I want to find a simpler form or closed form for the following integral: $ \int_A \,\prod_{t=1}^T f_{\Gamma(a,\theta_t)}(x_t) \,d\mathbf{x} $ where $A$ is the simplex $ A = \{\mathbf{x} \in \mathbb{R}^T : \sum_{t=1}^T x_t = 1 \hbox{ and } x_t \geq 0\} $ and $f_{\Gamma(a,\theta_t)}$ is the Gamma density $ f_{\Gamma(a,\theta_t)}(x_t) = \theta_t^a x_t^{a-1} \exp(-x_t \theta_t) / \Gamma(a) . $ We require that $a$ and $\theta_t$ are positive real numbers.
It seems that this shouldn't be too hard, but I've really been stumped. I have tried various methods from the paper by Wolpert and Wolf mentioned in this blog post - but when I follow the algorithm described in that paper, I end up running into even harder analytical difficulties at the end, when it is required to take an inverse Laplace transform of some product.
I feel that there must be a lower-tech way to do this! Thanks in advance for any help.