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 (ββ,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 pn+rrβ(npn+rβ), n=0,1,β¦ are the moments
of a probability distribution on some interval [0,Ο] {if and only if}
pβ₯1 and pβ₯rβ₯0. The same statement holds for the binomial
coefficients (npn+rβ1β), n=0,1,β¦.Comment: 23 pages, LaTeX; Minor corrections and explanations added, literature
update. To appear in Transactions Amer. Math. So