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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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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


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 $\binom {pn+r-1}n$, $n=0,1,\ldots$.
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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:
  • 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:
  • 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:
  • MathSciNet review: 3551579