The root distribution of polynomials with a three-term recurrence

For any fixed positive integer $n$, we study the root distribution of a sequence of polynomials $H_{m}(z)$ satisfying the rational generating function $\sum_{m=0}^{\infty}H_{m}(z)t^{m}=\frac{1}{1+B(z)t+A(z)t^{n}}$ where $A(z)$ and $B(z)$ are any polynomials in $z$ with complex coefficients. We show that the roots of $H_{m}(z)$ which satisfy $A(z)\ne0$ lie on a specific fixed real algebraic curve for all large $m$.

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