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>0 is 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 th...
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| Format: | Article |
| Language: | English |
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SpringerOpen
2016-04-01
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| Series: | Journal of the Egyptian Mathematical Society |
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| Online Access: | http://www.sciencedirect.com/science/article/pii/S1110256X15000334 |
| 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>0
is 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. |
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| ISSN: | 1110-256X |