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

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We consider the sequence of polynomials Wₙ(x) defined by the recursion Wₙ(x)=(ax+b)Wₙ₋₁(x)+dWₙ₋₂(x), with initial values W₀(x)=1 and W₁(x)=t(x-r), where a,b,d,t,r are real numbers, a,t>0, and d<0. We show that every polynomial Wₙ(x) is distinct-real-rooted, and that the roots of the polynomial Wₙ(x) interlace the roots of the polynomial Wₙ₋₁(x). We find that, as n→∞, the sequence of smallest roots of the polynomials Wₙ(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≥2, the sequence of ith smallest roots of the polynomials Wₙ(x) converges. Similarly, there is a number to which, for every positive integer i≥2, the sequence of ith largest roots of the polynomials Wₙ(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.