No it isn't! The function G(x) := Gamma(x) * (1 + c * sin(2 * pi
* x)) with 0 < c < 1 is such an example. Both, the
Gamma-function and G have the properties f(x+1) = x * f(x) and f(1)
= 1. That's why one needs a third property to define the gamma
function uniquely.