Abstract
We consider the integro-differential equation I0+αf=xmf on the half-line. We show that there exists a density solution, which is then unique and can be expressed in terms of the Beta distribution, if and only if m>α. These density solutions extend the class of generalized one-sided stable distributions introduced in [29] and more recently investigated in [27]. We study various analytical aspects of these densities, and we solve the open problems about infinite divisibility formulated in [27].