This paper discusses some aspects of Taylor’s formulas in fractional calculus, focusing on use of Caputo’s definition. Such formulas consist of a polynomial expansion whose coefficients are values of the fractional derivative evaluated at its starting point multiplied by some coefficients determined through the Gamma function. The properties of fractional derivatives heavily affect the expansion’s coefficients. In the first part of the paper, we review the currently available formulas in fractional calculus with a particular focus on the Caputo derivative. In the second part, we prove why the notion of sequential fractional derivative (i.e., n-fold fractional derivative) is required to build Taylor expansions in terms of fractional derivatives. Such properties do not seem to appear in the literature. Furthermore, some new properties of the expansion coefficients are also shown together with some computational examples in Wolfram Mathematica.

本文讨论了分数阶微积分中泰勒公式的一些方面，重点是卡普托定义的使用。此类公式由多项式展开组成，其系数是在其起点计算的分数导数值乘以通过 Gamma 函数确定的一些系数。分数阶导数的性质严重影响展开系数。在本文的第一部分中，我们回顾了分数阶微积分中当前可用的公式，特别关注 Caputo 导数。在第二部分中，我们证明为什么需要序贯分数阶导数（即 n 重分数阶导数）的概念来构建分数阶导数的泰勒展开式。这些性质似乎没有出现在文献中。此外，还展示了展开系数的一些新属性以及 Wolfram Mathematica 中的一些计算示例。