Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

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

 
 

 

Zeros of Gegenbauer and Hermite polynomials and connection coefficients


Authors: Iván Area, Dimitar K. Dimitrov, Eduardo Godoy and André Ronveaux
Journal: Math. Comp. 73 (2004), 1937-1951
MSC (2000): Primary 33C45, 26C10
DOI: https://doi.org/10.1090/S0025-5718-04-01642-4
Published electronically: March 23, 2004
MathSciNet review: 2059744
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper, sharp upper limit for the zeros of the ultraspherical polynomials are obtained via a result of Obrechkoff and certain explicit connection coefficients for these polynomials. As a consequence, sharp bounds for the zeros of the Hermite polynomials are obtained.


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

  • 1. S. Ahmed, M. E. Muldoon, and R. Spigler, Inequalities and numerical bound for zeros of ultraspherical polynomials, SIAM J. Math. Anal. 17 (1986) 1000-1007. MR 88b:33015
  • 2. T. S. Chihara, An Introduction to Orthogonal Polynomials, Gordon and Breach, New York, 1978. MR 58:1979
  • 3. D. K. Dimitrov, On a conjecture concerning monotonicity of zeros of ultraspherical polynomials, J. Approx. Theory 85 (1996) 88-97. MR 97e:33011
  • 4. D. K. Dimitrov, Connection coefficients and zeros of orthogonal polynomials, J. Comput. Appl. Math. 133 (2001), 331-340. MR 2002k:33007
  • 5. D. K. Dimitrov and R. O. Rodrigues, On the behaviour of zeros of Jacobi polynomials, J. Approx. Theory 116 (2002) 224-239. MR 2003f:33008
  • 6. D. K. Dimitrov, Convexity of the extreme zeros of Gegenbauer and Laguerre polynomials, J. Comput. Appl. Math. 153 (2003) 171-180.
  • 7. À. Elbert, Some recent results on the zeros of Bessel functions and orthogonal polynomials, J. Comput. Appl. Math. 133 (2001) 65-83. MR 2002j:33003
  • 8. À. Elbert and A. Laforgia, Upper bounds for the zeros of ultraspherical polynomials, J. Approx. Theory 61 (1990) 88-97. MR 91e:33006
  • 9. À. Elbert and A. Laforgia, Asymptotic formulas for the ultraspherical polynomials $P_{n}^{\lambda}(x)$ and their zeros for large values of $\lambda$, Proc. Amer. Math. Soc. 114 (1992) 371-377. MR 92e:33011
  • 10. À. Elbert, A. Laforgia and L. G. Rodonó, On the zeros of Jacobi polynomials, Acta Math. Hung. 64 (1994) 351-359. MR 95d:33005
  • 11. À. Elbert and P. D. Siafarikas, Monotonicity properties of the zeros of ultraspherical polynomials, J. Approx. Theory 97 (1999) 31-39. MR 2000a:33012
  • 12. J. Favard, Sur les polynomes de Tchebycheff, C. R. Acad. Sci. Paris 200 (1935) 2052-2053.
  • 13. K. J. Förster and K. Petras, On estimates for the weights in Gaussian quadrature in the ultraspherical case, Math. Comp. 55 (1990) 243-264. MR 91d:65043
  • 14. L. Gatteschi, New inequalities for the zeros of Jacobi polynomials, SIAM J. Math. Anal. 18(6) (1987), 1549-1562. MR 88m:33021
  • 15. E. Godoy, A. Ronveaux, A. Zarzo, and I. Area, Minimal recurrence relations for connection coefficients between classical orthogonal polynomials: Continuous case, J. Comput. Appl. Math. 84 (1997) 257-275. MR 99g:33023
  • 16. E. K. Ifantis and P. D. Siafarikas, A differential equation for the zeros of Bessel functions, Appl. Anal. 20 (1985) 296-281. MR 87f:33017
  • 17. E. K. Ifantis and P. D. Siafarikas, Differential inequalities on the greatest zero of Laguerre and ultraspherical polynomials, in: Actas del VI Simposium on Polinomios Ortogonales Y Aplicaciones, Gijon (1989), 187-197.
  • 18. E. K. Ifantis and P. D. Siafarikas, Differential inequalities for the positive zeros of Bessel functions, J. Comput. Appl. Math. 30 (1990) 139-143. MR 91d:33002
  • 19. M. E. H. Ismail, Monotonicity of zeros of orthogonal polynomials, in: D. Stanton, ed., $q$-Series and Partitions (Springer-Verlag, New York 1989) 177-190. MR 90j:33009
  • 20. M. E. H. Ismail, and J. Letessier, Monotonicity of zeros of ultraspherical polynomials, in: M. Alfaro, J. S. Dehesa, F. J. Marcellán, J. L. Rubio de Francia, and J. Vinuesa, eds., Orthogonal Polynomials and Their Applications, Lecture Notes in Mathematics, Vol. 1329 (Springer-Verlag, Berlin, 1988) 329-330. MR 89f:00027
  • 21. M. E. H. Ismail and X. Li, Bound on the extreme zeros of orthogonal polynomials, Proc. Amer. Math. Soc. 115 (1992) 131-140. MR 92h:33019
  • 22. A. Laforgia, A monotonic property for the zeros of ultraspherical polynomials, Proc. Amer. Math. Soc. 83 (1981) 757-758. MR 82j:33016
  • 23. A. Laforgia, Monotonicity properties for the zeros of orthogonal polynomials and Bessel functions, in: C. Brezinski, A. Draux, A. P. Magnus, P. Maroni, and A. Ronveaux, eds., Polynômes Orthogonaux et Applications, Lecture Notes in Mathematics, Vol. 1171 (Springer-Verlag, Berlin, 1985) 267-277. MR 87h:33017
  • 24. N. Obrechkoff, On the distribution of zeros of algebraic equations (in Bulgarian), Annuaire Univ. Sofia 15-16 (1918-1919, 1919-1920) 1-14, 1-11, 1-4.
  • 25. A. Ronveaux, A. Zarzo, and E. Godoy, Recurrence relation for connection coefficients between two families of orthogonal polynomials, J. Comp. Appl. Math. 62 (1995) 67-73.
  • 26. G. Szego, Orthogonal Polynomials, 4th ed., Amer. Math. Soc. Coll. Publ., Vol. 23, Providence, RI, 1975.
  • 27. R. Vidunas, Contiguous relations of hypergeometric series, J. Comp. Appl. Math. (to appear).
  • 28. H. S. Wall and M. Wetzel, Quadratic forms and convergence regions for continued fractions, Duke Math. J. 11 (1944) 89-102. MR 6:151b

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2000): 33C45, 26C10

Retrieve articles in all journals with MSC (2000): 33C45, 26C10


Additional Information

Iván Area
Affiliation: Departamento de Matemática Aplicada II, E.T.S.E. Telecomunicación, Universidade de Vigo, Campus Lagoas–Marcosende, 36200 Vigo, Spain
Email: area@dma.uvigo.es

Dimitar K. Dimitrov
Affiliation: Departamento de Ciências de Computação e Estatística, IBILCE, Universidade Estadual Paulista, 15054-000 São José do Rio Preto, SP, Brazil
Email: dimitrov@dcce.ibilce.unesp.br

Eduardo Godoy
Affiliation: Departamento de Matemática Aplicada II, E.T.S.I. Industriales, Universidade de Vigo, Campus Lagoas–Marcosende, 36200 Vigo, Spain
Email: egodoy@dma.uvigo.es

André Ronveaux
Affiliation: Departement de Mathematique, Unité D’Analyse Mathématique et de Mécanique, Université Catholique de Louvain, Bâtiment Marc de Hemptinne, Chemin du Cyclotron 2, B-1348 Louvain-la-Neuve, Belgium
Email: ronveaux@math.ucl.ac.be

DOI: https://doi.org/10.1090/S0025-5718-04-01642-4
Keywords: Orthogonal polynomials, zeros of Gegenbauer polynomials, zeros of Hermite polynomials, connection coefficients
Received by editor(s): October 14, 2002
Published electronically: March 23, 2004
Article copyright: © Copyright 2004 American Mathematical Society

American Mathematical Society