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.(0.1) E α ( a ( t + s ) α ) = E α ( a t α ) E α ( a s α ) , t , s ⩾ 0 , where a is a real constant and α > 0 . In this note it is proved that the above property is unavailable unless α = 1 or a = 0 . Moreover, a new equality on E α ( a t α ) is developed, whose limit state as α ↑ 1 is just the property (0.1).