On generating functions of Hausdorff moment sequences

The class of generating functions for completely monotone sequences (moments of finite positive measures on [0, 1]) has an elegant characterization as the class of Pick functions analytic and positive on (−∞, 1). We establish this and another such characterization and develop a variety of consequences. In particular, we characterize generating functions for moments of convex and concave probability distribution functions on [0, 1]. Also we provide a simple analytic proof that for any real p and r with p > 0, the Fuss-Catalan or Raney numbers r pn+r ( pn+r n ) , n = 0, 1, . . . are the moments of a probability distribution on some interval [0, τ ] if and only if p ≥ 1 and p ≥ r ≥ 0. The same statement holds for the binomial coefficients ( pn+r−1 n ) , n = 0, 1, . . ..