Representing the Gamma Function

1. Jan 24, 2014

hover

I'm not sure if this is a "general" math question but I do think it is an interesting one.

The Gamma Function, $\Gamma(t)$, has many interesting definitions. It can take on the form of an integral to an infinite product. There is one particular definition, however, that I am trying to understand that doesn't make sense to me. Take a look at the following link:

http://en.wikipedia.org/wiki/Gamma_function#Alternative_definitions

The definition for Gamma that confuses me is the one that mentions generalized Laguerre polynomials on that page. We can see that Gamma is a function of 't' and that 'n' is part of the summation. My question is this, what is x suppose to be? What defines it? To me, x appears to be there for no reason. I hope I am just overlooking something simple and someone can point it out to me.

Thanks!

2. Jan 24, 2014

mathman

My guess. It should have included lim x -> 0, but I really don't know.

3. Jan 24, 2014

hover

Hm... I tried making x approach 0 in Mathematica but that isn't it unfortunately. I've been messing with setting the value of x and plotting it. It seems like when x is "around" 1, the equation appears to converge towards the Gamma function. It is still strange though and is slightly irritating. What the heck is x suppose to be??? I'm wondering if this is nothing more than a good approximation for Gamma or whether this actually equals Gamma for t<1/2 for when t is real.