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

الأصل · EN

This paper is concerned with the distribution in the complex plane of the roots of a polynomial sequence {Wₙ(x)}ₙ≥₀ given by a recursion Wₙ(x)=aWₙ₋₁(x)+(bx+c)Wₙ₋₂(x), with W₀(x)=1 and W₁(x)=t(x-r), where a>0, b>0, and c,t,r. Our results include proof of the distinct-real-rootedness of every such polynomial Wₙ(x), derivation of the best bound for the zero-set {x Wₙ(x)=0for some n≥1}, and determination of three precise limit points of this zero-set. Also, we give several applications from combinatorics and topological graph theory.