A New Family of Fractional Counting Probability Distributions

TL;DR

This paper introduces a new family of fractional counting probability distributions based on a three-parameter generalized Mittag-Leffler function, with applications in quantum states and combinatorial numbers.

Contribution

It develops a novel fractional counting process and related mathematical objects, extending classical distributions and quantum states with fractional generalizations.

Findings

Reproduces known Poisson and fractional Poisson distributions

Introduces fractional Bell and Stirling numbers

Develops stretched quantum coherent states

Abstract

A new family of fractional counting processes based on a three-parameter generalized Mittag-Leffler function was introduced and studied. As applications we develop a fractional generalized compound process, introduce and develop fractional generalized Bell polynomials and numbers, fractional generalized Stirling numbers of the second kind, and a new family of quantum coherent states. Stretched quantum coherent states, which are a generalization of the famous Schr\"odinger-Glauber coherent states, were also introduced and studied. In particular cases, the presented results reproduce known equations for Poisson and fractional Poisson probability distributions, Bell numbers and fractional Bell numbers, Stirling numbers and fractional Stirling numbers of the second kind, as well as for known quantum coherent states.