Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



Zeros of classical orthogonal polynomials of a discrete variable

Authors: Iván Area, Dimitar K. Dimitrov, Eduardo Godoy and Vanessa G. Paschoa
Journal: Math. Comp. 82 (2013), 1069-1095
MSC (2010): Primary 33C45; Secondary 26C10
Published electronically: November 16, 2012
MathSciNet review: 3008850
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we obtain sharp bounds for the zeros of classical orthogonal polynomials of a discrete variable, considered as functions of a parameter, by using a theorem of A. Markov and the so-called Hellmann-Feynman theorem. Comparisons with previous results for zeros of Hahn, Meixner, Kravchuk and Charlier polynomials are also presented.

References [Enhancements On Off] (What's this?)

  • 1. G. Andrews, R. Askey and R. Roy, Special Functions, Cambridge Univ. Press, Cambridge, 1999. MR 1688958 (2000g:33001)
  • 2. I. Area, D.K. Dimitrov, E. Godoy and A. Ronveaux, Zeros of Gegenbauer and Hermite polynomials and connection coefficients, Math. Comp. 73 (2004), 1937-1951. MR 2059744 (2005g:33011)
  • 3. I. Area, D.K. Dimitrov, E. Godoy and F.R. Rafaeli, Inequalities for zeros of Jacobi polynomials via Obrechkoff's theorem, Math. Comp. 81 (2012), 991-1004. MR 2869046
  • 4. A. Björck, Numerical Methods for Least Squares Problems, SIAM, Philadelphia, PA, 1996. MR 1386889 (97g:65004)
  • 5. A. Borodin, P.L. Ferrari, M. Prähofer, and T. Sasamoto, Fluctuation properties of the TASEP with periodic initial configuration, J. Stat. Phys. 129 (5-6) (2007), 1055-1080. MR 2363389 (2009g:82048)
  • 6. P.L. Chebyshev, Interpolation of equidistant nodes, and, on continued fractions, in: Complete Works, USSR Academy of Sciences Publishing House, Moscow-Leningrad, 1948; French transl., Oeuvres, Chelsea, New York, 1962.
  • 7. T.S. Chihara, An Introduction to Orthogonal Polynomials, Gordon and Breach, New York, 1978. MR 0481884 (58:1979)
  • 8. L. Chihara and D. Stanton, Zeros of generalized Krawtchouk polynomials, J. Approx. Theory, 60(1) (1990), 43-57. MR 1028893 (91a:33011)
  • 9. G. Dahlquist and A. Björck, Numerical Methods, Prentice Hall, Englewood Cliffs, 1974. MR 0368379 (51:4620)
  • 10. D.K. Dimitrov, On a conjecture concerning monotonicity of zeros of ultraspherical polynomials, J. Approx. Theory 85 (1996), 88-97. MR 1382052 (97e:33011)
  • 11. D.K. Dimitrov and G.P. Nikolov, Sharp bounds for the extreme zeros of classical orthogonal polynomials, J. Approx. Theory 162 (2010), 1793-1804. MR 2728047 (2011i:42049)
  • 12. D.K. Dimitrov and F.R. Rafaeli, Monotonicity of zeros of Jacobi polynomials, J. Approx. Theory 149 (2007), 15-29. MR 2371611 (2010c:33021)
  • 13. D.K. Dimitrov and F.R. Rafaeli, Monotonicity of zeros of Laguerre polynomials. J. Comput. Appl. Math. 233 (2009), 699-702. MR 2583005 (2011b:33015)
  • 14. Á. Elbert, Some recent results on the zeros of Bessel functions and orthogonal polynomials, J. Comput. Appl. Math. 133 (2001), 65-83. MR 1858270 (2002j:33003)
  • 15. A. Gelb, R.B. Platte, and W.S. Rosenthal, The discrete orthogonal polynomial least squares method for approximation and solving partial differential equations, Commun. Comput. Phys. 3 (2008), 734-758. MR 2401084 (2009b:65262)
  • 16. R.A. Horn and C.R. Johnson, Matrix Analysis, Cambridge Univ. Press, Cambridge, 1985. MR 832183 (87e:15001)
  • 17. M.E. Ismail, Classical and Quantum Orthogonal Polynomials in One Variable, Cambridge University Press, 2005. MR 2191786 (2007f:33001)
  • 18. M.E. Ismail and M.E. Muldoon, A discrete approach to monotonicity of zeros of orthogonal polynomials, Trans. Amer. Math. Soc. 323 (1991), 65-78. MR 1014251 (91c:33017)
  • 19. M.E. Ismail and X. Li, Bound on the extreme zeros of orthogonal polynomials, Proc. Amer. Math. Soc. 115 (1992), 131-140. MR 1079891 (92h:33019)
  • 20. K. Johansson, Discrete orthogonal polynomial ensembles and the Plancherel measure, Ann. of Math. (2) 153 (2001), 259-296. MR 1826414 (2002g:05188)
  • 21. S. Karlin and J. McGregor, Many server queueing processes with Poisson input and exponential service times, Pacific J. Math. 8 (1958), 87-118. MR 0097132 (20:3611)
  • 22. K. Jordaan and F. Toókos, Interlacing theorems for the zeros of some orthogonal polynomials from different sequences, Appl. Num. Math. 59 (2009) 2015-2022. MR 2536092 (2010i:33010)
  • 23. R. Koekoek and R.F. Swarttouw, The Askey-scheme of hypergeometric orthogonal polynomials and its $ q$-analogue, Delft University of Technology, Faculty of Technical Mathematics and Informatics, Report no. 98-17, 1998. On-line version.
  • 24. I. Krasikov, Bounds for zeros of the Charlier polynomials, Methods Appl. Anal. 9 (4) (2002) 599-610. MR 2006607 (2004i:33017)
  • 25. I. Krasikov, On extreme zeros of classical orthogonal polynomials, J. Comput. Appl. Math. 193 (2006), 168-182. MR 2228713 (2006m:42046)
  • 26. I. Krasikov and A. Zarkh, On zeros of discrete orthogonal polynomials, J. Approx. Theory, 156 (2009), 121-141. MR 2494548 (2010c:33028)
  • 27. K. Kristinsson and G.A. Dumonts, Cross-directional control on paper machines using Gram polynomials, Automatica 32 (4) (1996), 533-548. MR 1386700
  • 28. A.B.J. Kuijlaars and E.A. Rakhmanov, Zero distributions for discrete orthogonal polynomials, J. Comput. Applied Math. 99 (1998), 255-274. MR 1662699 (2000c:42025a)
  • 29. A. Markov, Sur les racines de certaines équations (second note), Math. Ann. 27 (1886), 177-182. MR 1510373
  • 30. R.J. Levit, The zeros of the Hahn polynomials, SIAM Rev., 9 (1967), 191-203. MR 0213627 (35:4485)
  • 31. A. Ronveaux, A. Zarzo, I. Area, and E. Godoy, Classical orthogonal polynomials: dependence of parameters, J. Comput. Appl. Math. 121 (2000), 95-112. MR 1780044 (2002a:33016)
  • 32. G. Szegő, Orthogonal Polynomials, 4th ed., Amer. Math. Soc. Coll. Publ., Vol. 23, Providence, RI, 1975. MR 0372517 (51:8724)
  • 33. E.A. van Doorn and A.I. Zeifman, On the speed of convergence to stationarity of the Erlang loss system, Queueing Systems 63 (2009), 241-252. MR 2576013 (2011a:60324)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2010): 33C45, 26C10

Retrieve articles in all journals with MSC (2010): 33C45, 26C10

Additional Information

Iván Area
Affiliation: Departamento de Matemática Aplicada II, E.E. Telecomunicación, Universidade de Vigo, Campus Lagoas-Marcosende, 36310 Vigo, Spain

Dimitar K. Dimitrov
Affiliation: Departamento de Ciências de Computação e Estatística, IBILCE, Universidade Estadual Paulista, 15054-000 São José do Rio Preto, SP, Brazil

Eduardo Godoy
Affiliation: Departamento de Matemática Aplicada II, E.E. Industrial, Universidade de Vigo, Campus Lagoas-Marcosende, 36310 Vigo, Spain

Vanessa G. Paschoa
Affiliation: Departamento de Matemática Aplicada, IMECC, Universidade Estadual de Campinas (UNICAMP), 13083-859 Campinas, SP, Brazil
Email: van{\textunderscore}

Keywords: Orthogonal polynomials of a discrete variable, Zeros, Charlier polynomials, Kravchuk polynomials, Meixner polynomials, Hahn polynomials, Gram polynomials
Received by editor(s): September 16, 2011
Published electronically: November 16, 2012
Additional Notes: This research was supported by the joint project CAPES(Brazil)/DGU(Spain), Grants 160/08 and PHB2007–0078, by the Brazilian foundations CNPq under Grant 305622/2009–9 and FAPESP under Grant 2009/13832–9 and by the Ministerio de Ciencia e Innovación of Spain under grant MTM2009–14668–C02–01, co-financed by the European Community fund FEDER
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society