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Analysis of generalized Poisson distributions associated with higher Landau levels. (English) Zbl 1333.81454

Summary: To a higher Landau level corresponds a generalized Poisson distribution arising from generalized coherent states. In this paper, we write down the atomic decomposition of this probability distribution and express its probability mass function as a \(_2F_2\)-hypergeometric polynomial. Then, we prove that it is not infinitely divisible in contrast with the Poisson distribution corresponding to the lowest Landau level. We also derive a Lévy-Khintchine-type representation of its characteristic function when the latter does not vanish and deduce that the representative measure is a quasi-Lévy measure. By considering the total variation of this last measure, we obtain the characteristic function of a new infinitely divisible discrete probability distribution for which we also compute the probability mass function.

MSC:

81V80 Quantum optics
60E99 Distribution theory
81V70 Many-body theory; quantum Hall effect
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
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