Connections between discriminants and the root distribution of polynomials with rational generating function

Abstract

Let $H_{m}(z)$ be a sequence of polynomials whose generating function $\sum_{m=0}^{\infty}H_{m}(z)t^{m}$ is the reciprocal of a bivariate polynomial $D(t,z)$. We show that in the three cases $D(t,z)=1+B(z)t+A(z)t^{2}$, $D(t,z)=1+B(z)t+A(z)t^{3}$ and $D(t,z)=1+B(z)t+A(z)t^{4}$, where $A(z)$ and $B(z)$ are any polynomials in $z$ with complex coefficients, the roots of $H_{m}(z)$ lie on a portion of a real algebraic curve whose equation is explicitly given. The proofs involve the $q$-analogue of the discriminant, a concept introduced by Mourad Ismail.