## On generating functions of Hausdorff moment sequences

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- by Jian-Guo Liu and Robert L. Pego PDF
- Trans. Amer. Math. Soc.
**368**(2016), 8499-8518 Request permission

## 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 $\binom {pn+r-1}n$, $n=0,1,\ldots$.## References

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## Additional Information

**Jian-Guo Liu**- Affiliation: Departments of Physics and Mathematics, Duke University, Durham, North Carolina 27708
- MR Author ID: 233036
- ORCID: 0000-0002-9911-4045
- Email: jliu@phy.duke.edu
**Robert L. Pego**- Affiliation: Department of Mathematical Sciences and Center for Nonlinear Analysis, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213
- MR Author ID: 137455
- ORCID: 0000-0001-8502-2820
- Email: rpego@cmu.edu
- Received by editor(s): January 29, 2014
- Received by editor(s) in revised form: February 26, 2014, and October 13, 2014
- Published electronically: February 2, 2016
- Additional Notes: This material is based upon work supported by the National Science Foundation under grants DMS 1211161 and RNMS11-07444 (KI-Net) and partially supported by the Center for Nonlinear Analysis (CNA) under National Science Foundation grant 0635983.
- © Copyright 2016 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**368**(2016), 8499-8518 - MSC (2010): Primary 44A60; Secondary 60E99, 62E10, 05A15
- DOI: https://doi.org/10.1090/tran/6618
- MathSciNet review: 3551579