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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Characterizations of generalized Hermite and
sieved ultraspherical polynomials


Author: Holger Dette
Journal: Trans. Amer. Math. Soc. 348 (1996), 691-711
MSC (1991): Primary 33C45
MathSciNet review: 1311912
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Abstract: A new characterization of the generalized Hermite polyno-
mials and of the orthogonal polynomials with respect to the measure
$|x|^\gamma (1-x^2)^{1/2}dx$ is derived which is based on a ``reversing property" of the coefficients in the corresponding recurrence formulas and does not use the representation in terms of Laguerre and Jacobi polynomials. A similar characterization can be obtained for a generalization of the sieved ultraspherical polynomials of the first and second kind. These results are applied in order to determine the asymptotic limit distribution for the zeros when the degree and the parameters tend to infinity with the same order.


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Additional Information

Holger Dette
Affiliation: Institut für Mathematische Stochastik, Technische Universität Dresden\ Mommsenstr. 13, 01062 Dresden, Germany
Email: dette@math.tu-dresden.de

DOI: http://dx.doi.org/10.1090/S0002-9947-96-01438-9
PII: S 0002-9947(96)01438-9
Keywords: Generalized Hermite polynomials, sieved ultraspherical polynomials, Stieltjes transform, continued fractions, asymptotic zero distribution
Received by editor(s): June 5, 1994
Received by editor(s) in revised form: January 10, 1995
Article copyright: © Copyright 1996 American Mathematical Society