×

Exceptional zeros of polynomials satisfying a three-term recurrence. (English) Zbl 1381.30005

Summary: Let \(P_m(z)\) be a sequence of polynomials satisfying the recurrence relation \(P_m(z) + B(z) P_{m - 1}(z) + A(z) P_{m - 2}(z) = 0\). Tran showed that with a suitable initial condition, the zeros of \(P_m(z)\) lie on a fixed explicit curve \(\mathcal{C}\) on the complex plane. We study a more general initial condition in which we can count the finite number of zeros lying off \(\mathcal{C}\).

MSC:

30C15 Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral)
26C10 Real polynomials: location of zeros
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Andrews, G. E.; Askey, R.; Roy, R., Special Functions, Encycl. Math. Appl. (1999), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0920.33001
[2] McNamee, J. M., Sturm Sequences and Greatest Commom Divisors, Numerical Methods for Roots of Polynomials - Part I, 37-52 (2007), Elsevier Science · Zbl 1143.65002
[3] Tran, K., Connections between discriminants and the root distribution of polynomials with rational generating function, J. Math. Anal. Appl., 410, 330-340 (2014) · Zbl 1307.12002
[4] Tran, K., The root distribution of polynomials with a three-term recurrence, J. Math. Anal. Appl., 421, 878-892 (2015) · Zbl 1296.30010
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.