A new blackbody radiation law based on fractional calculus and its application to NASA COBE data

•Fractional calculus is applied to the equation proposed by M. Planck.•A new Planck distribution is described by the Mittag-Leffler function.•Role of fractional parameter pp is almost same as μμ in Bose–Einstein distribution.•NASA COBE data are analyzed by a new Planck distribution with pp.

By applying fractional calculus to the equation proposed by M. Planck in 1900, we obtain a new blackbody radiation law described by a Mittag-Leffler (ML) function. We have analyzed NASA COBE data by means of a non-extensive formula with a parameter (q−1)(q−1), a formula proposed by Ertik et al. with a fractional parameter (α−1)(α−1), and our new formula including a parameter (p−1)(p−1), as well as the Bose–Einstein distribution with a dimensionless chemical potential μμ. It can be said that one role of the fractional parameter (p−1)(p−1) is almost the same as that of chemical potential (μ)(μ) as well as that of the parameter (q−1)(q−1) in the non-extensive approach.