Representing the Gamma Function

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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!

 

[mathman]

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

 

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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.