On the invalidity of semigroup property for the Mittag–Leffler function with two parameters | ||
| Journal of the Egyptian Mathematical Society | ||
| Volume 24, Issue 2, 2016, Pages 200-203 PDF (301.68 K) | ||
| DOI: 10.1016/j.joems.2015.05.003 | ||
| Author | ||
| S. K. Elagan1, 2 | ||
| 1Department of Mathematics, Faculty of Science, Menofiya University, Shebin Elkom 32511, Egypt | ||
| 2Department of Mathematics and Statistics, Faculty of Science, Taif University, Taif, El-Haweiah, P.O. Box 888, 21974, Saudi Arabia | ||
| Abstract | ||
| It is shown that the following property E α,β a (s + t) αβ = E α,β ( as αβ ) E α,β ( at αβ ) , s, t ≥ 0 , a ∈ R , α, β > 0 (1) is true only when α = β = 1 , and a = 0 , β = 1 or β = 2 . Moreover, a new equality on E α,β ( at αβ ) 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 t β−1 E α,β ( at α ) . Finally, we showed that all results in [16] are special cases of our results when β = 1 . | ||
| Keywords | ||
| Mittag–Leffler function; Caputo fractional derivative; Semigroup property | ||
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