A Bochner type characterization theorem for exceptional orthogonal polynomials

It was recently conjectured that every system of exceptional orthogonal polynomials is related to a classical orthogonal polynomial system by a sequence of Darboux transformations. In this paper we prove this conjecture, which paves the road to a complete classification of all exceptional orthogonal polynomials. In some sense, this paper can be regarded as the extension of Bochner's result for classical orthogonal polynomials to the exceptional class. As a supplementary result, we derive a canonical form for exceptional operators based on a bilinear formalism, and prove that every exceptional operator has trivial monodromy at all primary poles.
M.A.G.F. acknowledges the financial support of the Spanish MINECO through a Severo Ochoa FPI scholarship. The work of M.A.G.F. is supported in part by the ERC Starting Grant 633152 and the ICMAT-Severo Ochoa project SEV-2015-0554 . The research of D.G.U. has been supported in part by Spanish MINECO-FEDER Grants MTM2012-31714 and MTM2015-65888-C4-3 and by the ICMAT-Severo Ochoa project SEV-2015-0554 . The research of the third author (RM) was supported in part by NSERC grant RGPIN-228057-2009 . D.G.U. would like to thank Dalhousie University for their hospitality during his visit in the Spring semester of 2014 where many of the results in this paper were obtained.