Nuij's theorem states that if a polynomial
$p\in \mathbb{R}[z]$
is hyperbolic (i.e., has only real roots), then
$p+s{{p}^{'}}$
is also hyperbolic for any
$s\in \mathbb{R}$
. We study other perturbations of hyperbolic polynomials of the form
${{p}_{a}}(z,s)\,\,:=\,\,\,p\,(z)+\,\sum\nolimits_{k=1}^{d}{{{a}_{k}}{{s}^{k}}{{p}^{(k)}}(z)}$
. We give a full characterization of those
$a=({{a}_{1}},...,{{a}_{d}})\,\in \,{{\mathbb{R}}^{d}}$
for which
${{p}_{a}}(z,s)$
is a pencil of hyperbolic polynomials. We also give a full characterization of those
$a=({{a}_{1}},...,{{a}_{d}})\,\in \,{{\mathbb{R}}^{d}}$
for which the associated families
$ $
admit universal determinantal representations. In fact, we show that all these sequences come fromspecial symmetric Toeplitz matrices.