A note on property of the Mittag-Leffler function

Recently the authors have found in some publications that the following property (0.1) of Mittag-Leffler function is taken for granted and used to derive other properties.equation(0.1)Eα(a(t+s)α)=Eα(atα)Eα(asα),t,s⩾0, where a   is a real constant and α>0α>0. In this note it is proved that the above property is unavailable unless α=1α=1 or a=0a=0. Moreover, a new equality on Eα(atα)Eα(atα) is developed, whose limit state as α↑1α↑1 is just the property (0.1).