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GómezUllate Otaiza, David and Kamran, Niky and Milson, Robert (2012) On orthogonal polynomials spanning a nonstandard flag. In Algebraic aspects of darboux transformations, quantum integrable systems and supersymmetric quantum mechanics. Contemporary mathematics (563). Amer Mathematical Soc. ISBN 9780821875841

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

http://www.ams.org/  Publisher 
http://arxiv.org/abs/1101.5584  Organisation 
Abstract
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 nonstandard 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 nonsingular weights.
Item Type:  Book Section 

Additional Information:  © Amer Mathematical Soc. 
Uncontrolled Keywords:  Shapeinvariant potentials; Quasiexact solvability; Differentialequation; Laguerrepolynomials; Systems; Supersymmetry 
Subjects:  Sciences > Physics > PhysicsMathematical 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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