Nuij type pencils of hyperbolic polynomials

Nuij’s theorem states that if a polynomial $p\in \mathbb{R}[z]$ is hyperbolic (i.e., has only real roots) then $p+s{p}^{\mathrm{\prime }}$ is also hyperbolic for any $s\in $\mathbb{R}$$\mathrm{.}Westudyotherperturbationsofhyperbolicpolynomialsoftheform$p_a(z,s): =p(z) +\sum_{k=1}^d a_ks^kp^{(k)}(z)$. We give a full characterization of those $a=(a_1,\dots ,a_d)\in {\mathbb{R}}^d$ for which $p_a(z,s)$ is a pencil of hyperbolic polynomials. We give also a full characterization of those $a=(a_1,\dots ,a_d)\in {\mathbb{R}}^d$ for which the associated families $p_a(z,s)$ admit universal determinantal representations. In fact we show that all these sequences come from special symmetric Toeplitz matrices.