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

Elagan, S. K.

doi:10.1016/j.joems.2015.05.003

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

Author

S. K. Elagan ¹^{, 2}

¹ Department of Mathematics, Faculty of Science, Menofiya University, Shebin Elkom 32511, Egypt

² Department of Mathematics and Statistics, Faculty of Science, Taif University, Taif, El-Haweiah, P.O. Box 888, 21974, Saudi Arabia

10.1016/j.joems.2015.05.003

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