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

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 W₀(x)=1 and W₁(x)=t(x-r), where a, b, d, t, r are real numbers, with a, t>0 and d<0. It is known that every polynomial W_n(x) is distinct-real-rooted. We find that, as n→∞, the smallest root of the polynomial W_n(x) converges decreasingly to a real number, and that the largest root converges increasingly to a real number. Moreover, by using the Dirichlet approximation theorem, we prove that for every integer j≥2, the jth smallest root of the polynomial W_n(x) converges as n→∞, and so does the jth largest root. It turns out that these two convergence points are independent of the numbers t, r, and the integer j. We obtain explicit expressions for the above four limit points.