Generalized Taylor's formula for power fractional derivatives

We establish a new generalized Taylor's formula for power fractional derivatives with nonsingular and nonlocal kernels, which includes many known Taylor's formulas in the literature. Moreover, as a consequence, we obtain a general version of the classical mean value theorem. We apply our main result to approximate functions in Taylor's expansions at a given point. The explicit interpolation error is also obtained. The new results are illustrated through examples and numerical simulations.
Comment: This is a preprint of a paper whose final and definite form is published in 'Bolet\'{\i}n de la Sociedad Matem\'{a}tica Mexicana' at [https://www.springer.com/journal/40590]