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

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

We study the possible growth of $PF_r$ g.f.'s analytic in the unit disk and describe the proximate orders of growth of these functions. Some classical function theory results concerning orders of growth are generalized to the case of proximate orders.

石头-剪刀-布曼德博集合

A rock-paper-scissors Mandelbrot set

本文构造了基于石头-剪刀-布运算的三维实代数上的曼德博集合类似物，引入有限维实代数函数全纯性概念，证明其等价于解推广的柯西-黎曼方程组，并提供分形动画生成与游戏探索代码。

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.

Pólya–Schur 问题与自由概率

Pólya--Schur problems and free probability

本文在 Pólya–Schur 纲领与 Voiculescu 自由概率理论之间建立桥梁，证明任何自由（加法）无穷可分分布可作为 Appell 多项式根分布的弱极限，并推广到自由乘法无穷可分分布、矩形自由卷积及一般实根多项式，从而将热流与自由布朗运动的联系推广到任意自由 Lévy 过程。

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…