The limit of the zero set of polynomials of the Fibonacci type

Abstract

The paper concerns the limit of the zero set Sn of the Fibonacci-type polynomials Gn(x)=Gn(a,b,x) given by G0(x)=a, G1(x)=x+b, and Gn+2(x)=xGn+1(x)+Gn(x). It presents an explicit set L consisting of at most two points such that Sn converges to the union of [−2i,2i] and L in the Hausdorff metric. The proof is based on the asymptotic analysis of the eigenvalues of special non-Hermitian perturbations of Hermitian Toeplitz matrices.