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# On generating functions of Hausdorff moment sequences (14-CNA-001)

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 rpn+r(pn+rn), 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−1n), n=0,1,….