In this paper, we introduce the \(\psi \)-Hilfer pseudo-fractional operator, motivated by the \(\psi \)-Hilfer fractional derivative and the theory of pseudo-analysis. We investigate a wide class of important and essential results for pseudo-fractional calculus in a semiring \(([a, b], \oplus , \odot )\) and some particular cases are discussed. Specifically, we present a class of pseudo-fractional operators which are particular cases of the \(\psi \)-Hilfer pseudo-fractional operator. In addition, we present the pseudo-Leibniz-type rules I and II and pseudo-Leibniz rules and some particular cases of Leibniz-type rules I and II are discussed. Finally, we obtain formulas for the Hilfer pseudo-fractional derivative, for the pseudo-Laplace transform, and for the g-integration by parts of the \(\psi \)-Hilfer pseudo-fractional operator.

中文翻译：

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$$ \ psi $$ψ-希尔弗（Hilfer）伪分数算子：关于分数演算的新结果

在本文中，我们介绍了\（\ psi \）- Hilfer伪分数算子，其受\（\ psi \）- Hilfer分数导数和伪分析理论的启发。我们研究半环\（（[[a，b]，\ oplus，\ odot）\）中伪分数阶微积分的一系列重要的重要结果，并对某些特殊情况进行了讨论。具体来说，我们提供了一类伪分数运算符，它们是\（\ psi \）的特殊情况-Hilfer伪分数运算符。另外，我们给出了伪莱布尼兹型规则I和II和伪莱布尼兹规则，并讨论了莱布尼兹型规则I和II的一些特殊情况。最后，我们得到公式的Hilfer的伪分数阶导数，对伪拉普拉斯变换，并为克由的部分-整合\（\ PSI \） -Hilfer伪小数运算符。