ca.analysis and odes - How many ways can we characterize gamma function?

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)=$$lim n!n^s/[s(s+1)...(s+n)] $$ $$nrightarrow +infty$$

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

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

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

May it help!