Abstract In this paper, we address the one-parameter families of the fractional integrals and derivatives defined on a finite interval. First we remind the reader of the known fact that under some reasonable conditions, there exists precisely one unique family of the fractional integrals, namely, the well-known Riemann-Liouville fractional integrals. As to the fractional derivatives, their natural definition follows from the fundamental theorem of the Fractional Calculus, i.e., they are introduced as the left-inverse operators to the Riemann-Liouville fractional integrals. Until now, three families of such derivatives were suggested in the literature: the Riemann-Liouville fractional derivatives, the Caputo fractional derivatives, and the Hilfer fractional derivatives. We clarify the interconnections between these derivatives on different spaces of functions and provide some of their properties including the formulas for their projectors and the Laplace transforms. However, it turns out that there exist infinitely many other families of the fractional derivatives that are the left-inverse operators to the Riemann-Liouville fractional integrals. In this paper, we focus on an important class of these fractional derivatives and discuss some of their properties.

中文翻译：

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分数阶导数和分数阶微积分基本定理

摘要 在本文中，我们讨论了在有限区间上定义的分数积分和导数的单参数族。首先我们提醒读者一个已知的事实，在某些合理条件下，恰好存在一个唯一的分数积分族，即众所周知的黎曼-刘维尔分数积分。对于分数阶导数，它们的自然定义遵循分数阶微积分的基本定理，即作为黎曼-刘维尔分数积分的左逆算子引入。迄今为止，文献中提出了三类此类导数：黎曼-刘维尔分数阶导数、Caputo 分数阶导数和 Hilfer 分数阶导数。我们阐明了这些导数在不同函数空间上的相互联系，并提供了它们的一些性质，包括它们的投影仪和拉普拉斯变换的公式。然而，事实证明存在无数其他分数阶导数族，它们是 Riemann-Liouville 分数阶积分的左逆算子。在本文中，我们关注这些分数导数中的一个重要类别并讨论它们的一些性质。