Abstract In this paper, we introduce and study two counting processes by considering state dependency on the order of fractional derivative as well as on the exponent of backward shift operator involved in the governing difference-differential equations of the state probabilities of space-time fractional Poisson process. The Adomian decomposition method is employed to obtain their state probabilities and then their Laplace transforms are evaluated. Also, the compound versions of these state dependent models are studied and the corresponding governing fractional integral equations of their state probabilities are obtained.

中文翻译：

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时空分数泊松过程的状态相关版本

摘要 在本文中，我们通过考虑状态对分数阶导数的依赖以及对时空分数泊松状态概率的控制差分微分方程中涉及的后移算子的指数的状态依赖性，介绍和研究了两个计数过程。过程。采用 Adomian 分解方法获得它们的状态概率，然后评估它们的拉普拉斯变换。此外，研究了这些状态相关模型的复合版本，并获得了它们状态概率的相应控制分数积分方程。