I am trying to create an example that applies fully parametric estimation. I am using a Gamma-Poisson distribution where the random variable is a Poisson random variable with mean $\lambda$ which has a Gamma distribution with parameters $\alpha$ and $\beta$. Also denoted as $X \sim \textrm{Gamma-Poisson}(\alpha,\beta)$ with probability mass function
\begin{equation*} f(x) = \frac{\Gamma{(x+\beta)}\alpha^{x}}{\Gamma(\beta)(1+\alpha)^{\beta+x}x!} \;\;\; x=0,1,2,... \end{equation*}
I am familiar with solving for MLE's but not entirely sure with this distribution. Currently what I have is below but I'm not sure about the $\Gamma$ function.
\begin{align*} L(\theta) &= \prod_{i=1}^{n} \frac{\Gamma{(x_i+\beta)}\alpha^{x_i}}{\Gamma(\beta)(1+\alpha)^{\beta+x_i}x_i!} \\ \textrm{ln} \; L(\theta) &= \sum_{i=1}^{n} \textrm{ln} \left(\frac{\Gamma{(x_i+\beta)}\alpha^{x_i}}{\Gamma(\beta)(1+\alpha)^{\beta+x_i}x_i!}\right) \\ &= \sum_{i=1}^{n} \big[\textrm{ln}\:\Gamma{(x_i+\beta)} + x_i\:\textrm{ln}\:\alpha - \textrm{ln}\:\Gamma(\beta) - (\beta+x_i)\:\textrm{ln}\:(1+\alpha) - \textrm{ln}\:(x_i!)\big] \\ & \; \vdots \\ \frac{\partial}{\partial\alpha}\;\textrm{ln}\;L(\theta) &= \dots = 0 \\ \hat{\alpha} &= \\ \frac{\partial}{\partial\beta}\;\textrm{ln}\;L(\theta) &= \dots = 0 \\ \hat{\beta} &= \end{align*}
