On the proximate order of growth of generating functions of Polya frequency sequences

Domains of holomorphy of generating functions of Polya frequency sequences of finite order

A domain $G\subset \bar\mathbb C$ is the domain of holomorphy of the generating function of a Pólya frequency sequence of order $r$ if and only if it satisfies the following conditions: (A) $G$ contains the point $z=0$, (B) $G$ is symmetric with respect to the real axis, (C) $T=dist(0,\partial G)\in \partial G$.

One helpful property of functions generating Pólya frequency sequences

In this work we study the solutions of the equation $z^pR(z^k)=α$ with nonzero complex $α$, integer $p,k$ and $R(z)$ generating a (possibly doubly infinite) totally positive sequence. It is shown that the zeros of $z^pR(z^k)-α$ are simple (or at most double in the case of real $α^k$) and split evenly among the sectors $\{\frac jk π\le\operatorname{Arg} z\le\frac {j+1}k π\}$, $j=0,\dots, 2k-1$. Our approach rests on the fact that $z(\ln z^{p/k}R(z) )'$ is an $\mathcal R$-function (i.e. maps the upper half of the complex plane into itself). This result guarantees the same localization to zeros o…

Multiple positivity and the Riemann zeta-function

In the paper it is discussed the relations of the Riemann $ζ-$function to classes of generating functions of multiply positive sequences according to Schoenberg (also called Pólya frequency sequences). Key words: multiply positive sequences, totally positive sequences, Pólya frequency sequences, Laguerre-Pólya class, Riemann $ζ-$function.

Hausdorff 矩序列与超几何函数

Hausdorff moment sequences and hypergeometric functions

本文完整刻画了系数为 Hausdorff 矩序列的超几何函数，并利用 Phragmèn-Lindelöf 型定理给出充分条件，进而得到平移超几何函数全星形的充要条件。

Bell-shaped sequences

A nonnegative real function $f$ is said to be bell-shaped if it converges to zero at $\pm\infty$ and the $n$th derivative of $f$ changes sign $n$ times for every $n = 0, 1, 2, \ldots$ In a similar way, we may say that a nonnegative sequence $a_k$ is bell-shaped if it converges to zero and the $n$th iterated difference of $a_k$ changes sign $n$ times for every $n = 0, 1, 2, \ldots$ Bell-shaped functions were recently characterised by Thomas Simon and the first author. In the present paper we provide an analogous description of bell-shaped sequences. More precisely, we identify bell-shaped seque…