On generating functions of Hausdorff moment sequences

Abstract:

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 $(-\infty ,1)$. 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 $\frac {r}{pn+r}\binom {pn+r}{n}$, $n=0,1,\ldots$, are the moments of a probability distribution on some interval $[0,\tau ]$ if and only if $p\ge 1$ and $p\ge r\ge 0$. The same statement holds for the binomial coefficients $\binom {pn+r-1}n$, $n=0,1,\ldots$.