Involve: A Journal of Mathematics

  • Involve
  • Volume 11, Number 3 (2018), 501-518.

Zeros of polynomials with four-term recurrence

Khang Tran and Andres Zumba

Full-text: Access denied (no subscription detected)

However, an active subscription may be available with MSP at

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


Given real numbers b,c, we form the sequence of polynomials {Hm(z)}m=0 satisfying the four-term recurrence

H m ( z ) + c H m 1 ( z ) + b H m 2 ( z ) + z H m 3 ( z ) = 0 , m 1 ,

with the initial conditions H0(z)=1 and H1(z)=H2(z)=0. We find necessary and sufficient conditions on b and c under which the zeros of Hm(z) are real for all m, and provide an explicit real interval on which m=0Z(Hm) is dense, where Z(Hm) is the set of zeros of Hm(z).

Article information

Involve, Volume 11, Number 3 (2018), 501-518.

Received: 14 March 2017
Revised: 7 June 2017
Accepted: 19 June 2017
First available in Project Euclid: 20 December 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 30C15: Zeros of polynomials, rational functions, and other analytic functions (e.g. zeros of functions with bounded Dirichlet integral) {For algebraic theory, see 12D10; for real methods, see 26C10} 26C10: Polynomials: location of zeros [See also 12D10, 30C15, 65H05] 11C08: Polynomials [See also 13F20]

generating functions hyperbolic polynomials recursive sequence


Tran, Khang; Zumba, Andres. Zeros of polynomials with four-term recurrence. Involve 11 (2018), no. 3, 501--518. doi:10.2140/involve.2018.11.501.

Export citation


  • R. Bates and R. Yoshida, “Quadratic hyperbolicity preservers and multiplier sequences”, Rocky Mountain J. Math. 46:1 (2016), 51–72.
  • S. Beraha, J. Kahane, and N. J. Weiss, “Limits of zeroes of recursively defined polynomials”, Proc. Nat. Acad. Sci. U.S.A. 72:11 (1975), 4209.
  • S. Beraha, J. Kahane, and N. J. Weiss, “Limits of zeros of recursively defined families of polynomials”, pp. 213–232 in Studies in foundations and combinatorics, edited by G.-C. Rota, Adv. in Math. Suppl. Stud. 1, Academic Press, New York, 1978.
  • J. Borcea and P. Brändén, “Pólya–Schur master theorems for circular domains and their boundaries”, Ann. of Math. $(2)$ 170:1 (2009), 465–492.
  • J. Borcea, R. Bøgvad, and B. Shapiro, “On rational approximation of algebraic functions”, Adv. Math. 204:2 (2006), 448–480.
  • R. Boyer and W. M. Y. Goh, “On the zero attractor of the Euler polynomials”, Adv. in Appl. Math. 38:1 (2007), 97–132.
  • R. Boyer and W. M. Y. Goh, “Polynomials associated with partitions: asymptotics and zeros”, pp. 33–45 in Special functions and orthogonal polynomials, edited by D. Dominici and R. S. Maier, Contemp. Math. 471, Amer. Math. Soc., Providence, RI, 2008.
  • A. Bunton, N. Jacobs, S. Jenkins, C. McKenry, Jr., A. Piotrowski, and L. Scott, “Nonreal zero decreasing operators related to orthogonal polynomials”, Involve 8:1 (2015), 129–146.
  • T. Craven and G. Csordas, “Composition theorems, multiplier sequences and complex zero decreasing sequences”, pp. 131–166 in Value distribution theory and related topics, edited by G. Barsegian et al., Adv. Complex Anal. Appl. 3, Kluwer, Boston, 2004.
  • O. Eğecioğlu, T. Redmond, and C. Ryavec, “From a polynomial Riemann hypothesis to alternating sign matrices”, Electron. J. Combin. 8:1 (2001), art. id. 36.
  • T. Forgács and K. Tran, “Polynomials with rational generating functions and real zeros”, J. Math. Anal. Appl. 443:2 (2016), 631–651.
  • A. D. Sokal, “Chromatic roots are dense in the whole complex plane”, Combin. Probab. Comput. 13:2 (2004), 221–261.
  • K. Tran, “Connections between discriminants and the root distribution of polynomials with rational generating function”, J. Math. Anal. Appl. 410:1 (2014), 330–340.
  • K. Tran, “The root distribution of polynomials with a three-term recurrence”, J. Math. Anal. Appl. 421:1 (2015), 878–892.