Czechoslovak Mathematical Journal, first online, pp. 1-12


Criterion of the reality of zeros in a polynomial sequence satisfying a three-term recurrence relation

Innocent Ndikubwayo

Received December 4, 2018.   Published online February 24, 2020.

Abstract:  This paper establishes the necessary and sufficient conditions for the reality of all the zeros in a polynomial sequence $\{P_i\}_{i=1}^{\infty}$ generated by a three-term recurrence relation $P_i(x)+ Q_1(x)P_{i-1}(x) +Q_2(x) P_{i-2}(x)=0$ with the standard initial conditions $P_0(x)=1, P_{-1}(x)=0,$ where $Q_1(x)$ and $Q_2(x)$ are arbitrary real polynomials.
Keywords:  recurrence relation; polynomial sequence; support; real zeros
Classification MSC:  12D10, 26C10, 30C15
DOI:  10.21136/CMJ.2020.0535-18

PDF available at:  Springer   Institute of Mathematics CAS

References:
[1] S. Beraha, J. Kahane, N. J. Weiss: Limits of zeros of recursively defined families of polynomials. Studies in Foundations and Combinatorics. Adv. Math., Suppl. Stud. 1, Academic Press, New York (1978), 213-232. MR 0520560 | Zbl 0477.05034
[2] N. Biggs: Equimodular curves. Discrete Math. 259 (2002), 37-57. DOI 10.1016/S0012-365X(02)00444-2 | MR 1948772 | Zbl 1008.05060
[3] P. Brändén: Unimodality, log-concavity, real-rootedness and beyond. Handbook of Enumerative Combinatorics Discrete Mathematics and Its Applications, CRC Press, Boca Raton (2015), 437-483. DOI 10.1201/b18255-10 | MR 3409348 | Zbl 1327.05051
[4] L. Carleson, T. W. Gamelin: Complex Dynamics. Universitext: Tracts in Mathematics, Springer, New York (1993). DOI 10.1007/978-1-4612-4364-9 | MR 1230383 | Zbl 0782.30022
[5] K. Dilcher, K. B. Stolarsky: Zeros of the Wronskian of a polynomial. J. Math. Anal. Appl. 162 (1991), 430-451. DOI 10.1016/0022-247X(91)90160-2 | MR 1137630 | Zbl 0748.30007
[6] V. P. Kostov: Topics on Hyperbolic Polynomials in One Variable. Panoramas et Synthèses 33, Société Mathématique de France, Paris (2011). MR 2952044 | Zbl 1259.12001
[7] V. P. Kostov, B. Shapiro, M. Tyaglov: Maximal univalent disks of real rational functions and Hermite-Biehler polynomials. Proc. Am. Math. Soc. 139 (2011), 1625-1635. DOI 10.1090/S0002-9939-2010-10778-5 | MR 2763752 | Zbl 1223.26033
[8] Q. I. Rahman, G. Schmeisser: Analytic Theory of Polynomials. London Mathematical Society Monographs 26, Oxford University Press, Oxford (2002). MR 1954841 | Zbl 1072.30006
[9] K. Tran: Connections between discriminants and the root distribution of polynomials with rational generating function. J. Math. Anal. Appl. 410 (2014), 330-340. DOI 10.1016/j.jmaa.2013.08.025 | MR 3109843 | Zbl 1307.12002
[10] K. Tran: The root distribution of polynomials with a three-term recurrence. J. Math. Anal. Appl. 421 (2015), 878-892. DOI 10.1016/j.jmaa.2014.07.066 | MR 3250512 | Zbl 1296.30010

Affiliations:   Innocent Ndikubwayo, Department of Mathematics, Stockholm University, Kräftriket, SE-106 91 Stockholm, Sweden; Department of Mathematics, Makerere University, P.O Box 7062 Kampala, Uganda, e-mail: innocent@math.su.se, ndikubwayo@cns.mak.ac.ug


 
PDF available at: