Analysis of generalized probability distributions associated with higher Landau levels

TL;DR

This paper explores the properties of generalized probability distributions linked to higher Landau levels, including their atomic decomposition, divisibility, and the derivation of a new infinitely divisible distribution.

Contribution

It introduces a novel generalized distribution for higher Landau levels, analyzes its divisibility properties, and constructs a new infinitely divisible distribution with explicit weights.

Findings

The distribution is not infinitely divisible for higher Landau levels.

The characteristic function can be expressed via hypergeometric polynomials.

A new infinitely divisible distribution with computed weights is derived.

Abstract

To a higher Landau Level corresponds a generalization of the Poisson distribution arising from generalized coherent states. In this paper, we write down the atomic decomposition of this probability measure and expressed its weights through 2F2 hypergeometric polynomials. Then, we prove that it is not infinitely divisible in opposite to the Poisson distribution corresponding to the lowest Landau level. We also derive the Levy-Kintchine representation of its characteristic function when the latter does not vanish and deduce that the representative measure is signed. By considering the total variation of the last measure, we obtain the characteristic function of a new infinitely divisible discrete probability distribution for which we compute also the weights.