# 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?

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For the Beta function, the variable of integration should be $t$ not $x$. The definition is in fact $$B(x,y)=\int_0^1 t^{x-1}(1-t)^{y-1} dt$$ How would you generalize then? –  Spenser Dec 3 '12 at 13:53
yes, of course. I've already edit it. –  stepkamipt Dec 3 '12 at 15:02
You need to change it in your definition of $F$ also. You can't integrate with the arguments of the function. For example, what happens when you evaluate it? Say $F(1,1,...,1)$? Then you integrate with respect to $d1d1...d1$ which makes no sense. Note also that the integral of the Beta function has powers $x-1$ and $y-1$, not $x$ and $y$. –  Spenser Dec 3 '12 at 22:24

What you can look at is the Selberg integral. It is a generalization of the Beta function and is defined by

\begin{eqnarray} S_n(\alpha,\beta,\gamma) &=& \int_0^1\cdots\int_0^1\prod_{i=1}^n t_i^{\alpha-1}(1-t_i)^{\beta-1}\prod_{1\leq i<j\leq n}|t_i-t_j|^{2\gamma}dt_1\cdots dt_n \\ &=& \prod_{j=0}^{n-1}\frac{\Gamma(\alpha+j\gamma)\Gamma(\beta+j\gamma)\Gamma(1+(j+1)\gamma)}{\Gamma(\alpha+\beta+(n+j-1)\gamma)\Gamma(1+\gamma)} \end{eqnarray}

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$$t_1^{x_1-1}(1-t_1)^{x_2-1}(1-t_1-t_2)^{x_3-1}\cdots(1 - t_1 - \cdots-t_{n-1})^{x_n-1} .$$