Factorial of negative numbers

In mathematics, Factorial of a positive number 'n' is defined as -

n! (pronounced as: n factorial) = n * (n-1) * (n-2) *............* 2 * 1....

where.. '*' is multiplication sign.

0! is taken as 1.
Actually, 0! is not 'taken' as 1. It is one as you will see shortly..

Factorial of positive numbers have great application in mathematics. Permutations and combinations is one (where we find the number of ways a group of objects can be arranged or selected in given group of objects). Expansion of terms like (a+b)^n (binomial theorem) are computed using factorials. 

If you search up this on Wikipedia, you will find a lot of data on it. 

Lets discuss about factorial of negative numbers.

a factorial can always be written as -

 n! = (n+1)! / (n+1)  .......... (since (n+1)! = (n+1) * n!)

(This is one way to calculate 0! ... just put n = 0 , and we get - 0! = 1 )

now, one can easily see the problem with factorial of a negative number : 

lets compute (-1)! this way... here, n =-1

hence, (-1)! = (-1+1)! / (-1+1)   ....................................... ( 0! = 1 )

                   = 1/0 

which is not defined.

If we go on this way. for every number 'n' , (-n)! =  not defined.

therefore, factorial of negative number is not possible.

But what if we define it some other way?

Why can't we take factorial of negative number as - 

(-n)! = -n * (-n + 1) * (-n+2) * ................ * -2 * -1

or 

(-n)! = (-1)^n * n!

The Idea is to deal with factorial differently when using for a positive number as compared to a negative number. Different identities and different formulas for negative numbers and positive numbers will exist.

I am no mathematician, but this can be used. May be it can not be used. It's just an idea. I can not say if it will have any applications, but it's not always about application. Its about developing the subject.

Lets just give it a thought. If it works, its great but if it does not, we can always leave it....