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00925njm 22002777a 4500 dc Gómez-Ullate Otaiza, David author Kamran, Niky author Milson, Robert author 2012 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. 978-0-8218-7584-1 10.1090/conm/563/11164 https://hdl.handle.net/20.500.14352/45574 http://dx.doi.org/10.1090/conm/563/11164 http://www.ams.org/ http://arxiv.org/abs/1101.5584 On orthogonal polynomials spanning a non-standard flag