Gómez-Ullate Otaiza, DavidKamran, NikyMilson, Robert2023-06-192023-06-192013-081615-337510.1007/s10208-012-9128-6https://hdl.handle.net/20.500.14352/34711© Springer. 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 105490-2011. The research of RM was supported in part by NSERC grant RGPIN-228057-2009.Exceptional orthogonal polynomial systems (X-OPSs) arise as eigenfunctions of Sturm-Liouville problems, but without the assumption that an eigenpolynomial of every degree is present. In this sense, they generalize the classical families of Hermite, Laguerre, and Jacobi, and include as a special case the family of CPRS orthogonal polynomials introduced by Cariena et al. (J. Phys. A 41:085301, 2008). We formulate the following conjecture: every exceptional orthogonal polynomial system is related to a classical system by a Darboux-Crum transformation. We give a proof of this conjecture for codimension 2 exceptional orthogonal polynomials (X-2-OPs). As a by-product of this analysis, we prove a Bochner-type theorem classifying all possible X-2-OPSs. The classification includes all cases known to date plus some new examples of X-2-Laguerre and X-2-Jacobi polynomials.engjournal articlehttp://dx.doi.org/10.1007/s10208-012-9128-6http://link.springer.comhttp://arxiv.org/abs/1203.6857open access51-73Shape-invariant potentialsQuasi-exact solvabilityX-L laguerreDarboux transformationsDifferential-equationSupersymmetryOperatorsFísica-Modelos matemáticosFísica matemática