On generating functions of Hausdorff 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 $\frac{r}{pn+r}\left(\genfrac{}{}{0ex}{}{pn+r}{n}\right)$, $n=0,1,\dots$ are the moments of a probability distribution on some interval $[0,\ddot{I}„]$ {if and only if} $p\ge 1$ and $p\ge r\ge 0$. The same statement holds for the binomial coefficients $\left(\genfrac{}{}{0ex}{}{pn+r-1}{n}\right)$, $n=0,1,\dots$. A corrigendum (Trans. Amer. Math.Soc., to appear) has been included as an appendix, correcting gaps in the proof of Lemma 3.

25 pages, LaTeX; Corrigendum added, to appear in Transactions Amer. Math. Soc