A Fractional Calculus on Arbitrary Time Scales: Fractional   Differentiation and Fractional Integration
  We introduce a general notion of fractional (noninteger) derivative for functions defined on arbitrary time scales. The basic tools for the time-scale fractional calculus (fractional differentiation and fractional integration) are then developed. As particular cases, one obtains the usual time-scale Hilger derivative when the order of differentiation is one, and a local approach to fractional calculus when the time scale is chosen to be the set of real numbers. 
Nadia Benkhettou, Artur M. C. Brito da Cruz, Delfim F. M. Torres
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