Taylor’s formula involving generalized fractional derivatives

摘要

In this paper, we establish a generalized Taylor expansion of a given function f in the formf(x)=∑j=0mcjα,ρ(xρ−aρ)jα+em(x)with m∈N0, cjα,ρ∈R, x > a > 0 and 0 < α ≤ 1. In case ρ=α=1, this expression coincides with the classical Taylor formula. The coefficients cjα,ρ, j=0,…,m as well as the residual term em(x) are given in terms of the generalized Caputo-type fractional derivatives. Several examples and applications of these results for the approximation of functions and for solving some fractional differential equations in series form are given in illustration.