The exponentiated exponential Poisson distribution revisited

The main aim of this article is to characterize and investigate the three parameter exponentiated exponential Poisson probability distribution ${\rm EEP}(\alpha, \beta, \lambda)$ by giving explicit closed form expressions for its characteristic function $\phi_\xi(t)$ and moment generating function $M_\xi(t)$, and finally, to show that the existing series and integral form expressions for positive integer order moments $\mathbb E\xi^\nu, \nu \in \mathbb N$ are in fact valid for all $\nu>1-\alpha, \alpha>0$.