The root distribution of polynomials with a three-term recurrence

TL;DR

This paper investigates the root distribution of polynomials generated by a specific rational function, showing that roots lie on a fixed algebraic curve as the degree grows large.

Contribution

It establishes that roots of polynomials with a three-term recurrence and rational generating functions asymptotically lie on a fixed algebraic curve.

Findings

Roots lie on a fixed real algebraic curve for large degrees.

The root distribution is characterized for polynomials with a three-term recurrence.

The result holds for any polynomials A(z) and B(z) with complex coefficients.

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 $A(z)\ne0$ lie on a specific fixed real algebraic curve for all large $m$.