Gamma-Poisson distribution arises as a continuous mixture of
Poisson distributions, where the mixing distribution
of the Poisson rate \(\lambda\) is a gamma distribution.
When \(X \sim \mathrm{Poisson}(\lambda)\)
and \(\lambda \sim \mathrm{Gamma}(\alpha, \beta)\), then \(X \sim \mathrm{GammaPoisson}(\alpha, \beta)\).
Probability mass function
$$
f(x) = \frac{\Gamma(\alpha+x)}{x! \, \Gamma(\alpha)} \left(\frac{\beta}{1+\beta}\right)^x \left(1-\frac{\beta}{1+\beta}\right)^\alpha
$$
Cumulative distribution function is calculated using recursive algorithm that employs the fact that
\(\Gamma(x) = (x - 1)!\). This enables re-writing probability mass function as
$$
f(x) = \frac{(\alpha+x-1)!}{x! \, \Gamma(\alpha)} \left( \frac{\beta}{1+\beta} \right)^x \left( 1- \frac{\beta}{1+\beta} \right)^\alpha
$$
what makes recursive updating from \(x\) to \(x+1\) easy using the properties of factorials
$$
f(x+1) = \frac{(\alpha+x-1)! \, (\alpha+x)}{x! \,(x+1) \, \Gamma(\alpha)} \left( \frac{\beta}{1+\beta} \right)^x \left( \frac{\beta}{1+\beta} \right) \left( 1- \frac{\beta}{1+\beta} \right)^\alpha
$$
and let's us efficiently calculate cumulative distribution function as a sum of probability mass functions
$$F(x) = \sum_{k=0}^x f(k)$$