Root geometry of polynomial sequences II: Type (1,0)

TL;DR

This paper studies the roots of a polynomial sequence defined by a specific recurrence, proving they are real, interlaced, and converge to explicit limits, revealing deep root structure and convergence properties.

Contribution

It provides a comprehensive analysis of the root behavior of the polynomial sequence, including interlacing, convergence, and explicit limit points, extending understanding of polynomial root geometry.

Findings

All polynomials in the sequence have distinct real roots.

Roots of consecutive polynomials interlace.

Sequences of roots converge to explicit limit points.

Abstract

We consider the sequence of polynomials $W_n(x)$ defined by the recursion $W_n(x)=(ax+b)W_{n-1}(x)+dW_{n-2}(x)$, with initial values $W_0(x)=1$ and $W_1(x)=t(x-r)$, where $a,b,d,t,r$ are real numbers, $a,t>0$, and $d<0$. We show that every polynomial $W_n(x)$ is distinct-real-rooted, and that the roots of the polynomial $W_n(x)$ interlace the roots of the polynomial $W_{n-1}(x)$. We find that, as $n\to\infty$, the sequence of smallest roots of the polynomials $W_n(x)$ converges decreasingly to a real number, and that the sequence of largest roots converges increasingly to a real number. Moreover, by using the Dirichlet approximation theorem, we prove that there is a number to which, for every positive integer $i\ge2$, the sequence of $i$th smallest roots of the polynomials $W_n(x)$ converges. Similarly, there is a number to which, for every positive integer $i\ge2$, the sequence of $i$th largest roots of the polynomials $W_n(x)$ converges. It turns out that these two convergence points are independent of the numbers $t$ and $r$, as well as $i$. We derive explicit expressions for these four limit points, and we determine completely when some of these limit points coincide.