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The Δ-power function and fractional Δ-integrals and fractional Δ-differential are defined, and then thedefinitions and properties of Δ-Mittag-Leffler function are given. The properties of fractional Δ-integrals and fractional Δ-differential on time scales are discussed in detail. After that, theexistence of the solution and the dependency of the solution upon theinitial value for Cauchy type problem with fractional Δ-derivativeare studied. Also the explicit solutions to homogeneous fractional Δ-differential equations and nonhomogeneous fractional Δ-differentialequations are derived by using Laplace transform method

Fractional Cauchy Problem with Riemann-Liouville Fractional Delta Derivative on Time Scales