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.