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

 

Multiple orthogonal polynomials for classical weights


Authors: A. I. Aptekarev, A. Branquinho and W. Van Assche
Journal: Trans. Amer. Math. Soc. 355 (2003), 3887-3914
MSC (2000): Primary 33C45, 42C05
Published electronically: June 10, 2003
MathSciNet review: 1990569
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Abstract: A new set of special functions, which has a wide range of applications from number theory to integrability of nonlinear dynamical systems, is described. We study multiple orthogonal polynomials with respect to $p > 1$ weights satisfying Pearson's equation. In particular, we give a classification of multiple orthogonal polynomials with respect to classical weights, which is based on properties of the corresponding Rodrigues operators. We show that the multiple orthogonal polynomials in our classification satisfy a linear differential equation of order $p+1$. We also obtain explicit formulas and recurrence relations for these polynomials.


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Additional Information

A. I. Aptekarev
Affiliation: Keldysh Institute of Applied Mathematics, Russian Academy of Sciences, Miusskaya Square 4, Moscow 125047, Russia
Email: aptekaa@rfbr.ru

A. Branquinho
Affiliation: Departamento de Matemática da FCTUC, FCTUC, Universidade de Coimbra, Apartado 3008, 3000 Coimbra, Portugal
Email: ajplb@mat.uc.pt

W. Van Assche
Affiliation: Department of Mathematics, Katholieke Universiteit Leuven, Celestijnenlaan 200B, B-3001 Leuven, Belgium
Email: walter@wis.kuleuven.ac.be

DOI: http://dx.doi.org/10.1090/S0002-9947-03-03330-0
PII: S 0002-9947(03)03330-0
Received by editor(s): February 13, 2001
Published electronically: June 10, 2003
Additional Notes: This work was conducted in the framework of project INTAS-2000-272. The research was carried out while the first author was visiting the Universidade de Coimbra with a grant from Fundação para a Ciência e Tecnologia PRAXIS XXI/2654/98/BCC and the Katholieke Universiteit Leuven with senior fellowship F/99/009 of the research counsel. The first author is supported by grant RFBR 00-15-96132 and RFBR 02-01-00564. The second author is supported by Centro de Matemática da Universidade de Coimbra (CMUC). The third author is supported by grant G.0184.02 of FWO-Vlaanderen
Article copyright: © Copyright 2003 American Mathematical Society