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On generating functions of Hausdorff moment sequences


Authors: Jian-Guo Liu and Robert L. Pego
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
Published electronically: February 2, 2016
MathSciNet review: 3551579
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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 \,$.


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Additional Information

Jian-Guo Liu
Affiliation: Departments of Physics and Mathematics, Duke University, Durham, North Carolina 27708
Email: jliu@phy.duke.edu

Robert L. Pego
Affiliation: Department of Mathematical Sciences and Center for Nonlinear Analysis, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213
Email: rpego@cmu.edu

DOI: https://doi.org/10.1090/tran/6618
Keywords: Completely monotone sequence, Fuss-Catalan numbers, complete Bernstein function, infinitely divisible, canonical sequence, exchangeable trials, concave distribution function, random matrices
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.
Article copyright: © Copyright 2016 American Mathematical Society