I am trying to evaluate the following integral:
$\displaystyle \int_0^\infty \frac{\theta^{\sum x_i + \alpha -1}e^{-\theta(n + 1/\beta)}}{x_1!\cdots x_n!\Gamma(\alpha)\beta^{\alpha}}d\theta$
This integral comes from Bayes estimation involving a Poisson random sample and a Gamma random variable $\Theta$.
The above integral will be the marginal pdf given $\Theta = \theta$.
I tried Integration by parts but this seems to give me a recursive result.
Using : $u = \theta^{\sum x_i + \alpha -1}, v = -e^{-\theta(n + 1/\beta)}$.
I know the result should be $\displaystyle \frac{\Gamma(\sum x_i + \alpha)}{x_1!\cdots x_n! \Gamma(\alpha)\beta^\alpha(n+1/\beta)^{\sum x_i + \alpha}}$
Perhaps there is a clearer substitution to use or a result?
The desired result follows from$$\Gamma\bigg(\alpha+\sum_ix_i\bigg)=\int_0^\infty u^{\alpha+\sum_ix_i-1}e^{-u}du=\frac{\int_0^\infty\theta^{\alpha+\sum_ix_i-1}e^{-\theta(n+1/\beta)}d\theta}{(n+1/\beta)^{\alpha+\sum_ix_i}}.$$