I have the following PMF: \begin{equation} P(\alpha^{(k)} \ast x = r) = \begin{cases} A_{k}^{x}, & \text{if $r = 0$}.\\ \frac{1}{r} \sum_\limits{l=1}^{r}{l{r \choose l}A_{k}^{x-1}B_{k}^{l}C_{k}^{r-1}{x \choose l}}, & \text{if $r \geq 1$}, \end{cases} \end{equation} where $A_k = \frac{1 - \alpha^{k}}{1 - \alpha^{k+1}}$, $B_k = \frac{\alpha^{k}(1-\alpha)^2}{(1-\alpha^{k+1})^2}$, and $C_k = \frac{\alpha(1-\alpha^{k})}{1-\alpha^{k+1}}$.
$r$, $k$, and $x$ are all integers and $\alpha \in (0,1)$.
For fixed $\alpha$ and $k$, if $x = 0$ or $x = 1$, the PMF sum to $1$ when we vary $r$ ($r = 0, 1, 2,\ldots$). However, when $x>1$, the PMF does not sum to $1$. I checked this computationally, but I need to obtain a formal proof.
Source:
Awale, M., Ramanathan, T. V., and Kale, M. 2017. Coherent forecasting in integer-valued AR(1) models with geometric marginals. Journal of Data Science 15:95–114.