Mellin-Barnes integrals for stable distributions and their convolutions

In this paper we first survey the method of Mellin-Barnes integrals to represent the $alpha$ stable Levy distributions in probability theory. These integrals are known to be useful for obtaining convergent and asymptotic series representations of the corresponding probability density functions. Our original contribution concerns the Mellin-Barnes representatioin of the convolution between two stable probability densities of different Levy index, which turns to be a probability law of physical interest, even it iis no longer stable and self-similar/ A particular but interesting case of convolution is obtained combining the Cauchy-Lorentz density with the Gaussian density that yields the so-called Voigt profile. Our machinery can be applied to derive the fundamental solutions of space-fractional diffuusion equations of two orders.