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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

The complementary polynomials and the Rodrigues operator of classical orthogonal polynomials


Authors: Roberto S. Costas-Santos and Francisco Marcellán Español
Journal: Proc. Amer. Math. Soc. 140 (2012), 3485-3493
MSC (2010): Primary 33C45; Secondary 34B24, 42C05
Published electronically: February 20, 2012
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Abstract: From the Rodrigues representation of polynomial eigenfunctions of a second order linear hypergeometric-type differential (difference or $ q$-differ-
ence) operator, complementary polynomials for classical orthogonal polynomials are constructed using a straightforward method. Thus a generating function in a closed form is obtained.

For the complementary polynomials we present a second order linear hyper-
geometric-type differential (difference or $ q$-difference) operator, a three-term recursion and Rodrigues formulas which extend the results obtained by H. J. Weber for the standard derivative operator.


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Additional Information

Roberto S. Costas-Santos
Affiliation: Departamento de Matemáticas, Facultad de Ciencias, Universidad de Alcalá, 28871 Alcalá de Henares, Spain
Email: rscosa@gmail.com, roberto.costas@uah.es

Francisco Marcellán Español
Affiliation: Departamento de Matemáticas, Universidad Carlos III de Madrid, Avenida de la Universidad 30, 28911 Leganés, Spain
Email: pacomarc@ing.uc3m.es

DOI: http://dx.doi.org/10.1090/S0002-9939-2012-11229-8
PII: S 0002-9939(2012)11229-8
Keywords: Classical orthogonal polynomials, Rodrigues operator, complementary polynomials, generating formula
Received by editor(s): February 11, 2011
Received by editor(s) in revised form: April 8, 2011
Published electronically: February 20, 2012
Additional Notes: The first author acknowledges financial support from Dirección General de Investigación del Ministerio de Ciencia e Innovación of Spain under grant MTM2009-12740-C03-01 and from the program of postdoctoral grants (Programa de becas postdoctorales)
The second author acknowledges financial support from Dirección General de Investigación del Ministerio de Ciencia e Innovación of Spain under grant MTM 2009-12740-C03-01.
Communicated by: Walter Van Assche
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.