On some properties of the Mittag-Leffler function $E_α(-t^α)$, completely monotone for $t > 0$ with $0 < α< 1$

We analyse some peculiar properties of the function of the Mittag-Leffler (M-L) type, $e_α(t):= E_α(-t^α)$ for $0 <α< 1$ and $t > 0$, which is known to be completely monotone (CM) with a non negative spectrum of frequencies and times, suitable to model fractional relaxation processes. We first note that these two spectra coincide so providing a universal scaling property of this function. Furthermore, we consider the problem of approximating our M-L function with simpler CM functions for small and large times. We provide two different sets of elementary CM functions that are asymptotically equivalent to $e_α(t)$ as $t \to 0$ and $t \to \infty$.

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