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On orthogonal polynomials spanning a non-standard flag



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Gómez-Ullate Otaiza, David and Kamran, Niky and Milson, Robert (2012) On orthogonal polynomials spanning a non-standard flag. In Algebraic aspects of darboux transformations, quantum integrable systems and supersymmetric quantum mechanics. Contemporary mathematics (563). Amer Mathematical Soc. ISBN 978-0-8218-7584-1

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Official URL: http://dx.doi.org/10.1090/conm/563/11164


We survey some recent developments in the theory of orthogonal polynomials defined by differential equations. The key finding is that there exist orthogonal polynomials defined by 2nd order differential equations that fall outside the classical families of Jacobi, Laguerre, and Hermite polynomials. Unlike the classical families, these new examples, called exceptional orthogonal polynomials, feature non-standard polynomial flags; the lowest degree polynomial has degree m > 0. In this paper we review the classification of codimension m = 1 exceptional polynomials, and give a novel, compact proof of the fundamental classification theorem for codimension 1 polynomial flags. As well, we describe the mechanism or rational factorizations of 2nd order operators as the analogue of the Darboux transformation in this context. We finish with the example of higher codimension generalization of Jacobi polynomials and perform the complete analysis of parameter values for which these families have non-singular weights.

Item Type:Book Section
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© Amer Mathematical Soc.
Conferencia: Jairo Charris Seminar on Algebraic Aspects of Darboux Transformations, Quantum Integrable Systems and Supersymmetric Quantum Mechanics (2010 . Santa Marta, Colombia).
We thank Ferenc Tookos for useful comments and suggestions. The research of DGU was supported in part by MICINN-FEDER grant MTM2009-06973 and CUR-DIUE grant 2009SGR859. The research of NK was supported in part by NSERC grant RGPIN 105490-2004. The research of RM was supported in part by NSERC grant RGPIN-228057-2004.

Uncontrolled Keywords:Shape-invariant potentials; Quasi-exact solvability; Differential-equation; Laguerre-polynomials; Systems; Supersymmetry
Subjects:Sciences > Physics > Physics-Mathematical models
Sciences > Physics > Mathematical physics
ID Code:30804
Deposited On:12 Jun 2015 08:46
Last Modified:10 Dec 2018 15:09

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