I was deriving the formula for volume of a n- dimensional Hypersphere when I came across something interesting. It seems like we can define $(0.5)!$ without resorting to the Gamma function.
Define:
$$C(n) = \int_{-π/2}^{π/2} cos^n\theta\ d\theta = 2\int_{0}^{π/2} cos^n\theta\ d\theta$$
Using reduction formula or Walli's product it can be written as:
$$C(n) = \frac{π}{2^n}{n \choose \frac{n}{2}}$$
for all(*not strictly all because negative integral factorials aren't defined) $n\in\Bbb R$ (the reason is explained in the next para).
This is because the second equation for $C(n)$ is strictly for even reals but we can extend it to *all reals by defining the values for fractional and negative factorials.
For example:
Set $n =1$ in both of the equations. We then get:
$$C(1)= 2 = \frac{π}{2}\cdot\frac{1}{\left(\frac{1}{2}\right)!\left(\frac{1}{2}\right)!}$$
From here we can write:
$$\left(\frac{1}{2}\right)! = \frac{\sqrt π}{2}$$ which is exactly what we would get even from Gamma function.
The Problem:
The problem occurs when we set $n = -1$ in both the equations.
$$C(-1) = \frac{2π(-1)!}{\left(\left(-\frac{1}{2}\right)!\right)^2} = 2\prod_{j=1}^{∞}\frac{2j}{2j-1}$$
The rightmost equality follows from the Walli's product which in this case(for $n = -1$) is clearly tending to infinity.
I want that this equation gives me the value of $(-0.5)!$ which is $\sqrt π$ (known from the Gamma function) but I can't wrap my head around how to deal with the fraction of infinite product and an undefined factorial.
Maybe it can be written as a sort of a limit but I am completely clueless even about that. So, how can I proceed further (keeping in mind the fact that we don't want to use the Gamma function).
Any kind of help will be highly appreciated!