Some Applications of the Fractional Poisson Probability Distribution

TL;DR

This paper explores the fractional Poisson distribution's applications in physics and mathematics, introducing new quantum states, generalized combinatorial numbers, and polynomial sequences, linking them to classical results in the limit.

Contribution

It introduces fractional generalizations of Bell and Stirling numbers, new quantum coherent states, and polynomial sequences, expanding the mathematical and physical understanding of fractional Poisson processes.

Findings

Introduction of fractional Bell and Stirling numbers.

Development of new quantum coherent states.

Connections to classical polynomials and functions in the limit.

Abstract

Physical and mathematical applications of fractional Poisson probability distribution have been presented. As a physical application, a new family of quantum coherent states has been introduced and studied. As mathematical applications, we have discovered and developed the fractional generalization of Bell polynomials, Bell numbers, and Stirling numbers. Appearance of fractional Bell polynomials is natural if one evaluates the diagonal matrix element of the evolution operator in the basis of newly introduced quantum coherent states. Fractional Stirling numbers of the second kind have been applied to evaluate skewness and kurtosis of the fractional Poisson probability distribution function. A new representation of Bernoulli numbers in terms of fractional Stirling numbers of the second kind has been obtained. A representation of Schlafli polynomials in terms of fractional Stirling numbers of the second kind has been found. A new representations of Mittag-Leffler function involving fractional Bell polynomials and fractional Stirling numbers of the second kind have been discovered. Fractional Stirling numbers of the first kind have been introduced and studied. Two new polynomial sequences associated with fractional Poisson probability distribution have been launched and explored. The relationship between new polynomials and the orthogonal Charlier polynomials has also been investigated. In the limit case when the fractional Poisson probability distribution becomes the Poisson probability distribution, all of the above listed developments and implementations turn into the well-known results of quantum optics, the theory of combinatorial numbers and the theory of orthogonal polynomials of discrete variable.