具有仅含非正实零点生成函数的多项式变形归一化波赫哈默序列
Polynomially Deformed Normalized Pochhammer Sequences Having Generating Functions With Only Real Non-positive Zeros
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中文总结 AI 辅助
本文研究多项式变形归一化波赫哈默序列的开放问题，建立相关必要与充分条件，给出示例并提出新开放问题。
中文摘要 AI 辅助
对于给定的实数a>0和给定的实多项式P_n∈ℝ[x]（次数为n=0,1,2,…），易证∑_{k=0}^∞ [(a)_k/k!]P_n(k)z^k = S_{n,a}(z)/(1-z)^{a+n}（|z|<1），其中S_{n,a}是次数不超过n的实多项式。此处(a)_k = a(a+1)…(a+k-1)，(a)_0=1，即上升阶乘，又称波赫哈默符号。本文研究以下开放问题：描述次数为n=0,1,2,…的实多项式P_n∈ℝ[x]的集合，使得对应多项式S_{n,a}的所有零点均为非正实数。当a=1时，该问题已在文献[vish]中被研究。本文建立了若干新的必要条件与充分条件，给出了多个重要示例，并提出了若干开放问题。
英文摘要
For a given real number $a>0$ and a given real polynomial $P_n\in \mathbb{R}[x]$ of degree $n=0, 1, 2, \ldots$ it is easy to see that $ \sum_{k=0}^\infty \frac{(a)_k}{k!} P_n(k) z^k =\frac{S_{n, a}(z)}{(1-z)^{a+ n}}, \ |z|<1, $ where $S_{n, a}$ is a real polynomial of degree not greater than $n.$ Here $(a)_k =a(a+1)\cdot \ldots \cdot (a+k-1),\ (a)_0 = 1,$ is the rising factorial, or the Pochhammer symbol. We consider the following open problem: to describe the set of real polynomials $P_n\in \mathbb{R}[x]$ of degree $n=0, 1, 2, \ldots,$ such that the corresponding polynomial $S_{n, a}$ has all real non-positive zeros. In the case $a=1$ this problem has been studied in \cite{vish}. We establish several new necessary conditions and several sufficient conditions, present a number of important examples, and formulate several open problems.
