Series expansion in fractional calculus and fractional differential equations

Fractional calculus is the calculus of differentiation and integration of non-integer orders. In a recently paper (Annals of Physics 323 (2008) 2756-2778), the Fundamental Theorem of Fractional Calculus is highlighted. Based on this theorem, in this paper we introduce fractional series expansion method to fractional calculus. We define a kind of fractional Taylor series of an infinitely fractionally-differentiable function. Further, based on our definition we generalize hypergeometric functions and derive corresponding differential equations. For finitely fractionally-differentiable functions, we observe that the non-infinitely fractionally-differentiability is due to more than one fractional indices. We expand functions with two fractional indices and display how this kind of series expansion can help to solve fractional differential equations.