Why is $(-1)!!$ equal to $1$?

Question

The Mathematica software returns $(-1)!!$ as $1$. But, why is it so? We know that, for example, $7!!$ is the product of all odd numbers to $7$, i.e., $7!! = 7 * 5 * 3 * 1$, so it seems that $(-1)!!$ doesn't make sense. We can also think about factorial in terms of gamma function, and again gamma function is not defined for negative numbers. So, why do authors assume $(-1)!! = 1$?

Answer 1

See the Wikipedia article on the double factorial, in particular the section on extensions to negative arguments:

The ordinary factorial, when extended to the gamma function, has a pole at each negative integer, preventing the factorial from being defined at these numbers. However, the double factorial of odd numbers may be extended to any negative odd integer argument by inverting its recurrence relation $$ n!!=n\times (n-2)!! $$ to give $$ n!!=\frac{(n+2)!!}{n+2}\,. $$ Using this inverted recurrence, $−1!! = 1$, $−3!! = −1$, and $−5!! = \frac13$; negative odd numbers with greater magnitude have fractional double factorials. In particular, this gives, when $n$ is an odd number, $$ (-n)!!\times n!!=(-1)^{\frac {n-1}{2}}\times n\,. $$

^{Source: Wikipedia contributors, "Extensions: Negative arguments" in "Double factorial". Wikipedia, The Free Encyclopedia, https://en.wikipedia.org/w/index.php?title=Double_factorial&oldid=963313730#Negative_arguments (accessed June 24, 2020).}