On the invalidity of semigroup property for the Mittag–Leffler function with two parameters

It is shown that the following propertyequation(1)Eα,β(a(s+t)αβ)=Eα,β(asαβ)Eα,β(atαβ),s,t≥0,a∈R,α,β>0Eα,β(a(s+t)αβ)=Eα,β(asαβ)Eα,β(atαβ),s,t≥0,a∈R,α,β>0is true only when α=β=1α=β=1, and a=0,β=1a=0,β=1 or β=2β=2. Moreover, a new equality on Eα, β(atαβ) is developed, whose limit state as α↑1 and β > α is just the above property (1) and if β=1β=1, then the result is the same as in [16]. Also, it is proved that this equality is the characteristic of the function tβ−1Eα,β(αat)tβ−1Eα,β(atα). Finally, we showed that all results in [16] are special cases of our results when β=1β=1.