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. E alpha (a(t + s)(alpha)) = E(alpha) (at(alpha)) E(alpha) (as(alpha)) t, s >= 0, (0.1) where a is a real constant and alpha > 0. In this note it is proved that the above property is unavailable unless alpha = 1 or a = 0. Moreover, a new equality on E(alpha) (at(alpha)) is developed, whose limit state as alpha up arrow 1 is just the property (0.1). (C) 2010 Elsevier Inc. All rights reserved.