Suppose that $Y_1...,Y_n$ are independant gamma random variables: $Y_i \sim \Gamma(\alpha_i,\beta_i)$, with density $f_i(t) = \frac{t^{\alpha_i-1} e^{-\frac{x}{\beta}}}{\Gamma(\alpha) \beta^{\alpha}}$.
Suppose that $a_1,...a_m$ and $b_1,...,b_m$ are strictly positives reals.
I want to compute:
$$\mathbb E \left(\prod\limits_{j=1}^m \left(b_j + Y_1 +...+ Y_n\right)^{-a_j}\right)$$
My first attemp was expanding the function inside the expectation into a multivairate taylor serie in $(Y_1,...,Y_n)$ and then use the fact that $\mathbb E(Y_i^k)$ is known. But it is quite inneficient.
Maybe from the structure of the function we could expand in a 'power serie' with non-integer powers (which would probably depend on $a_1,...,a_m$), and the serie would converge more efficiently (since $\mathbb E(Y_i^a)$ is known for any $a$, even $a$ that are not integers).
How can i do that ?
