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A conjecture on exceptional orthogonal polynomials

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Gómez-Ullate Otaiza, David and Kamran, Niky and Milson, Robert (2013) A conjecture on exceptional orthogonal polynomials. Foundations of computational mathematics, 13 (4). pp. 615-666. ISSN 1615-3375

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Official URL: http://dx.doi.org/10.1007/s10208-012-9128-6




Abstract

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.


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© 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.

Uncontrolled Keywords:Shape-invariant potentials; Quasi-exact solvability; X-L laguerre; Darboux transformations; Differential-equation; Supersymmetry; Operators
Subjects:Sciences > Physics > Physics-Mathematical models
Sciences > Physics > Mathematical physics
ID Code:30772
Deposited On:11 Jun 2015 10:02
Last Modified:10 Dec 2018 15:09

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