Foundations of Computational Mathematics

, Volume 13, Issue 4, pp 615–666 | Cite as

A Conjecture on Exceptional Orthogonal Polynomials

  • David Gómez-Ullate
  • Niky KamranEmail author
  • Robert Milson


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 Cariñena 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 (X2-OPs). As a by-product of this analysis, we prove a Bochner-type theorem classifying all possible X2-OPSs. The classification includes all cases known to date plus some new examples of X2-Laguerre and X2-Jacobi polynomials.


Exceptional orthogonal polynomials Sturm–Liouville problems Darboux–Crum transformation Bochner theorem 

Mathematics Subject Classification

33E99 33C4 34A30 34A05 



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.


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Copyright information

© SFoCM 2012

Authors and Affiliations

  • David Gómez-Ullate
    • 1
  • Niky Kamran
    • 2
    Email author
  • Robert Milson
    • 3
  1. 1.Departamento de Física Teórica IIUniversidad Complutense de MadridMadridSpain
  2. 2.Department of Mathematics and StatisticsMcGill UniversityMontrealCanada
  3. 3.Department of Mathematics and StatisticsDalhousie UniversityHalifaxCanada

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