Generalization of the fractional Poisson distribution

TL;DR

This paper introduces a generalized fractional Poisson distribution using the Mittag-Leffler function, extending the standard model and providing algebraic moment calculations, with potential interpretations for new parameters.

Contribution

It proposes a new generalized fractional Poisson distribution incorporating an additional parameter and derives its moments algebraically, expanding the existing models.

Findings

Distribution contains the standard fractional Poisson as a subset

Algebraic calculation of raw moments using Bell polynomials

Proposes a possible interpretation for the new parameter eta"

Abstract

A generalization of the Poisson distribution based on the generalized Mittag-Leffler function $E_{\alpha, \beta}(\lambda)$ is proposed and the raw moments are calculated algebraically in terms of Bell polynomials. It is demonstrated, that the proposed distribution function contains the standard fractional Poisson distribution as a subset. A possible interpretation of the additional parameter $\beta$ is suggested.