multiplying gamma function over different input variables

Question

I have a function (the Gamma function) defined as:

$$ F(\alpha) = \int_{0}^{\infty} x^{\alpha-1} e^{-x}dx $$

Now, I want to multiply this function evaluated at two points as:

$$ F(\alpha_0) = \int_{0}^{\infty} x^{\alpha_0-1} e^{-x}dx $$

and $$ F(\alpha_1) = \int_{0}^{\infty} x^{\alpha_1-1} e^{-x}dx $$

Is there a way to get an integral expression in terms of the original gamma function? So, basically can I say something like

$$ F(\alpha_0) F(\alpha_1) = F(\alpha_0 + \alpha_1) $$

or something like that?

Answer 1

One intuition on the $\Gamma$ function is the identity $$\Gamma(n) = (n-1)!$$ for integer $n \geq 1$. It is certainly easy to see that $$ n! \times m! \neq (n+m)! $$