Gómez-Ullate Otaiza, DavidKamran, NikyMilson, Robert2023-06-202023-06-202010-050021-904510.1016/j.jat.2009.11.002https://hdl.handle.net/20.500.14352/44631© 2009 Elsevier Inc. We thank Peter Crooks for reviewing the manuscript and providing many useful remarks. We are also grateful to Jorge Arvesú, Norrie Everitt, Mourad Ismail, Francisco Marcellán, Lance Littlejohn and André Ronveaux for their helpful comments on the manuscript. This work was supported in part by the Ramón y Cajal program of the Spanish Ministry of Science and Technology; the Dirección General de Investigación [grants MTM2006-00478 and MTM2006- 14603 to D.G.U.] and the National Science and Engineering Reserach Council of Canada [grants RGPIN 105490- 2004 to N.K. and RGPIN-228057-2004 to R.M.]We prove an extension of Bochner's classical result that characterizes the classical polynomial families as eigenfunctions of a second-order differential operator with polynomial coefficients. The extended result involves considering differential operators with rational coefficients and the requirement is that they have a numerable sequence of polynomial eigenfunctions p(1), p(2),.. of all degrees except for degree zero. The main theorem of the paper provides a characterization of all such differential operators. The existence of such differential operators and polynomial sequences is based on the concept of exceptional polynomial subspaces, and the converse part of the main theorem rests on the classification of codimension one exceptional subspaces under projective transformations, which is performed in this paper.engjournal articlehttp://dx.doi.org/10.1016/j.jat.2009.11.002http://www.sciencedirect.com/http://arxiv.org/abs/0805.3376open access51-73Quasi-exact solvabilityOrthogonal polynomialsDifferential-equationFísica-Modelos matemáticosFísica matemática