Riemann-Liouville differintegral
The classical form of fractional calculus is given by the Riemann-Liouville differintegral, essentially what has been described above. The theory for periodic functions, therefore including the 'boundary condition' of repeating after a period, is the Weyl differintegral. It is defined on Fourier series, and requires the constant Fourier coefficient to vanish (so, applies to functions on the unit circle integrating to 0).
By contrast the Grunwald-Letnikov differintegral starts with the derivative.
from wiki