Publication 14-CNA-001
**On generating functions of Hausdorff moment sequences **

Jian-Guo Liu

Departments of Physics and Mathematics

Duke University

Durham, NC 27708

Robert L. Pego

Department of Mathematical Sciences

Carnegie Mellon University

Pittsburgh, PA 15213

**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} $p\ge1$ and $p\ge r\ge 0$.
The same statement holds for the binomial coefficients $\binom{pn+r-1}n$,
$n=0,1,\ldots$.

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