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

It is shown that the following property(1)Eα,β(a(s+t)αβ)=Eα,β(asαβ)Eα,β(atαβ),s,t≥0,a∈R,α,β>0is true only when α=β=1, and a=0,β=1 or β=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=1, then the result is the s...

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Bibliographic Details
Published inJournal of the Egyptian Mathematical Society Vol. 24; no. 2; pp. 200 - 203
Main Author Elagan, S.K.
Format Journal Article
LanguageEnglish
Published Elsevier B.V 01.04.2016
SpringerOpen
Subjects
Caputo fractional derivative
Mittag–Leffler function
Semigroup property
Mittag–Leffler function
Caputo fractional derivative
Semigroup property
34A12
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Summary:It is shown that the following property(1)Eα,β(a(s+t)αβ)=Eα,β(asαβ)Eα,β(atαβ),s,t≥0,a∈R,α,β>0is true only when α=β=1, and a=0,β=1 or β=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=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(atα). Finally, we showed that all results in [16] are special cases of our results when β=1.
ISSN:1110-256X
2090-9128
DOI:10.1016/j.joems.2015.05.003