# What is the moment generating function of Dirichlet distribution?

I want to find the moment generating function (or the Laplace transform) of the Dirichlet distribution. I know the moments can be found using the gamma functions as follows :$$E\left[\prod_{i=1}^K x_i^{\beta_i}\right]=\frac{B\left(\boldsymbol{\alpha}+\boldsymbol{\beta}\right)}{B\left(\boldsymbol{\alpha}\right)}=\frac{\Gamma\left(\sum_{i=1}^{n}\alpha_{i}\right)}{\Gamma\left(\sum_{i=1}^{n}\alpha_{i}+\beta_{i}\right)}\times\prod_{i=1}^{n}\frac{\Gamma\left(\alpha_{i}+\beta_{i}\right)}{\Gamma\left(\alpha_{i}\right)},$$ but what I am really interested in is the functional form of the MGF (or the Laplace transform) so that it can be used to find other sampling distributions thereof or any other transformation of the Dirichlet also.

You may find the MGF of the Dirichlet by consulting pages 15 and 16, here:

https://mast.queensu.ca/~communications/Papers/msc-jiayu-lin.pdf

This work is not mine. It is simply a reference I found online.

You may compute it along the following lines: let $E_n$ be the set: $$E_n = \left\{(x_1,\ldots,x_n): x_i\geq 0, \sum_{i=1}^{n}x_i=1\right\}\subset\mathbb{R}^n.$$ Then: $$\int_{E_n} x_1^{\alpha_1-1}\cdot\ldots\cdot x_{n}^{\alpha_n-1}\,d\mu = \frac{\prod_{i=1}^{n}\Gamma(\alpha_i)}{\Gamma(\alpha_1+\ldots+\alpha_n)}$$ for any choice of the $\alpha_i$ such that $\text{Re}(\alpha_i)>0$.