@article{gomez-ullateExtendedClassOrthogonal2009,
 abstract = {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 X1-Jacobi and X1-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,$ınfty$), respectively, and they are a basis of the corresponding L2 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 \pi\i=1$ınfty$, then it must be either the X1-Jacobi or the X1-Laguerre Sturm--Liouville problem. A Rodrigues-type formula can be derived for both of the X1 polynomial sequences.},
 author = {Gómez-Ullate, David and Kamran, Niky and Milson, Robert},
 copyright = {https://www.elsevier.com/tdm/userlicense/1.0/},
 doi = {10.1016/j.jmaa.2009.05.052},
 file = {/Users/david/Zotero/storage/CGCC3MSI/Gómez-Ullate et al. - 2009 - An extended class of orthogonal polynomials defined by a Sturm–Liouville problem.pdf},
 issn = {0022247X},
 journal = {Journal of Mathematical Analysis and Applications},
 langid = {english},
 month = {November},
 number = {1},
 pages = {352--367},
 title = {An Extended Class of Orthogonal Polynomials Defined by a Sturm--Liouville Problem},
 urldate = {2025-10-19},
 volume = {359},
 year = {2009}
}