Asymptotics of individual eigenvalues of a class of large Hessenberg Toeplitz matrices

Abstract. We study the asymptotic behavior of individual eigenvalues of the n-by-n truncations of certain innite Hessenberg Toeplitz matrices as n goes to innity. The generating function of the Toeplitz matrices is supposed to be of the form a(t) = t1(1 t)f(t) (t 2 T), where is a positive real number but not an integer and f is a smooth function in H1. The classes of generating functions considered here and in a recent paper by Dai, Geary, and Kadano are overlapping, and in the overlapping cases, our results imply in particular a rigorous justication of an asymptotic formula which was conjectured by Dai, Geary, and Kadano on the basis of numerical computations