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

Abstract

It is known that the Mittag-Leffler (ML) function, Eα(z), a non-local extension of the Euler exponential function ez, does not enjoy the semigroup property while ez does. The purpose of this note is to show that Eα(z) does, however, for real t, s with s small enough, enjoy the approximate semigroup property, Eα(t+s)≈Eα(t)(1+Eα,α(t)/Eα(t))s. This follows from an approximation of limh→0+Eα(±(u+h)α)/Eα(±uα) , 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 0h≤0.1, with virtually no distinction from identity for 0h≤0.01.