Abstract
In this lecture we discuss properties of zeros of orthogonal polynomials. We review properties that have been used to derive bounds for the zeros of orthogonal polynomials. Topics to be covered include Markov’s theorem on monotonicity of zeros and its generalisations, the proof of a conjecture by Askey and its extensions, interlacing properties of zeros, Sturm’s comparison theorem and convexity of zeros.
Keywords
- Orthogonal polynomials
- Zeros
- Jacobi polynomials
- Monotonicity of zeros
- Interlacing of zeros
- Stieltjes interlacing
- Bounds for zeros
Mathematics Subject Classification (2000)
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsReferences
I. Area, D.K. Dimitrov, E. Godoy, F.R. Rafaeli, Inequalities for zeros of Jacobi polynomials via Obrechkoff’s theorem. Math. Comput. 81, 991–1912 (2012)
R. Askey, Graphs as an aid to understanding special functions. Asymptotic Comput. Anal. Lect. Notes Pure Appl. 124, 3–33 (1990)
A. Deaño, A. Gil, J. Segura, New inequalities from classical Sturm theorems. J. Approx. Theory 131, 208–243 (2004)
D.K. Dimitrov, G.P. Nikolov, Sharp bounds for the extreme zeros of classical orthogonal polynomials. J. Approx. Theory 162, 1793–1804 (2010)
D.K. Dimitrov, F.R. Rafaeli, Monotonicity of zeros of Laguerre polynomials. J. Comput. Appl. Math., 223, 699–702 (2009)
D.K. Dimitrov, R.O. Rodrigues, On the behaviour of zeros of Jacobi and Gegenbauer polynomials. J. Approx. Theory 116, 224–239 (2002)
D.K. Dimitrov, A. Sri Ranga. Monotonicity of the zeros of orthogonal Laurent polynomials. Methods Appl. Anal. 9, 9–12 (2002)
D.K. Dimitrov, M.V. Mello, F.R. Rafaeli, Monotonicity of zeros of Jacobi-Sobolev type orthogonal polynomials. Appl. Numer. Math. 60, 263–276 (2010)
D.K. Dimitrov, M.E.H. Ismail, F.R. Rafaeli, Interlacing of zeros of orthogonal polynomials under modification of the measure. J. Approx. Theory 175, 64–76 (2013)
K. Driver, K. Jordaan, Bounds for extreme zeros of some classical orthogonal polynomials. J. Approx. Theory 164, 1200–1204 (2012)
K. Driver, K. Jordaan, N. Mbuyi, Interlacing of the zeros of Jacobi polynomials with different parameters. Numer. Algorithms 49, 143–152 (2008)
K. Driver, A. Jooste, K. Jordaan, Stieltjes interlacing of zeros of Jacobi polynomials from different sequences. Electron. Trans. Numer. Anal. 38, 317–326 (2011)
Á. Elbert, A. Laforgia, Upper bounds for the zeros of ultraspherical polynomials. J. Approx. Theory. 61, 88–97 (1990)
Á. Elbert, A. Laforgia, L.G. Rodonó, On the zeros of Jacobi polynomials. Acta Math. Hungar. 64(4), 351–359 (1994)
W.H. Foster, I. Krasikov, Inequalities for real-root polynomials and entire functions. Adv. Appl. Math. 29, 102–114 (2002)
G. Freud, Orthogonal Polynomials (Pergamon, Oxford, 1971)
W. Hahn, Bericht über die Nullstellen der Laguerrschen und der Hermiteschen Polynome. Jahresber. Deutsch. Math.-Verein. 44, 215–236 (1933)
D. Hilbert, Über die Diskriminante der im Endlichen abbrechenden hypergeometrischen Reihe. J. Reine. Angew. Math. 103, 337–345 (1888)
E. Hille, Über die Nulstellen der Hermiteschen Polynome. Jahresber. Deutsch. Math.-Verein. 44, 162–165 (1933)
M.E.H. Ismail, The variation of zeros of certain orthogonal polynomials. Adv. Appl. Math. 8, 111–118 (1987)
M.E.H. Ismail, An electrostatic model for zeros of general orthogonal polynomials. Pac. J. Math. 193, 355–369 (2000)
M.E.H. Ismail, More on electrostatic models for zeros of orthogonal polynomials. J. Nonlinear Funct. Anal. Optim. 21, 43–55 (2000)
M.E.H. Ismail, Classical and Quantum Orthogonal Polynomials in One Variable. Encyclopedia of Mathematics and its Applications, vol. 98 (Cambridge University Press, Cambridge, 2005)
M.E.H. Ismail, M.E. Muldoon, A discrete approach to monotonicity of zeros of orthogonal polynomials. Trans. Am. Math. Soc. 323, 65–78 (1991)
M.E.H. Ismail, X. Li, Bounds on the extreme zeros of orthogonal polynomials. Proc. Am. Math. Soc. 115, 131–140 (1992)
M.E.H. Ismail, R Zhang, On the Hellmann-Feynman theorem and the variation of zeros of certain special functions. Adv. Appl. Math. 9, 439–446 (1988)
K. Jordaan, F. Tookós, Convexity of the zeros of some orthogonal polynomials and related functions. J. Comp. Anal. Appl. 233, 762–767 (2009)
I. Krasikov, Bounds for zeros of the Laguerre polynomials. J. Approx. Theory 121, 287–291 (2003)
I. Krasikov, On zeros of polynomials and allied functions satisfying second order differential equations. East J. Approx. 9, 41–65 (2003)
R.J. Levit, The zeros of the Hahn polynomials. SIAM Rev. 9(2), 191–203 (1967)
D.S. Lubinsky, Quadrature identities for interlacing and orthogonal polynomials. Proc. Am. Math. Soc. 144, 4819–4829 (2016)
F. Marcellán, F.R. Rafaeli, Monotonicity and Asymptotics of zeros of Laguerre-Sobolev-type orthogonal polynomials of higher order derivatives. Proc. Am. Math. Soc. 139(11), 3929–3936 (2011)
A. Markov, Sur les racines de certaines equations (Second note). Math. Ann. 27, 177–182 (1886)
M.E. Muldoon, Properties of zeros of orthogonal polynomials and related functions. J. Comput. Appl. Math. 48, 167–186 (1993)
G.P. Nikolov, R. Uluchev, in Inequalities for Real-Root Polynomials. Proof of a Conjecture of Foster and Krasikov, in ed. by D.K. Dimitrov, G.P. Nikolov, R. Uluchev. Approximation Theory: A volume dedicated to B. Bojanov (Marin Drinov Academic Publishing House, Sofia, 2004), pp. 201–216
P. Paule, Contiguous relations and creative telescoping, Technical report, RISC, Austria, 2001
R. Vidũnas, Contiguous relations of hypergeometric series. J. Comput. Appl. Math. 153(1–2), 507–519 (2003)
J. Segura, Interlacing of the zeros of contiguous hypergeometric functions. Numer. Algorithms 49, 387–407 (2008)
B. Simon, in Orthogonal Polynomials on the Unit Circle, Part 1: Classical Theory. American Mathematical Society Colloquium Publications, vol. 54 (American Mathematical Society, Providence, 2005)
T.J. Stieltjes, Sur quelques théorèmes d’algèbre. C. R. Acad. Sci. 100, 439–440 (1885). Ouvres Complètes 1, 440–441
T.J. Stieltjes, Sur les polynômes de Jacobi. C. R. Acad. Sci. 100, 620–622 (1885). Ouvres Complètes 1, 442–444
C. Sturm, Memoire sur les équations différentielles du second ordre. J. Math. Pures Appl. 1, 106–186 (1836)
G. Szegő, in Orthogonal Polynomials. AMS Colloquium Publications, vol. 23 (American Mathematical Society, Providence, 1975)
N. Takayame, Gröbner basis and the problem of contiguous relations. Jpn J. Appl. Math. 6, 147–160 (1989)
R. Vidũnas, T. Koornwinder, Webpage of the NWO project. Algorithmic methods for special functions by computer algebra (2000). http://www.science.uva.nl/~thk/specfun/compalg.html
H.S. Wall, M. Wetzel, Quadratic forms and convergence regions for continued fractions. Duke Math. J. 11, 983–1000 (1944)
B. Wendroff, On orthogonal polynomials. Proc. Am. Math. Soc. 12, 554–555 (1961)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2020 Springer Nature Switzerland AG
About this paper
Cite this paper
Jordaan, K. (2020). Zeros of Orthogonal Polynomials. In: Foupouagnigni, M., Koepf, W. (eds) Orthogonal Polynomials. AIMSVSW 2018. Tutorials, Schools, and Workshops in the Mathematical Sciences . Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-36744-2_17
Download citation
DOI: https://doi.org/10.1007/978-3-030-36744-2_17
Published:
Publisher Name: Birkhäuser, Cham
Print ISBN: 978-3-030-36743-5
Online ISBN: 978-3-030-36744-2
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)