This paper explores mixture distributions induced by a product of the positive stable random variable and a power of another positive random variable. The paper also considers the convolution of the stable density with a gamma density. These two constructs, mixing and convolution, suffice to generate a rich family of distributions. An example is the positive Linnik distribution, which is known to arise from a product involving stable and gamma random variables. We show that gamma-Linnik convolution gives the Mittag-Leffler distribution and the Mittag-Leffler Markov chain associated with the growth of random trees. Building on that, we construct a new family of distributions with explicit densities. A particular choice of parameters for this family yields the Lamperti-type laws associated with occupation times for Markov processes.