There is a fine little book (good luck finding a copy!) entitled The Gamma Function by Emil Artin. In it he shows that the Gamma function is distinguished by being "log convex."
As Retina has noted, x! = Γ(x+1) when x is a nonnegative integer.
As to the derivative: There is a wellknown asymptotic expansion [Stirling's asymptotic series] for ln(Γ(z)), z a complex variable. Taking the derivative term by term gives an asymptotic series for Γ'(z)/Γ(z).
