For a group of variables $x_i>0$ with constraint $\sum_{i=1}^{n} x_i$. We define tail sum of exponents $p_i>0$ to be $S_k=\sum_{i=k}^n p_i$, and $T_k=\sum_{i=k}^{n} x_i$. Define Integral operator $$\hat \int_k=\prod_{i=1}^k\int_0^{1-S_{i-1}} x_i^{p_i-1}dx_i$$ we can write multi-integral:
If we have $\mathbf{p'}=\mathbf{p}+(1,0,\ldots,0)$, then $$\Beta(\mathbf{p'})=\frac{p_0}{\sum p_i}\Beta(\mathbf{p})$$