Research and Innovation Articles
On some asymptotic properties of classical Hermite polynomials modified by a rational factor
Published 2018-03-06
Keywords
- Asymptotics properties,
- perturbed Hermite polynomials,
- Christoffel and Geronimus perturbations.
How to Cite
Molano Molano, L. A. (2018). On some asymptotic properties of classical Hermite polynomials modified by a rational factor. Revista Integración, Temas De matemáticas, 35(2), 149–161. https://doi.org/10.18273/revint.v35n2-2017002
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Abstract
In this paper we study some asymptotic properties of the sequence of monic polynomials orthogonal with respect to the measure dµ = x 2+a x2+b e −x 2 dx, where a, b > 0 and a 6= b. In this way we study the outer relative asymptotic with respect to the classical Hermite polynomials; besides, Mehler-Heine type formulas are analyzed.
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References
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[5] Dueñas H., Huertas E. and Marcellán F. "Asymptotic properties of Laguerre-Sobolev type orthogonal polynomials", Numer. Algorithms 60 (2012), No. 1, 51-73.
[6] Fejzullahu B.Xh., "Asymptotics for orthogonal polynomials with respect to the Laguerre measure modifed by a rational factor, Acta Sci. Math. (Szeged) 77 (2011), No. 1-2, 73-85.
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[8] Meijer H.G., "Determination of all coherent pairs", J. Approx. Theory 89 (1997), No. 3, 321-343.
[9] Moreno-Balcazar J.J., "Smallest zeros of some types of orthogonal polynomials: asymptotics", J. Comput. Appl. Math. 179 (2005), No. 1-2, 289-301.
[10] Szego G., Orthogonal Polynomials, Amer. Math. Soc. Colloq. Publ., Vol. 23, Amer. Math. Soc., 4th Edition, Providence, 1975.
[11] Van Assche W., Asymptotics for orthogonal polynomials, Lecture Notes in Mathematics, Vol. 1265, Springer-Verlag, Berlin, 1987.
[2] Alfaro M., Moreno-Balcázar J.J., Pérez T.E., Piñar M.A. and Rezola M.L., "Asymptotics of Sobolev orthogonal polynomials for Hermite coherent pairs", J. Comput. Appl. Math. 133 (2001), No. 1-2, 141-150.
[3] Chihara T.S., An Introduction to Orthogonal Polynomials, Gordon and Breach, New York, 1978.
[4] Delgado A.M. and Marcellán F., "On an extension of symmetric coherent pairs of
orthogonal polynomial", J. Comput. Appl. Math. 178 (2005), No. 1-2, 155-168.
[5] Dueñas H., Huertas E. and Marcellán F. "Asymptotic properties of Laguerre-Sobolev type orthogonal polynomials", Numer. Algorithms 60 (2012), No. 1, 51-73.
[6] Fejzullahu B.Xh., "Asymptotics for orthogonal polynomials with respect to the Laguerre measure modifed by a rational factor, Acta Sci. Math. (Szeged) 77 (2011), No. 1-2, 73-85.
[7] Iserles A., Koch P.E., Norsett S.P. and Sanz-Serna J. M., "On polynomials orthogonal with respect to certain Sobolev inner products", J. Approx. Theory 65 (1991), No. 2, 151-175.
[8] Meijer H.G., "Determination of all coherent pairs", J. Approx. Theory 89 (1997), No. 3, 321-343.
[9] Moreno-Balcazar J.J., "Smallest zeros of some types of orthogonal polynomials: asymptotics", J. Comput. Appl. Math. 179 (2005), No. 1-2, 289-301.
[10] Szego G., Orthogonal Polynomials, Amer. Math. Soc. Colloq. Publ., Vol. 23, Amer. Math. Soc., 4th Edition, Providence, 1975.
[11] Van Assche W., Asymptotics for orthogonal polynomials, Lecture Notes in Mathematics, Vol. 1265, Springer-Verlag, Berlin, 1987.