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

We analyse some peculiar properties of the functionof the Mittag-Leffler (M-L) type, e_ (t): = E_ (-t^) for 00, 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 (surprisingly) these two spectra coincide so providinga universal scaling property of this function, not well pointed out in the literature. 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_ (t) as t 0 and t +. The first set is given by the stretched exponential for small timesand the power law for large times, following a standard approach. For the second set we chose two rational CM functions in t^, obtained as the Pad\`e Approximants (PA) 0/1 to the convergent series in positive powers (as t 0) and to the asymptotic series in negative powers (as t), respectively. From numerical computations we are allowed to the conjecture that the second set provides upper and lower bounds to the Mittag-Leffler function.