New monotonicity and infinite divisibility properties for the Mittag-Leffler function and for the stable distributions

Hyperbolic complete monotonicity property ($\mathrm{H}\mathrm{C}\mathrm{M}$) is a way to check if a distribution is a generalized gamma ($\mathrm{G}\mathrm{G}\mathrm{C}$), hence is infinitely divisible. In this work, we illustrate to which extent the Mittag-Leffler functions $E_{\alpha },\alpha \in (0,2]$, enjoy the $\mathrm{H}\mathrm{C}\mathrm{M}$ 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 $x\mapsto E_{\alpha }\left(zx\right)$, are tightly linked to the stable distributions and to the generalized Cauchy kernel.