Abstract
We study the root distribution of some univariate polynomials satisfying a recurrence of order two with linear polynomial coefficients. We show that the set of non-isolated limits of zeros of the polynomials is either an arc, or a circle, or an interval, or a “lollipop.” As an application, we discover a sufficient and necessary condition for the universal real-rootedness of the polynomials, subject to certain sign condition on the coefficients of the recurrence. Moreover, we obtain the sharp bound for all the zeros when they are real.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Andrews, G.E., Askey, R., Ranjan, R.: Special Functions. Cambridge University Press, Cambridge (1999)
Beraha, S., Kahane, J.: Is the four-color conjecture almost false? J. Combin. Theory Ser. B 27(1), 1–12 (1979)
Beraha, S., Kahane, J., Weiss, N.J.: Limits of zeroes of recursively defined polynomials. Proc. Natl. Acad. Sci. USA 72(11), 4209 (1975)
Beraha, S., Kahane, J., Weiss, N.J.: Limits of zeroes of recursively defined families of polynomials. Adv. Math. Suppl. Stud. 1, 213–232 (1978)
Birkhoff, G.D.: A determinant formula for the number of ways of coloring a map. Ann. Math. (2) 14, 42–46 (1912)
Bleher, P., Mallison, R. Jr.: Zeros of sections of exponential sums. Int. Math. Res. Not. 2006, 1–49 (2006). Article ID 38937
Boyer, R., Goh, W.M.Y.: On the zero attractor of the Euler polynomials. Adv. Appl. Math. 38(1), 97–132 (2007)
Boyer, R., Goh, W.M.Y.: Appell polynomials and their zero attractors. Contemp. Math. 517, 69–96 (2010)
Brezinski, C., Driver, K.A., Redivo-Zaglia, M.: Quasi-orthogonality with applications to some families of classical orthogonal polynomials. Appl. Numer. Math. 48, 157–168 (2004)
Brown, J.I., Tufts, J.: On the roots of domination polynomials. Graphs Comb. 30, 527–547 (2014)
Goh, W., He, M.X., Ricci, P.E.: On the universal zero attractor of the Tribonacci-related polynomials. Calcolo 46, 95–129 (2009)
Gross, J.L., Mansour, T., Tucker, T.W., Wang, D.G.L.: Root geometry of polynomial sequences I: type (0,1). J. Math. Anal. Appl. 433(2), 1261–1289 (2016)
Gross, J.L., Mansour, T., Tucker, T.W., Wang, D.G.L.: Root geometry of polynomial sequences II: type (1,0). J. Math. Anal. Appl. 441(2), 499–528 (2016)
Jin, D.D.D., Wang, D.G.L.: Common zeros of polynomials satisfying a recurrence of order two. arXiv:1712.04231
Kuijlaars, A.B.J., McLaughlin, K.T.-R.: Asymptotic zero behavior of Laguerre polynomials with negative parameter. Constr. Approx. 20(4), 497–523 (2004)
Kuijlaars, A.B.J., van Assche, W.: The asymptotic zero distribution of orthogonal polynomials with varying recurrence coefficients. J. Approx. Theory 99(1), 167–197 (1999)
Liu, L.L., Wang, Y.: A unified approach to polynomial sequences with only real zeros. Adv. Appl. Math. 38, 542–560 (2007)
Liu, L.L., Wang, Y.: Positivity problems and conjectures in algebraic combinatorics. In: Arnold, V., Atiyah, M., Lax, P., Mazur, B. (eds.) Mathematics: Frontiers and Perspectives, pp. 295–319. American Mathematical Society, Providence (2000)
Tran, K.: Connections between discriminants and the root distribution of polynomials with rational generating function. J. Math. Anal. Appl. 410, 330–340 (2014)
Wang, D.G.L., Zhang, J.J.R.: Piecewise interlacing property of polynomials satisfying some recurrence of order two. arXiv:1712.04225
Wang, D.G.L., Zhang, J.J.R.: Root geometry of polynomial sequences III: type (1,1) with positive coefficients. arXiv:1712.06105
Wang, Y., Yeh, Y.-N.: Polynomials with real zeros and Pólya frequency sequences. J. Comb. Theory Ser. A 109, 63–74 (2005)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Fuad Kittaneh.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
D.G.L. Wang is supported by the General Program of National Natural Science Foundation of China (Grant No. 11671037). This paper is completed when the first author is a visiting scholar at MIT.
Rights and permissions
About this article
Cite this article
Wang, D.G.L., Zhang, J.J.R. Geometry of Limits of Zeros of Polynomial Sequences of Type (1,1). Bull. Malays. Math. Sci. Soc. 44, 785–803 (2021). https://doi.org/10.1007/s40840-020-00975-y
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40840-020-00975-y