1. Multiplier sequences and CZDS

Problem 1.1.
[A. Vishnyakova] Let $\{c_k\}_{k=0}^n\subset\mathbb{R}$ be given. Does there exist a real rooted polynomial, $p(x)$, with zeros outside $[0,n]$ such that $c_k=p(k)$, $k=0,1,\ldots,n$? 
Problem 1.2.
[G. Csordas] Let $f(z)$ be a meromorphic function. When is the sequence $\{f(k)\}_{k=0}^\infty$ a multiplier sequence?
Cite this as: AimPL: Stability and hyperbolicity, available at http://aimpl.org/hyperbolicpoly.