Zeros of orthogonal polynomials. (English) Zbl 1443.33017
Foupouagnigni, Mama (ed.) et al., Orthogonal polynomials. Proceedings of the 2nd AIMS-Volkswagen Stiftung workshop on introduction to orthogonal polynomials and applications, Douala, Cameroon, October 5–12, 2018. Cham: Birkhäuser. Tutor. Sch. Workshops Math. Sci., 441-456 (2020).
Summary: 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.
For the entire collection see [Zbl 1442.33005].
For the entire collection see [Zbl 1442.33005].
MSC:
| 33C45 | Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) |
| 42C05 | Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis |
Keywords:
orthogonal polynomials; zeros; Jacobi polynomials; monotonicity of zeros; interlacing of zeros; Stieltjes interlacing; bounds for zerosReferences:
| [1] | 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) · Zbl 1247.33013 · doi:10.1090/S0025-5718-2011-02553-6 |
| [2] | R. Askey, Graphs as an aid to understanding special functions. Asymptotic Comput. Anal. Lect. Notes Pure Appl. 124, 3-33 (1990) · Zbl 0694.33002 |
| [3] | A. Deaño, A. Gil, J. Segura, New inequalities from classical Sturm theorems. J. Approx. Theory 131, 208-243 (2004) · Zbl 1065.33004 · doi:10.1016/j.jat.2004.09.006 |
| [4] | D.K. Dimitrov, G.P. Nikolov, Sharp bounds for the extreme zeros of classical orthogonal polynomials. J. Approx. Theory 162, 1793-1804 (2010) · Zbl 1206.33008 · doi:10.1016/j.jat.2009.11.006 |
| [5] | D.K. Dimitrov, F.R. Rafaeli, Monotonicity of zeros of Laguerre polynomials. J. Comput. Appl. Math., 223, 699-702 (2009) · Zbl 1181.33010 · doi:10.1016/j.cam.2009.02.038 |
| [6] | D.K. Dimitrov, R.O. Rodrigues, On the behaviour of zeros of Jacobi and Gegenbauer polynomials. J. Approx. Theory 116, 224-239 (2002) · Zbl 1006.33003 · doi:10.1006/jath.2002.3671 |
| [7] | D.K. Dimitrov, A. Sri Ranga. Monotonicity of the zeros of orthogonal Laurent polynomials. Methods Appl. Anal. 9, 9-12 (2002) · Zbl 1033.42022 |
| [8] | 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) · Zbl 1191.33003 · doi:10.1016/j.apnum.2009.12.004 |
| [9] | 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) · Zbl 1337.30003 · doi:10.1016/j.jat.2013.07.007 |
| [10] | K. Driver, K. Jordaan, Bounds for extreme zeros of some classical orthogonal polynomials. J. Approx. Theory 164, 1200-1204 (2012) · Zbl 1262.33008 · doi:10.1016/j.jat.2012.05.014 |
| [11] | K. Driver, K. Jordaan, N. Mbuyi, Interlacing of the zeros of Jacobi polynomials with different parameters. Numer. Algorithms 49, 143-152 (2008) · Zbl 1169.30002 · doi:10.1007/s11075-008-9162-2 |
| [12] | K. Driver, A. Jooste, K. Jordaan, Stieltjes interlacing of zeros of Jacobi polynomials from different sequences. Electron. Trans. Numer. Anal. 38, 317-326 (2011) · Zbl 1293.33011 |
| [13] | Á. Elbert, A. Laforgia, Upper bounds for the zeros of ultraspherical polynomials. J. Approx. Theory. 61, 88-97 (1990) · Zbl 0697.33005 · doi:10.1016/0021-9045(90)90025-L |
| [14] | Á. Elbert, A. Laforgia, L.G. Rodonó, On the zeros of Jacobi polynomials. Acta Math. Hungar. 64(4), 351-359 (1994) · Zbl 0814.33006 · doi:10.1007/BF01874695 |
| [15] | W.H. Foster, I. Krasikov, Inequalities for real-root polynomials and entire functions. Adv. Appl. Math. 29, 102-114 (2002) · Zbl 1018.33007 · doi:10.1016/S0196-8858(02)00005-2 |
| [16] | G. Freud, Orthogonal Polynomials (Pergamon, Oxford, 1971) · Zbl 0226.33014 |
| [17] | W. Hahn, Bericht über die Nullstellen der Laguerrschen und der Hermiteschen Polynome. Jahresber. Deutsch. Math.-Verein. 44, 215-236 (1933) · Zbl 0010.30101 |
| [18] | D. Hilbert, Über die Diskriminante der im Endlichen abbrechenden hypergeometrischen Reihe. J. Reine. Angew. Math. 103, 337-345 (1888) · JFM 20.0154.03 |
| [19] | E. Hille, Über die Nulstellen der Hermiteschen Polynome. Jahresber. Deutsch. Math.-Verein. 44, 162-165 (1933) · JFM 60.0295.02 |
| [20] | M.E.H. Ismail, The variation of zeros of certain orthogonal polynomials. Adv. Appl. Math. 8, 111-118 (1987) · Zbl 0628.33001 · doi:10.1016/0196-8858(87)90009-1 |
| [21] | M.E.H. Ismail, An electrostatic model for zeros of general orthogonal polynomials. Pac. J. Math. 193, 355-369 (2000) · Zbl 1011.33011 · doi:10.2140/pjm.2000.193.355 |
| [22] | M.E.H. Ismail, More on electrostatic models for zeros of orthogonal polynomials. J. Nonlinear Funct. Anal. Optim. 21, 43-55 (2000) |
| [23] | 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) · Zbl 1082.42016 |
| [24] | 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) · Zbl 0718.33004 · doi:10.1090/S0002-9947-1991-1014251-8 |
| [25] | M.E.H. Ismail, X. Li, Bounds on the extreme zeros of orthogonal polynomials. Proc. Am. Math. Soc. 115, 131-140 (1992) · Zbl 0744.33005 · doi:10.1090/S0002-9939-1992-1079891-5 |
| [26] | 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) · Zbl 0684.33004 |
| [27] | K. Jordaan, F. Tookós, Convexity of the zeros of some orthogonal polynomials and related functions. J. Comp. Anal. Appl. 233, 762-767 (2009) · Zbl 1184.33005 · doi:10.1016/j.cam.2009.02.045 |
| [28] | I. Krasikov, Bounds for zeros of the Laguerre polynomials. J. Approx. Theory 121, 287-291 (2003) · Zbl 1018.33008 · doi:10.1016/S0021-9045(03)00029-7 |
| [29] | I. Krasikov, On zeros of polynomials and allied functions satisfying second order differential equations. East J. Approx. 9, 41-65 (2003) · Zbl 1110.33002 |
| [30] | R.J. Levit, The zeros of the Hahn polynomials. SIAM Rev. 9(2), 191-203 (1967) · Zbl 0154.06601 · doi:10.1137/1009032 |
| [31] | D.S. Lubinsky, Quadrature identities for interlacing and orthogonal polynomials. Proc. Am. Math. Soc. 144, 4819-4829 (2016) · Zbl 1351.41024 · doi:10.1090/proc/13099 |
| [32] | 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) · Zbl 1242.42021 · doi:10.1090/S0002-9939-2011-10806-2 |
| [33] | A. Markov, Sur les racines de certaines equations (Second note). Math. Ann. 27, 177-182 (1886) · doi:10.1007/BF01452056 |
| [34] | M.E. Muldoon, Properties of zeros of orthogonal polynomials and related functions. J. Comput. Appl. Math. 48, 167-186 (1993) · Zbl 0796.33005 · doi:10.1016/0377-0427(93)90321-2 |
| [35] | 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 · Zbl 0649.41002 |
| [36] | P. Paule, Contiguous relations and creative telescoping, Technical report, RISC, Austria, 2001 |
| [37] | R. Vidũnas, Contiguous relations of hypergeometric series. J. Comput. Appl. Math. 153(1-2), 507-519 (2003) · Zbl 1018.33001 · doi:10.1016/S0377-0427(02)00643-X |
| [38] | J. Segura, Interlacing of the zeros of contiguous hypergeometric functions. Numer. Algorithms 49, 387-407 (2008) · Zbl 1171.33008 · doi:10.1007/s11075-008-9211-x |
| [39] | 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) · Zbl 1082.42020 |
| [40] | T.J. Stieltjes, Sur quelques théorèmes d’algèbre. C. R. Acad. Sci. 100, 439-440 (1885). Ouvres Complètes 1, 440-441 · JFM 17.0062.01 |
| [41] | T.J. Stieltjes, Sur les polynômes de Jacobi. C. R. Acad. Sci. 100, 620-622 (1885). Ouvres Complètes 1, 442-444 · JFM 17.0062.02 |
| [42] | C. Sturm, Memoire sur les équations différentielles du second ordre. J. Math. Pures Appl. 1, 106-186 (1836) |
| [43] | G. Szegő, in Orthogonal Polynomials. AMS Colloquium Publications, vol. 23 (American Mathematical Society, Providence, 1975) · Zbl 0305.42011 |
| [44] | N. Takayame, Gröbner basis and the problem of contiguous relations. Jpn J. Appl. Math. 6, 147-160 (1989) · Zbl 0691.68032 · doi:10.1007/BF03167920 |
| [45] | 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 |
| [46] | H.S. Wall, M. Wetzel, Quadratic forms and convergence regions for continued fractions. Duke Math. J. 11, 983-1000 (1944) · Zbl 0060.16504 · doi:10.1215/S0012-7094-44-01110-5 |
| [47] | B. · doi:10.1090/S0002-9939-1961-0131120-2 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
