  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.
