Gómez-Ullate Otaiza, DavidKamran, NikyMilson, Robert2023-06-202023-06-202009-11-010022-247X10.1016/j.jmaa.2009.05.052https://hdl.handle.net/20.500.14352/44632© 2009 Elsevier Inc. All rights reserved. We are grateful to Jorge Arvesú, Mourad Ismail, Francisco Marcellán and André Ronveaux for their helpful comments. A special note of thanks goes to Norrie Everitt for his suggestions and remarks regarding operator domains and the limit point/circle analysis, and to Lance Littlejohn for comments regarding classical polynomials with negative integer parameters. The research of DGU is supported in part by the Ramón y Cajal program of the Spanish ministry of Science and Technology and by the DGI under grants MTM2006-00478 and MTM2006-14603. The research of NK is supported in part by NSERC grant RGPIN 105490-2004. The research of RM is supported in part by NSERC grant RGPIN-228057-2004.We present two infinite sequences of polynomial eigenfunctions of a Sturm-Liouville problem. As opposed to the classical orthogonal polynomial systems, these sequences start with a polynomial of degree one. We denote these polynomials as X(1)-Jacobi and X(1)-Laguerre and we prove that they are orthogonal with respect to a positive definite inner product defined over the compact interval [-1, 1] or the half-line [0, infinity), respectively, and they are a basis of the corresponding L(2) Hilbert spaces. Moreover, we prove a converse statement similar to Bochner's theorem for the classical orthogonal polynomial systems: if a self-adjoint second-order operator has a complete set of polynomial eigenfunctions {p(i)}(i=1)(infinity), then it must be either the X(1)-Jacobi or the X(1)-Laguerre Sturm-Liouville problem. A Rodrigues-type formula can be derived for both of the X(1) polynomial sequences.engjournal articlehttp://dx.doi.org/10.1016/j.jmaa.2009.05.052http://www.sciencedirect.comhttp://arxiv.org/abs/0807.3939open access51-73Differential-equationBochnerTheoremFísica-Modelos matemáticosFísica matemática