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

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.