Analog of Beta-function What is the multi-dimensional analogue of the Beta-function called? The Beta-function being $$B(x,y) = \int_0^1 t^x (1-t)^y dt$$ 
I have a function 
$$F(x_1, x_2,\ldots, x_n) = \int_0^1\cdots\int_0^1t_1^{x_1}t_2^{x_2}\cdots(1 - t_1 - \cdots-t_{n-1})^{x_n}dx_1\ldots dx_n$$ 
and I don't know what it is called or how to integrate it. 
I have an idea that according to the Beta-function: $$F(x_1, \ldots,x_n) = \dfrac{\Gamma(x_1)\cdots\Gamma(x_n)}{\Gamma(x_1 + \cdots + x_n)}$$
Is there any analogue for this integral such as Gamma-function form for Beta-function?  

Maybe the integrand has the form
$$ t_1^{x_1-1}(1-t_1)^{x_2-1}(1-t_1-t_2)^{x_3-1}\cdots(1 - t_1 - \cdots-t_{n-1})^{x_n-1} .$$
