Zeros of Gegenbauer and Hermite polynomials and connection coefficients
HTML articles powered by AMS MathViewer
- by Iván Area, Dimitar K. Dimitrov, Eduardo Godoy and André Ronveaux PDF
- Math. Comp. 73 (2004), 1937-1951 Request permission
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
- Shafique Ahmed, Martin E. Muldoon, and Renato Spigler, Inequalities and numerical bounds for zeros of ultraspherical polynomials, SIAM J. Math. Anal. 17 (1986), no. 4, 1000–1007. MR 846402, DOI 10.1137/0517070
- T. S. Chihara, An introduction to orthogonal polynomials, Mathematics and its Applications, Vol. 13, Gordon and Breach Science Publishers, New York-London-Paris, 1978. MR 0481884
- Dimitar K. Dimitrov, On a conjecture concerning monotonicity of zeros of ultraspherical polynomials, J. Approx. Theory 85 (1996), no. 1, 88–97. MR 1382052, DOI 10.1006/jath.1996.0030
- Dimitar K. Dimitrov, Connection coefficients and zeros of orthogonal polynomials, Proceedings of the Fifth International Symposium on Orthogonal Polynomials, Special Functions and their Applications (Patras, 1999), 2001, pp. 331–340. MR 1858291, DOI 10.1016/S0377-0427(00)00653-1
- Dimitar K. Dimitrov and Romildo O. Rodrigues, On the behaviour of zeros of Jacobi polynomials, J. Approx. Theory 116 (2002), no. 2, 224–239. MR 1911080, DOI 10.1006/jath.2002.3671
- D. K. Dimitrov, Convexity of the extreme zeros of Gegenbauer and Laguerre polynomials, J. Comput. Appl. Math. 153 (2003) 171–180.
- Á. Elbert, Some recent results on the zeros of Bessel functions and orthogonal polynomials, Proceedings of the Fifth International Symposium on Orthogonal Polynomials, Special Functions and their Applications (Patras, 1999), 2001, pp. 65–83. MR 1858270, DOI 10.1016/S0377-0427(00)00635-X
- Árpád Elbert and Andrea Laforgia, Upper bounds for the zeros of ultraspherical polynomials, J. Approx. Theory 61 (1990), no. 1, 88–97. MR 1047150, DOI 10.1016/0021-9045(90)90025-L
- Árpád Elbert and Andrea Laforgia, Asymptotic formulas for ultraspherical polynomials $P_n^\lambda (x)$ and their zeros for large values of $\lambda$, Proc. Amer. Math. Soc. 114 (1992), no. 2, 371–377. MR 1089404, DOI 10.1090/S0002-9939-1992-1089404-X
- Á. Elbert, A. Laforgia, and Lucia G. Rodonò, On the zeros of Jacobi polynomials, Acta Math. Hungar. 64 (1994), no. 4, 351–359. MR 1275650, DOI 10.1007/BF01874695
- Árpád Elbert and Panayiotis D. Siafarikas, Monotonicity properties of the zeros of ultraspherical polynomials, J. Approx. Theory 97 (1999), no. 1, 31–39. MR 1676306, DOI 10.1006/jath.1996.3259
- J. Favard, Sur les polynomes de Tchebycheff, C. R. Acad. Sci. Paris 200 (1935) 2052–2053.
- Klaus-Jürgen Förster and Knut Petras, On estimates for the weights in Gaussian quadrature in the ultraspherical case, Math. Comp. 55 (1990), no. 191, 243–264. MR 1023758, DOI 10.1090/S0025-5718-1990-1023758-1
- Luigi Gatteschi, New inequalities for the zeros of Jacobi polynomials, SIAM J. Math. Anal. 18 (1987), no. 6, 1549–1562. MR 911648, DOI 10.1137/0518111
- 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), no. 2, 257–275. MR 1475378, DOI 10.1016/S0377-0427(97)00137-4
- E. K. Ifantis and P. D. Siafarikas, A differential equation for the zeros of Bessel functions, Applicable Anal. 20 (1985), no. 3-4, 269–281. MR 814954, DOI 10.1080/00036818508839574
- 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.
- E. K. Ifantis and P. D. Siafarikas, Differential inequalities for the positive zeros of Bessel functions, J. Comput. Appl. Math. 30 (1990), no. 2, 139–143. MR 1062318, DOI 10.1016/0377-0427(90)90022-R
- Mourad E. H. Ismail, Monotonicity of zeros of orthogonal polynomials, $q$-series and partitions (Minneapolis, MN, 1988) IMA Vol. Math. Appl., vol. 18, Springer, New York, 1989, pp. 177–190. MR 1019851, DOI 10.1007/978-1-4684-0637-5_{1}4
- 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. MR 973417, DOI 10.1007/BFb0083349
- Mourand E. H. Ismail and Xin Li, Bound on the extreme zeros of orthogonal polynomials, Proc. Amer. Math. Soc. 115 (1992), no. 1, 131–140. MR 1079891, DOI 10.1090/S0002-9939-1992-1079891-5
- Andrea Laforgia, A monotonic property for the zeros of ultraspherical polynomials, Proc. Amer. Math. Soc. 83 (1981), no. 4, 757–758. MR 630050, DOI 10.1090/S0002-9939-1981-0630050-X
- A. Laforgia, Monotonicity properties for the zeros of orthogonal polynomials and Bessel functions, Orthogonal polynomials and applications (Bar-le-Duc, 1984) Lecture Notes in Math., vol. 1171, Springer, Berlin, 1985, pp. 267–277. MR 838993, DOI 10.1007/BFb0076553
- 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.
- 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.
- G. Szegő, Orthogonal Polynomials, 4th ed., Amer. Math. Soc. Coll. Publ., Vol. 23, Providence, RI, 1975.
- R. Vidūnas, Contiguous relations of hypergeometric series, J. Comp. Appl. Math. (to appear).
- Albert Eagle, Series for all the roots of the equation $(z-a)^m=k(z-b)^n$, Amer. Math. Monthly 46 (1939), 425–428. MR 6, DOI 10.2307/2303037
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
- MR Author ID: 308699
- 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
- Received by editor(s): October 14, 2002
- Published electronically: March 23, 2004
- © Copyright 2004 American Mathematical Society
- Journal: Math. Comp. 73 (2004), 1937-1951
- MSC (2000): Primary 33C45, 26C10
- DOI: https://doi.org/10.1090/S0025-5718-04-01642-4
- MathSciNet review: 2059744