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