's not against the rules if it doesn't agree with the factual results in the even case! The situation is the same with the series:
$$1 + 2 + 3 + \cdots = - \frac{1}{12}$$
which, despite seeming "non-sense" at first glance, actually finds applications even in physics.
PS For ordinary computations (e.g. in combinatorics) you should use the ordinary formula.