Let ${\mathcal A}$ be the ${\mathcal L}^q-$functional of a stable Lévy process starting from one and killed when crossing zero. We observe that ${\mathcal A}$ can be represented as the independent quotient of two infinite products of renormalized Beta random variables. The proof relies on Markovian time change, the Lamperti transform, and an explicit computation on perpetuities of hypergeometric Lévy processes previously obtained by Kuznetsov and Pardo. This representation allows to retrieve several factorizations previously obtained by various authors, and also to derive new ones. We emphasize the connections between ${\mathcal A}$ and more standard positive random variables. We also investigate the law of Riemannian integrals of stable subordinators. Finally, we derive several distributional properties of ${\mathcal A}$ related to infinite divisibility, self-decomposability, and the generalized Gamma convolutions.