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 $n=0,1,2,\dots$ 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 $(a{)}_k=a(a+1)\cdot \dots \cdot (a+k-1),(a{)}_0=1,$ 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 $n=0,1,2,\dots ,$ 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.