In this paper, a fractional dynamics approach is used to characterize the observed accelerating expansion of the universe. We claim that the evolution of accelerating expansion obeys an α-exponential function, which is the fundamental solution of linear fractional order dynamical equation. We find that the Hubble constant is 67.8807, 68.2546, and 67.9119 for all redshift z < 1.5, z < 1, and z < 0.1 based on the dataset collected by the Supernova Cosmology Project. Furthermore, we verify that the expansion rate of our universe is speeding up and actually obeys a Mittag-Leffler law.

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