Let $\alpha_1,...,\alpha_n>0$.
Let $$v_n=\int\limits_{\substack{x_1,\ldots,x_n\geq0\\x_1+\cdots+x_n\leq1}}x_{1}^{\alpha_1 -1}\cdots x_{n}^{\alpha_n-1}dx_1\cdots dx_n.$$
i. Express $v_n$ in function of $v_{n-1}$ and show that $v_n=\frac{\Gamma(\alpha_1)\cdots \Gamma(\alpha_n)}{\Gamma(1+\alpha_1+\cdots+\alpha_n)}$.
ii. Let $a>0$. Compute $$u_{n}(a)=\int\limits_{\substack{x_1,\ldots,x_n\geq0\\x_1+\cdots+x_n\leq a}}x_{1}^{\alpha_1 -1}\cdots x_{n}^{\alpha_n-1}dx_1\cdots dx_n.$$
The hint I have is to use the previous exercise, where I proved that $\mathrm{B}(x,y)= \frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)}$ where $\mathrm{B}$ is the beta function. But I have no idea how to start. Can someone help me?
Hint:
$0 \le x_1+x_2+ \cdots+ +x_n < a \quad \Rightarrow \quad 0 \le \frac{x_1}{a}+ \cdots +\frac{x_n}{a} <1$.
So change accordingly the integration variables ..
So, for the first, we have $$ I(a_{\,1} , \cdots ,a_{\,n} ) = \int\limits_{\left\{ {\matrix{ {0 \le x_{\,1} , \cdots ,x_{\,n} } \cr {\left( {0 \le } \right)x_{\,1} + \cdots + x_{\,n} \le 1} \cr } } \right.} {x_{\,1} ^{a_{\,1} - 1} \cdots x_{\,n} ^{a_{\,n} - 1} dx_{\,1} \cdots dx_{\,n} } $$
For $n=1$ it is $$ I(a_{\,1} ) = \int\limits_{0 \le x_{\,1} \le 1} {x_{\,1} ^{a_{\,1} - 1} dx_{\,1} } = {1 \over {a_{\,1} }}\quad $$
We can then start a recursion by noting that for a general $2 \le n$ we have $$ \eqalign{ & I(a_{\,1} , \cdots ,a_{\,n} ) = \int\limits_{\left\{ {\matrix{ {0 \le x_{\,1} , \cdots ,x_{\,n} } \cr {\left( {0 \le } \right)x_{\,1} + \cdots + x_{\,n} \le 1} \cr } } \right.} {x_{\,1} ^{a_{\,1} - 1} \cdots x_{\,n} ^{a_{\,n} - 1} dx_{\,1} \cdots dx_{\,n} } = \cr & = \int\limits_{\left\{ {\matrix{{0 \le x_{\,1} , \cdots ,x_{\,n} } \cr {\left( {0 \le } \right)x_{\,1} + \cdots + x_{\,n - 1} \le 1 - x_{\,n} = u} \cr {0 \le u \le 1} \cr } } \right.} {x_{\,1} ^{a_{\,1} - 1} \cdots x_{\,n - 1} ^{a_{\,n - 1} - 1} x_{\,n} ^{a_{\,n} - 1} dx_{\,1} \cdots dx_{\,n - 1} dx_{\,n} } = \cr & = \int_{u = 0}^1 {u^{a_{\,1} + \cdots + a_{\,n - 1} - \left( {n - 1} \right) + \left( {n - 1} \right)} \left( {1 - u} \right)^{a_{\,n} - 1} du} \int\limits_{\left\{ {\matrix{ {0 \le x_{\,1} /u, \cdots ,x_{\,n} /u} \cr {\left( {0 \le } \right){{x_{\,1} } \over u} + \cdots + {{x_{\,n - 1} } \over u}\, \le \,1} \cr } } \right.} {\left( {{{x_{\,1} } \over u}} \right)^{a_{\,1} - 1} \cdots \left( {{{x_{\,n - 1} } \over u}} \right)^{a_{\,n - 1} - 1} d\left( {{{x_{\,1} } \over u}} \right) \cdots d\left( {{{x_{\,n - 1} } \over u}} \right)} = \cr & = B\left( {a_{\,1} + \cdots + a_{\,n - 1} + 1,\;a_{\,n} } \right) I\left( {a_{\,1} , \cdots ,a_{\,n - 1} } \right) \cr} $$ Therefrom we get $$ \eqalign{ & I\left( {a_{\,1} } \right) = {1 \over {a_{\,1} }} = {{\Gamma \left( {a_{\,1} } \right)} \over {\Gamma \left( {a_{\,1} + \;1} \right)}} \cr & I\left( {a_{\,1} ,a_{\,2} } \right) = {{\Gamma \left( {a_{\,1} + 1} \right)\Gamma \left( {\;a_{\,2} } \right)} \over {\Gamma \left( {a_{\,1} + \;a_{\,2} + 1} \right)}} {{\Gamma \left( {a_{\,1} } \right)} \over {\Gamma \left( {a_{\,1} + \;1} \right)}} = \cr & = {{\Gamma \left( {a_{\,1} } \right)\Gamma \left( {\;a_{\,2} } \right)} \over {\Gamma \left( {a_{\,1} + \;a_{\,2} + 1} \right)}} = {{\Gamma \left( {a_{\,1} } \right)\Gamma \left( {\;a_{\,2} } \right)} \over {\left( {a_{\,1} + \;a_{\,2} } \right)\Gamma \left( {a_{\,1} + \;a_{\,2} } \right)}} \cr & \quad \vdots \cr} $$