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

We consider the sequence of polynomials Wn(x)Wn(x) defined by the recursion Wn(x)=(ax+b)Wn−1(x)+dWn−2(x)Wn(x)=(ax+b)Wn−1(x)+dWn−2(x), with initial values W0(x)=1W0(x)=1 and W1(x)=t(x−r)W1(x)=t(x−r), where a, b, d, t, r   are real numbers, with a,t>0a,t>0 and d<0d<0. It is known that every polynomial Wn(x)Wn(x) is distinct-real-rooted. We find that, as n→∞n→∞, the smallest root of the polynomial Wn(x)Wn(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≥2j≥2, the j  th smallest root of the polynomial Wn(x)Wn(x) converges as n→∞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.