How many ways can we characterize gamma function? - MathOverflow

How many ways can we characterize gamma function?

2010-02-15T21:25:16Z

First let's state a well-known characterization of gamma function.

If f is a positive function on positive real numbers such that: (1).f(x+1)=xf(x); (2).f(1)=1; (3).logf is convex, then f(x) is gamma function.

Now here I'm wondering how many ways can we characterize gamma function like the above? Especially if we consider it as a function on complex plane with poles.

ps: I'm not asking different ways to express gamma function explicitly, but the abstraction of it.

Answer by Gu Yejun for How many ways can we characterize gamma function?

2010-03-07T16:19:49Z

maybe I can give you some help. Gamma function is also called the second Euler integral.

Here comes some characterizations.

a f(s)= $$t(x)=\int_{0}^{+\infty}{t^(s-1)}{exp(-t)}dt$$ s>0

b f(s)=$$\lim n!n^s/[s(s+1)...(s+n)] $$ $$n\rightarrow +\infty$$

c $$B（p，q）=\Gamma(p)\Gamma(q)/\Gamma(pq）$$ p>0 q>0

d $$\Gamma(2s)=2^(2s-1）\Gamma(s)\Gamma(s+1/2)/\sqrt(2\pi) $$ s>0

e $$\Gamma(s)\Gamma(1-s)=\pi/sin(s\pi)$$ 0

May it help!

Answer by vonjd for How many ways can we characterize gamma function?

2010-03-07T18:38:44Z

Have a look here: http://dlmf.nist.gov/5/