Generalized Taylor formulas involving generalized fractional derivatives

Abstract

In this paper, we establish a generalized Taylor expansion of a given function $f$ in the form $\displaystyle{f(x) = \sum_{j=0}^m c_j^{α,ρ}\left(x^ρ-a^ρ\right)^{jα} + e_m(x)}$ \noindent with $m\in \mathbb{N}$, $c_j^{α,ρ}\in \mathbb{R}$, $x>a$ and $0< α\leq 1$. In case $ρ= α= 1$, this expression coincides with the classical Taylor formula. The coefficients $c_j^{α,ρ}$, $j=0,\dots,m$ as well as an estimation of $e_m(x)$ are given in terms of the generalized Caputo-type fractional derivatives. Some applications of these results for approximation of functions and for solving some fractional differential equations in series form are given in illustration.