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A Bochner type characterization theorem for exceptional orthogonal polynomials

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García Ferrero, María Ángeles and Gómez-Ullate Oteiza, David and Milson, Robert (2019) A Bochner type characterization theorem for exceptional orthogonal polynomials. Journal of mathematical analysis and applications, 472 (1). pp. 584-626. ISSN 0022-247X

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Official URL: http://dx.doi.org/10.1016/j.jmaa.2018.11.042


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Abstract

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.


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©2018 Elsevier Inc.
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.

Uncontrolled Keywords:Shape invariant potentials; Scattering-amplitude; Darboux transformations; Hermite; Zeros; Extensions; Families; Charlier; Meixner; Orthogonal polynomials; Darboux transformations; Sturm-Liouville problems; Trivial monodromy
Subjects:Sciences > Physics > Physics-Mathematical models
Sciences > Physics > Mathematical physics
ID Code:51778
Deposited On:12 Mar 2019 18:26
Last Modified:13 Mar 2019 11:54

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