Some Approximate Division and Semigroup Identities for the Mittag-Leffler Function

Abstract

It is known that the Mittag-Leffler (ML) function, E_α(z), a non-local extension of the Euler exponential function e^z, does not enjoy the semigroup property while e^z does. The purpose of this note is to show that E_α(z) does, however, for real t, s with s small enough, enjoy the approximate semigroup property, E_α(t+s)≈E_α(t)(1+E_α,α(t)/E_α(t))s. This follows from an approximation of lim_h→0+E_α(±(u+h)^α)/E_α(±u^α) , which also yields related expressions for α→1⁻, and is obtained from a recently proposed universal difference quotient representation for fractional derivatives. Graphical demonstrations are presented to show that the approximations are 'reasonably accurate' for 0<h≤0.1, with virtually no distinction from identity for 0<h≤0.01.