There exist only a few known examples of subordinators for which the transition probability density can be computed explicitly along side an expression for its Lévy measure and Laplace exponent. Such examples are useful in several areas of applied probability. For example, they are used in mathematical finance for modeling stochastic time change. They appear in combinatorial probability to construct sampling formulae, which in turn is related to a variety of issues in the theory of coalescence models. Moreover, they have also been extensively used in the potential analysis of subordinated Brownian motion in dimension d≥2. In this paper, we show that Kendall's classic identity for spectrally negative Lévy processes can be used to construct new families of subordinators with explicit transition probability semigroups. We describe the properties of these new subordinators and emphasize some interesting connections with explicit and previously unknown Laplace transform identities and with complete monotonicity properties of certain special functions.