Location of the zeros of polynomials satisfying three-term recurrence relations. I. General case with complex coefficients

Abstract

The finite sequences of polynomials { P n } n = 0 N generated from three-term recurrence relations with complex coefficients are considered. First a general method is presented which allows the determination the regions where all zeros of the polynomials in question are located. Next one way is followed, say ¦μ n¦ < ¦β n¦ , and the first results are established. In the second paper (J. Gilewicz and E. Leopold, Location of zeros of polynomials satisfying three-term recurrence relations. II. General case with complex coefficients, in preparation) the reverse way, ¦μ n¦ > ¦β n¦ , is followed. Subsequent papers (E. Leopold, J. Approx. Theory 43, 15–24 (1985); E. Leopold, Location of zeros of polynomials satisfying three-term recurrence relations. IV. Application to some polynomials and to generalized Bessel polynomials, in preparation) are devoted to some particular cases and to numerical applications.