Gómez-Ullate Otaiza, DavidGrandati, YvesMilson, Robert2023-06-192023-06-192014-01-101751-811310.1088/1751-8113/47/1/015203https://hdl.handle.net/20.500.14352/34707© IOP Publishing Ltd. The research of the first author (DGU) has been supported in part by Spanish MINECO-FEDER grants MTM2009-06973, MTM2012-31714, and the Catalan grant 2009SGR-859. The research of the third author (RM) was supported in part by NSERC grant RGPIN-228057-2009.We prove that every rational extension of the quantum harmonic oscillator that is exactly solvable by polynomials is monodromy free, and therefore can be obtained by applying a finite number of state-deleting Darboux transformations on the harmonic oscillator. Equivalently, every exceptional orthogonal polynomial system of Hermite type can be obtained by applying a Darboux-Crum transformation to the classical Hermite polynomials. Exceptional Hermite polynomial systems only exist for even codimension 2m, and they are indexed by the partitions λ of m. We provide explicit expressions for their corresponding orthogonality weights and differential operators and a separate proof of their completeness. Exceptional Hermite polynomials satisfy a 2l + 3 recurrence relation where l is the length of the partition λ. Explicit expressions for such recurrence relations are given.engjournal articlehttp://dx.doi.org/10.1088/1751-8113/47/1/015203http://iopscience.iop.orghttp://arxiv.org/abs/1306.5143open access51-73Shape-invariant potentialsQuasi-exact solvabilityOrthogonal polynomialsDarboux transformationsLaguerre-polynomialsMechanicsEquationFormulaFísica-Modelos matemáticosFísica matemática