  A generalization of the Gr\"{u}nwald difference approximation for fractional
derivatives in terms of a real sequence and its generating function is
presented. Properties of the generating function are derived for consistency
and order of accuracy for the approximation corresponding to the generator.
Using this generalization, some higher order Gr\"{u}nwald type approximations
are constructed and tested for numerical stability by using steady state
fractional differential problems. These higher order approximations are used in
Crank-Nicolson type numerical schemes to approximate the solution of space
fractional diffusion equations. Stability and convergence of these numerical
schemes are analyzed and are supported by numerical examples.
