Geodesic
The Poisson and compound Poisson distributions of order~$k$ and some of their properties
Zapiski Nauchnykh Seminarov POMI, Problems of the theory of probability distributions. Part VIII, Tome 130 (1983), pp. 175-180

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The Poisson distribution of order $k$ is studied. Its probability generating function is derived and is employed to show that the sum of i. r. v. distributed as Poisson of order $k$ is also distributed similarily. A generalized compound Poisson distribution is obtained and briefly studied. 
@article{ZNSL_1983_130_a18,
     author = {A. N. Philippou},
     title = {The {Poisson} and compound {Poisson} distributions of order~$k$ and some of their properties},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {175--180},
     publisher = {mathdoc},
     volume = {130},
     year = {1983},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1983_130_a18/}
}
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A. N. Philippou. The Poisson and compound Poisson distributions of order~$k$ and some of their properties. Zapiski Nauchnykh Seminarov POMI, Problems of the theory of probability distributions. Part VIII, Tome 130 (1983), pp. 175-180. http://geodesic.mathdoc.fr/item/ZNSL_1983_130_a18/