Tempered Fractional Poisson Processes and Fractional Equations with Z-Transform

TL;DR

This paper derives state probabilities for tempered space- and time-fractional Poisson processes using z-transform, introduces Gegenbauer fractional differential equations, and generalizes existing results in fractional Poisson processes.

Contribution

It introduces tempered fractional Poisson processes and Gegenbauer fractional differential equations, expanding the theoretical framework of fractional stochastic processes.

Findings

Derived state probabilities for tempered fractional Poisson processes

Introduced Gegenbauer type fractional differential equations and solutions

Generalized existing results on fractional Poisson processes

Abstract

In this article, we derive the state probabilities of different type of space- and time-fractional Poisson processes using z-transform. We work on tempered versions of time-fractional Poisson process and space-fractional Poisson processes. We also introduce Gegenbauer type fractional differential equations and their solutions using z-transform. Our results generalize and complement the result available on fractional Poisson processes in several directions.