  We consider the integro-differential equation ${\rm I}^{\alpha}_{0+}f= x^m f$
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
$m> \alpha.$ These density solutions extend the class of generalized one-sided
stable distributions introduced in Schneider (1987) and more recently
investigated in Pakes (2014). We study various analytical aspects of these
densities, and we solve the open problems about infinite divisibility
formulated in Pakes (2014).
