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 $f(x)=\sum _{j=0}^m{c}_j^{\alpha ,\rho }{\left(x^{\rho }-a^{\rho }\right)}^{j\alpha }+e_m(x)$ \noindent with $m\in \mathbb{N}$, $c_j^{\alpha ,\rho }\in \mathbb{R}$, $x>a$ and $0<\alpha \le 1$. In case $\rho =\alpha =1$, 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.

AMS-LaTeX2e, 20 pages

References in corpus (2)

• Time Fractional Quantum Mechanics
• Muntz type Theorems I

Cited by in corpus (3)

• Infinite series representation of fractional calculus: theory and applications
• A fractional Borel-Pompeiu type formula and a related fractional $\psi -$Fueter operator with respect to a vector-valued function
• On the fractional operators with respect to another function