Abstract
Hyperbolic complete monotonicity property (HCM) is a way to check if a distribution is a generalized gamma (GGC), hence is infinitely divisible. In this work, we illustrate to which extent the Mittag-Leffler functions Eα,α∈(0,2], enjoy the HCM property, and then intervene deeply in the probabilistic context. We prove that for suitable α and complex numbers z, the real and imaginary part of the functions x↦Eαzx, are tightly linked to the stable distributions and to the generalized Cauchy kernel.
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