In this article, convolution-type fractional derivatives generated by Dickman subordinator and inverse Dickman subordinator are discussed. The Dickman subordinator and its inverse are generalizations of stable and inverse stable subordinators, respectively. The series representations of densities of the Dickman subordinator and inverse Dickman subordinator are also obtained, which could be helpful for computational purposes. Moreover, the space and time-fractional Poisson-Dickman processes, space-fractional Skellam Dickman process and non-homogenous Poisson-Dickman process are introduced and their main properties are studied.

中文翻译：

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Dickman 从属子生成的广义分数导数及相关随机过程

本文讨论了由Dickman从属子和逆Dickman从属子生成的卷积型分数阶导数。迪克曼从属子及其逆分别是稳定和逆稳定从属子的推广。还获得了迪克曼从属子和逆迪克曼从属子的密度级数表示，这可能有助于计算目的。此外，还介绍了时空分数式Poisson-Dickman过程、空间分数式Skellam Dickman过程和非齐次Poisson-Dickman过程，并研究了它们的主要性质。