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

This paper concerns the distribution in the complex plane of the roots of a polynomial sequence {Wn(x)}n≥0{Wn(x)}n≥0 given by a recursion Wn(x)=aWn−1(x)+(bx+c)Wn−2(x)Wn(x)=aWn−1(x)+(bx+c)Wn−2(x), with W0(x)=1W0(x)=1 and W1(x)=t(x−r)W1(x)=t(x−r), where a>0a>0, b>0b>0, and c,t,r∈Rc,t,r∈R. Our results include proof of the distinct-real-rootedness of every such polynomial Wn(x)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.