Two families of functions related to the fractional powers of generators of strongly continuous contraction semigroups

Two families of functions on (0,∞) are related to the theory of fractional powers of generators of strongly continuous semigroups—namely the family (σα(·, t))t>0 of density functions of the one-sided stable semigroup of order α ∈ (0, 1), and a family (eα(·,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 eα(·,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 t →σα(x, t ), x >0, in terms of the derivative e α (·,μ).  2003 Elsevier Inc. All rights reserved