Two families of functions on (0,∞) are related to the theory of fractional powers of generators of strongly continuous semigroups—namely the family (σ_α(·,t))_t>0 of density functions of the one-sided stable semigroup of order α(0,1), and a family (e_α(·,μ))_μ>0 of Mittag–Leffler type functions. For the latter family we make this relation visible. We collect some important properties of these functions, and we improve a known result on the Laplace transform of the functions e_α(·,μ) and their derivatives in the sense that we enlarge the domains of the Laplace variable. We furthermore find an expression for the Laplace transform of the functions tσ_α(x,t), x>0, in terms of the derivative e′_α(·,μ).