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The fractional Planck distribution is calculated by applying the Caputofractional derivative with order $p$ ($p > 0$) to the equation proposed byPlanck in 1900. In addition, the integral representation of the Mittag--Lefflerfunction is employed to obtain a new formula for the fractional BEdistribution, which is then used to analyze the NASA COBE monopole data. Basedon this analysis, an identity $p\simeq e^{-\mu}$ is found, where $\mu$ is thedimensionless constant chemical potential that was introduced to the BEdistribution by the NASA COBE collaboration.

The effects of the chemical potential in a BE distribution and the fractional parameter in a distribution with Mittag-Leffler function