Location of the zeros of polynomials satisfying three-term recurrence relations with complex coefficients

Abstract

Two papers on the same subject were pubkushed by the authors in 1985:part I[1]and part III[2].The method announced for part II did not give satisfying results.However,we noticed that our method of estimationg the boundaries of zero-free regions for polynomials can be improved by introducing proper optimizing paramcters for each polynomial.The optimization is then performed over all these parameters at the end. It was showed that this finest method of optimimation is sharp in the following sense:it leads to the exact interval where all zeros of classical orthogonal polynomials are licated[3].In the present paper we give the complete proofs of these improved theorems of lication in quaestion.