关于豪斯多夫矩序列的生成函数
On generating functions of Hausdorff moment sequences
AI总结：
本文刻画了[0,1]上有限正测度矩（完全单调序列）生成函数为(-∞,1)上解析且正的Pick函数，推导其推论，刻画凸/凹概率分布矩的生成函数，并证明特定条件下Fuss-Catalan等数为概率分布矩。
摘要：
完全单调序列（[0,1]上有限正测度的矩）的生成函数类具有优雅的刻画，即Pick函数类在(-∞,1)上解析且为正。我们确立了这一结果及另一类刻画，并推导了多种推论。特别地，我们刻画了[0,1]上凸和凹概率分布函数的矩的生成函数。此外，我们提供了一个简单的解析证明：对于任意满足p>0的实数p和r，当且仅当p≥1且p≥r≥0时，Fuss-Catalan或Raney数\frac{r}{pn+r}\binom{pn+r}{n}（n=0,1,…）是某区间[0,τ]上概率分布的矩。同样的结论适用于二项式系数\binom{pn+r-1}{n}（n=0,1,…）。附录包含一篇待发表于《美国数学会汇刊》的勘误，修正了引理3证明中的漏洞。
英文摘要：
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.
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