On generating functions of Hausdorff moment sequences
Jian-Guo Liu, Robert L. Pego
TL;DR
This work characterizes generating functions of completely monotone (Hausdorff) moment sequences as Pick functions analytic on $(-\infty,1)$, with equivalent conditions expressed through $F$ and the upshifted $F_1(z)=zF(z)$, linking moments to integral representations $F(z)=\int_0^1 (1- tz)^{-1} d\mu(t)$.Building on this, the authors derive numerous consequences, including dilation, reflection, various composition rules, infinite divisibility, and convolution-group structures, all governed by Pick-function criteria and leading to constructive methods to generate new moment sequences from existing ones.They extend the framework to moments of convex and concave distribution functions on $[0,1]$, providing simple analytic proofs for when Fuss–Catalan (Raney) numbers and binomial sequences are moments of probability measures, and they develop canonical densities that describe these distributions; a corrigendum corrects gaps in a key lemma.Overall, the paper offers a unified, analytic approach to Hausdorff moment problems, connects discrete moment sequences with continuous density representations, and yields practical criteria for important families of combinatorial sequences to arise as moment sequences.
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 $[0,τ]$ {if and only if} $p\ge1$ and $p\ge r\ge 0$. The same statement holds for the binomial coefficients $\binom{pn+r-1}n$, $n=0,1,\ldots$. A corrigendum (Trans. Amer. Math.Soc., to appear) has been included as an appendix, correcting gaps in the proof of Lemma 3.