negative factorials

[wtf]

I was just reading that negative factorials are never defined, although they *can* be defined for all positive numbers based on the rationale that:

for all x's and y's such that x! = y
(x+1)!/y = x + 1

so for x = 2, y = 2, 2+1 = 3, 3! = 6, 6/2 = 3 = x+1

but this would not hold true for negative numbers:

for x = -1,,
x+1 = -1+1 = 0 / y = 0.

but i was thinking, that formula is silly, shouldn't it be a piecewise defined function so for negative domain the formula would be (x-1)!/y = x-1.

[mastemmer]

Take a look at
http://en.wikipedia.org/wiki/Gamma_function
The gamma function is a generalization of factorials, valid not only for integers.
Since gamma function has singularities for all negative integers, something similar can be said about factorials.

[atropos]

The factorial operator is just a special case of the gamma function.
In terms of this function, your formula

(x+1)/y = x+1

would be

G(z+1) = zG(z)

The complete definition of the gamma function is a complex integral over the half line (0, inf), and that can be extended to all complex numbers except the real negative integers.

Fun fact: (-0.5)! = 1.77245... (the square root of PI)

[wtf]

wow that really answers just about everything but my question. It says in the second sentence of the wikipedia article of Gamma functions that they are still ONLY defined for positive real numbers. My question was regarding a possible definition for negative numbers. I am not going to read the rest of the article.

@atropos ditto and interesting.

[atropos]

The gamma function IS defined for all real numbers except the negative integers.

If you integrate by parts, you wind up with

G(x+n) = (x+n - 1)(x+n -2)(x+n -3).....(x+1)(x) G(x)

and that formula is valid for ALL x =/= {0, -1, -2, -3,....}.

EDIT: I just checked the wiki page; the second sentence you were referring to was talking about convergence of the integral.