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To each hyperbolic Landau level of the Poincar\'e disc is attached ageneralized negative binomial distribution. In this paper, we compute themoment generating function of this distribution and supply its decomposition asa perturbation of the negative binomial distribution by a finitely-supportedmeasure. Using the Mandel parameter, we also discuss the nonclassical nature ofthe associated coherent states. Next, we determine the L\'evy-Kintchinedecomposition its characteristic function when the latter does not vanish anddeduce that it is quasi-infinitely divisible except for the lowest hyperbolicLandau level corresponding to the negative binomial distribution. Byconsidering the total variation of the obtained quasi-L\'evy measure, weintroduce a new infinitely-divisible distribution for which we derive thecharacteristic function.

Analysis of generalized negative binomial distributions attached to hyperbolic Landau levels