so I know the gamma function defined:

T(x) = [integral from 0 to infinity] t^(x-1)e^(-t)dt

converges for x > 0

I also know that T(x+1) = xT(x)

does this make it possible to extend the T(x) function for x <0? (except the negative integers), and if so, why is it okay to apply this identity even though T(x) is not defined (divergent) for x <= 0?

This is more of a conceptual question, so if anyone understands this I would love to hear an explanation. Thanks!