Generalized Taylor formulas involving generalized fractional derivatives

In this paper, we establish a generalized Taylor expansion of a given function $f$ in the form $\displaystyle{f(x) = \sum_{j=0}^m c_j^{\alpha,\rho}\left(x^\rho-a^\rho\right)^{j\alpha} + e_m(x)}$\text{noindent}with$m\in $\mathbb{N}$$,$c_j^{\alpha,\rho}\in $\mathbb{R}$$,$x>a$and$0< \alpha \leq 1$\mathrm{.}Incase$\rho = \alpha = 1$, this expression coincides with the classical Taylor formula. The coefficients $c_j^{\alpha ,\rho }$, $j=0,\dots ,m$ 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.