Hostname: page-component-7479d7b7d-t6hkb Total loading time: 0 Render date: 2024-07-12T21:05:51.976Z Has data issue: false hasContentIssue false
  • English
  • Français

Inequalities for Extreme Zeros of Some Classical Orthogonal and q-orthogonal Polynomials

Published online by Cambridge University Press:  28 January 2013

K. Driver  and
K. Jordaan
K. Driver*
Affiliation:
Department of Mathematics and Applied Mathematics, University of Cape Town 7701, RSA
K. Jordaan
Affiliation:
Department of Mathematics and Applied Mathematics, University of Pretoria, Pretoria, 0002, RSA
*
∗ Corresponding author. E-mail: kathy.driver@uct.ac.za
Get access

Abstract

Let {pn}∞n=0 be a sequence of orthogonal polynomials. We briefly review properties of pn that have been used to derive upper and lower bounds for the largest and smallest zero of pn. Bounds for the extreme zeros of Laguerre, Jacobi and Gegenbauer polynomials that have been obtained using different approaches are numerically compared and new bounds for extreme zeros of q-Laguerre and little q-Jacobi polynomials are proved.

Keywords

Bounds for extreme zeros of orthogonal and q-orthogonal polynomialscommon zeros of orthogonal polynomialsmonotonicityconvexityinterlacing of zerosseparation of zerosinequalities for zeros
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 8 , Issue 1: Harmonic analysis , 2013 , pp. 48 - 59
Copyright
© EDP Sciences, 2013

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Références

Ahmed, S., Laforgia, A., Muldoon, M. E.. On the spacing of the zeros of some classical orthogonal polynomials. J. London Math. Soc., second series 25 (1982), 246-252. CrossRefGoogle Scholar
Area, I., Dimitrov, D.K., Godoy, E., Ronveaux, A.. Zeros of Gegenbauer and Hermite polynomials and connection coefficients. Math. Comp., 73 (2004), 1937–1951. CrossRefGoogle Scholar
Area, I., Dimitrov, D.K., Godoy, E., Rafaeli, F.R.. Inequalities for zeros of Jacobi polynomials via Obrechkoff’s theorem. Math. Comp., 81 (2012), 991–1004. CrossRefGoogle Scholar
Beardon, A.F.. The theorems of Stieltjes and Favard. Lect. Notes Math., 11(1) (2011), 247–262. Google Scholar
Bottema, O.. Die Nulstellen gewisser durch Rekursionsformeln definierter Polynome. Proc. Amsterdam, 34(5) (1931), 681. Google Scholar
de Boor, C., Saff, EB. Finite sequences of orthogonal polynomials connected by a Jacobi matrix. Linear Algebra Appl., 75 (1986), 43–55. CrossRefGoogle Scholar
Deaño, A., Gil, A., Segura, J.. New inequalities from classical Sturm theorems. J. Approx. Theory, 131 (2004), 208–243. CrossRefGoogle Scholar
Deaño, A., Segura, J.. LG transformations and global inequalities for real zeros of Gauss hypergeometric functions. J. Approx. Theory, 48 (2007), 92–110. CrossRefGoogle Scholar
Dimitrov, D.K., Rafaeli, F.R.. Monotonicity of zeros of Laguerre polynomials. J. Comput. Appl. Math.. 223 (2009), 699–702. CrossRefGoogle Scholar
Dimitrov, D.K., Nikolov, G.P.. Sharp bounds for the extreme zeros of classical orthogonal polynomials. J. Approx. Theory, 162 (2010), 1793–1804. CrossRefGoogle Scholar
Dimitrov, D.K., Rodrigues, R.O.. On the behaviour of zeros of Jacobi and Gegenbauer polynomials. J. Approx. Theory., 116 (2002), 224–239. CrossRefGoogle Scholar
Driver, K., Jordaan, K.. Bounds for extreme zeros of some classical orthogonal polynomials. J. Approx. Theory., 164 (2012), 1200–1204. CrossRefGoogle Scholar
Elbert, Á., Laforgia, A.. Upper bounds for the zeros of ultraspherical polynomials. J. Approx. Theory., 61 (1990), 88–97. CrossRefGoogle Scholar
Elbert, Á., Laforgia, A., Rodonó, L.G.. On the zeros of Jacobi polynomials. Acta Math. Hungar., 64 (4) (1994), 351–359 CrossRefGoogle Scholar
Elbert, Á., Siafarikas, P.D.. Monotonicity properties of the zeros of ultraspherical polynomials. J. Approx. Theory., 97 (1999) 31-39. CrossRefGoogle Scholar
Erb, W., Tookós, F.. Monotonicity of extremal zeros of orthogonal polynomials and applications. Appl. Math. Comput., 217 (2011), 4771–4780. Google Scholar
Foster, W.H., Krasikov, I.. Inequalities for real-root polynomials and entire functions. Adv. Appl. Math., 29 (2002), 102–114. CrossRefGoogle Scholar
Gibson, P.C.. Common zeros of two polynomials in an orthogonal sequence. J. Approx. Theory, 105 (2000), 129–132. CrossRefGoogle Scholar
J. Gishe, F. Tookós. On the Sturm comparison and convexity theorem for difference and q-difference equations. Acta Scientiarum Mathematicarum. In press.
D.E. Gupta, M.E. Muldoon. Inequalities for the smallest zeros of Laguerre polynomials and their q -analogues. Journal of Inequalities in Pure and Applied Mathematics, 8 (2007), Issue 1, Article 24, 7 pp.
Hahn, W.. Bericht über die Nullstellen der Laguerrschen und der Hermiteschen Polynome. Jahresbericht der Deutschen Mathematiker-Vereinigung, 44 (1933), 215–236. Google Scholar
Hille, E.. Über die Nulstellen der Hermiteschen Polynome. Jahresbericht der Deutschen Mathematiker-Vereinigung, 44 (1933), 162–165. Google Scholar
E.K. Ifantis, P.D. Siafarikas. Differential inequalities on the greatest zero of Laguerre and ultraspherical polynomials in Actas del VI Simposium on Polinomios Orthogonales Y Aplicaciones, Gijon (1999) 187-197.
M.E.H. Ismail. Classical and Quantum Orthogonal Polynomials in One Variable, Encyclopedia of Mathematics and its Applications, Cambridge : Cambridge University Press, 98 (2005).
Ismail, M.E.H.. The variation of zeros of certain orthogonal polynomials. Advances in Appl. Math., 8 (1987), 111–118. CrossRefGoogle Scholar
M.E.H. Ismail. Monotonicity of zeros of orthogonal polynomials. Invited address in ”q-Series and Partitions”, edited by D. Stanton, IMA Volumes in Mathematics and its Applications, Vol. 18, Springer-Verlag, New York, 1989, 177–190.
Ismail, M.E.H.. An electrostatic model for zeros of general orthogonal polynomials. Pacific J. Math. 193 (2000), 355–369. CrossRefGoogle Scholar
Ismail, M.E.H.. More on elctrostatic models for zeros of orthogonal polynomials. J. Nonlinear Functional Analysis and Optimization, 21 (200, 43–55.
Ismail, M.E.H., Zhang, R, On the Hellmann-Feynman theorem and the variation of zeros of certain special functions. Adv. Appl. Math., 9 (1988), 439–446. CrossRefGoogle Scholar
Ismail, M.E.H., Muldoon, M.E., A discrete approach to monotonicity of zeros of orthogonal polynomials. Transactions Amer. Math. Soc., 323 (1991), 65–78. CrossRefGoogle Scholar
Ismail, M.E.H., Li, X.. Bounds on the extreme zeros of orthogonal polynomials. Proc. Amer. Math. Soc., 115 (1992) 131–140. CrossRefGoogle Scholar
Jordaan, K., Tookós, F.. Convexity of the zeros of some orthogonal polynomials and related functions. J. Comp. Anal. Appl., 233 (2009), 762–767. CrossRefGoogle Scholar
R. Koekoek, P.A. Lesky, R.F. Swarttouw. Hypergeometric orthogonal polynomials and their q-analogue. Springer Monographs in Mathematics, Springer Verlag, Berlin (2010).
Krasikov, I.. Bounds for zeros of the Laguerre polynomials. J. Approx. Theory., 121 (2003), 287–291. CrossRefGoogle Scholar
Krasikov, I.. On zeros of polynomials and allied functions satisfying second order differential equations. East J. Approx., 9 (2003), 41–65. Google Scholar
Markov, A.. Sur les racines de certaines équations (seconde note). Math. Ann., 27 (1886) 177–182. CrossRefGoogle Scholar
Moak, D.S.. The q-Analogue of the Laguerre Polynomials. J. Math. Anal. Appl., 81 (1981), 20–47. CrossRefGoogle Scholar
Muldoon, M.E.. Properties of zeros of orthogonal polynomials and related functions. J. Comput. Appl. Math., 48 (1993) 167–186. CrossRefGoogle Scholar
M.E. Muldoon. Convexity properties of special functions and their zeros. Milovanovic, G. V. (ed.), Recent progress in inequalities. Dedicated to Prof. Dragoslav S. Mitrinovic. Dordrecht : Kluwer Academic Publishers. Math. Appl., Dordr. 430 1998, 309–323 Publishers, Inc., Boston, 1991.
Neumann, E.R.. Zur Theorie der Laguerreschen Polynome. Jahresber. d. S.M.V., 30 (1921) 15. Google Scholar
G.P. Nikolov, R. Uluchev. Inequalities for real-root polynomials. Proof of a conjecture of Foster and Krasikov. in : D.K. Dimitrov, G.P. Nikolov, R. Uluchev (eds.), Approximation Theory : A volume dedicated to B. Bojanov. Marin Drinov Academic Publishing House, Sofia, 2004, pp. 201–216.
Siafarikas, P.D.. Inequalities for the zeros of the associated ultraspherical polynomials. Math. Inequal. Applic.(2), 2 (1999) 233–241. Google Scholar
Siegel, C.L.. Über einige Anwendungen diophantischer Approximationen. Abh. Preuss. Akad. Wiss., 1, (1929), 1–70. Google Scholar
Stieltjes, T.J.. Sur les racines de l équation Xn = 0. Acta Math., 9 (1886) 385–400. CrossRefGoogle Scholar
Sturm, C.. Memoire sur les équations différentielles du second ordre. J. Math. Pures Appl., 1 (1836) 106–186. Google Scholar
G. Szegő. Orthogonal Polynomials. American Mathematical Society Colloquium Publications, Volume XXIII, Providence, RI, fourth edition, 1975.
van Doorn, E.. Representations and bounds for zeros of orthogonal polynomials and eigenvalues of sign-symmetric tri-diagonal matrices. J. Approx. Theory., 51 (1987) 254–266. CrossRefGoogle Scholar
Vinet, L., Zhedanov, A.. A characterization of classical and semiclassical orthogonal polynomials from their dual polynomials. J. Comput. Appl. Math. 172 (2004), 41–48. CrossRefGoogle Scholar
Wall, H.S., Wetzel, M.. Quadratic forms and convergence regions for continued fractions. Duke Math. J., 11 (1944), 983–1000. CrossRefGoogle Scholar
G.N. Watson. A Treatise on the Theory of Bessel Functions 2nd ed., (Cambridge University Press, Cambridge 1966).