On generating functions of Hausdorff moment sequences

Jian-Guo Liu Duke University Robert L. Pego Carnegie Mellon University

Classical Analysis and ODEs mathscidoc:1702.05001

Transactions of the American Mathematical Society, 368, (12), 8499-8518, 2016.12
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 $[0,1]$, and we provide a simple analytic proof that for any real $p,r>0$, 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 $p\ge r$.
Completely monotone sequence, Fuss-Catalan numbers, complete Bernstein function, infinitely divisible, canonical sequence, exchangeable trials, concave distribution function, random matrices
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  title={On generating functions of Hausdorff moment sequences},
  author={Jian-Guo Liu, and Robert L. Pego},
  booktitle={Transactions of the American Mathematical Society},
Jian-Guo Liu, and Robert L. Pego. On generating functions of Hausdorff moment sequences. 2016. Vol. 368. In Transactions of the American Mathematical Society. pp.8499-8518. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170207212602488061264.
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