Taylor's formula involving generalized fractional derivatives

Abstract In this paper, we establish a generalized Taylor expansion of a given function f in the form f ( x ) = ∑ j = 0 m c j α , ρ ( x ρ − a ρ ) j α + e m ( x ) with m ∈ N 0 , c j α , ρ ∈ R , x > a > 0 and 0  ρ = α = 1 , this expression coincides with the classical Taylor formula. The coefficients c j α , 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.

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