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Representing the Gamma Function

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Jan24-14, 12:47 PM
P: 344
I'm not sure if this is a "general" math question but I do think it is an interesting one.

The Gamma Function, [itex]\Gamma(t)[/itex], 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:

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.

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Jan24-14, 01:57 PM
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P: 6,076
My guess. It should have included lim x -> 0, but I really don't know.
Jan24-14, 02:09 PM
P: 344
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.

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