On Taylor’s formulas in fractional calculus: overview and characterization for the Caputo derivative

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.