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 results available on fractional Poisson processes in several directions.

中文翻译：

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带 Z 变换的缓和分数 Poisson 过程和分数方程

摘要 在本文中，我们使用 z 变换推导出不同类型的空间和时间分数泊松过程的状态概率。我们研究时间分数泊松过程和空间分数泊松过程的缓和版本。我们还介绍了 Gegenbauer 型分数阶微分方程及其使用 z 变换的解。我们的结果概括和补充了几个方向上的分数泊松过程的可用结果。