The root distribution of polynomials with a three-term recurrence

Abstract

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 $A(z)\ne 0$ lie on a specific fixed real algebraic curve for all large $m$.

References in corpus (1)

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

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