Monotonicity and asymptotics of zeros of Laguerre-Sobolev-type orthogonal polynomials of higher order derivatives
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- by Francisco Marcellán and Fernando R. Rafaeli PDF
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Abstract:
In this paper we analyze the location of the zeros of polynomials orthogonal with respect to the inner product \begin{equation} \langle p,q\rangle = \displaystyle \int _{0}^{\infty }p(x)q(x)x^{\alpha }e^{-x}dx+Np^{(j)}(0)q^{(j)}(0), \end{equation} where $\alpha >-1$, $N\geq 0,$ and $j\in \mathbb {N}.$ In particular, we focus our attention on their interlacing properties with respect to the zeros of Laguerre polynomials as well as on the monotonicity of each individual zero in terms of the mass $N.$ Finally, we give necessary and sufficient conditions in terms of $N$ in order for the least zero of any Laguerre-Sobolev-type orthogonal polynomial to be negative.References
- Renato Álvarez-Nodarse and Juan J. Moreno-Balcázar, Asymptotic properties of generalized Laguerre orthogonal polynomials, Indag. Math. (N.S.) 15 (2004), no. 2, 151–165. MR 2071854, DOI 10.1016/S0019-3577(04)90012-2
- H. Bavinck, Differential operators having Sobolev-type Laguerre polynomials as eigenfunctions, Proc. Amer. Math. Soc. 125 (1997), no. 12, 3561–3567. MR 1422848, DOI 10.1090/S0002-9939-97-04091-4
- C. F. Bracciali, D. K. Dimitrov, and A. Sri Ranga, Chain sequences and symmetric generalized orthogonal polynomials, J. Comput. Appl. Math. 143 (2002), no. 1, 95–106. MR 1907785, DOI 10.1016/S0377-0427(01)00499-X
- Dimitar K. Dimitrov, Francisco Marcellán, and Fernando R. Rafaeli, Monotonicity of zeros of Laguerre-Sobolev-type orthogonal polynomials, J. Math. Anal. Appl. 368 (2010), no. 1, 80–89. MR 2609260, DOI 10.1016/j.jmaa.2010.02.038
- Dimitar K. Dimitrov, Mirela V. Mello, and Fernando R. Rafaeli, Monotonicity of zeros of Jacobi-Sobolev type orthogonal polynomials, Appl. Numer. Math. 60 (2010), no. 3, 263–276. MR 2602677, DOI 10.1016/j.apnum.2009.12.004
- Herbert Dueñas and Francisco Marcellán, The Laguerre-Sobolev-type orthogonal polynomials, J. Approx. Theory 162 (2010), no. 2, 421–440. MR 2581392, DOI 10.1016/j.jat.2009.07.006
- H. Dueñas and F. Marcellán, The Laguerre-Sobolev-type orthogonal polynomials. Holonomic equation and electrostatic interpretation, Rocky Mount. J. of Math. (2011). In press.
- R. Koekoek and H. G. Meijer, A generalization of Laguerre polynomials, SIAM J. Math. Anal. 24 (1993), no. 3, 768–782. MR 1215437, DOI 10.1137/0524047
- J. Koekoek, R. Koekoek, and H. Bavinck, On differential equations for Sobolev-type Laguerre polynomials, Trans. Amer. Math. Soc. 350 (1998), no. 1, 347–393. MR 1433121, DOI 10.1090/S0002-9947-98-01993-X
- Francisco Marcellán and Juan José Moreno Balcázar, Asymptotics and zeros of Sobolev orthogonal polynomials on unbounded supports, Acta Appl. Math. 94 (2006), no. 2, 163–192. MR 2273888, DOI 10.1007/s10440-006-9073-y
- F. Marcellán and A. Ronveaux, On a class of polynomials orthogonal with respect to a discrete Sobolev inner product, Indag. Math. (N.S.) 1 (1990), no. 4, 451–464. MR 1106092, DOI 10.1016/0019-3577(90)90013-D
- H. G. Meijer, Zero distribution of orthogonal polynomials in a certain discrete Sobolev space, J. Math. Anal. Appl. 172 (1993), no. 2, 520–532. MR 1201003, DOI 10.1006/jmaa.1993.1041
- Gábor Szegő, Orthogonal polynomials, 4th ed., American Mathematical Society Colloquium Publications, Vol. XXIII, American Mathematical Society, Providence, R.I., 1975. MR 0372517
Additional Information
- Francisco Marcellán
- Affiliation: Departamento de Matemáticas, Escuela Politécnica Superior, Universidad Carlos III de Madrid, 28911 Leganés, Spain
- Fernando R. Rafaeli
- Affiliation: Instituto de Matemática, Estatística e Computação Científica, Universidade Estadual de Campinas, São Paulo, Brazil
- Address at time of publication: Departamento de Matemática, Estatística e Computação/FCT, Universidade Estadual Paulista-UNESP, 19060-900 Presidente Prudente, São Paulo, Brazil
- Received by editor(s): November 25, 2009
- Received by editor(s) in revised form: September 7, 2010
- Published electronically: March 10, 2011
- Communicated by: Walter Van Assche
- © Copyright 2011
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 139 (2011), 3929-3936
- MSC (2010): Primary 42C05; Secondary 33C47
- DOI: https://doi.org/10.1090/S0002-9939-2011-10806-2
- MathSciNet review: 2823039