Polynomially Deformed Normalized Pochhammer Sequences Having Generating Functions With Only Real Non-positive Zeros

Abstract

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.