%0 Journal Article
%A Gómez-Ullate Otaiza, David
%A Kamran, Niky
%A Milson, Robert
%T An extended class of orthogonal polynomials defined by a Sturm-Liouville problem
%D 2009
%@ 0022-247X
%U https://hdl.handle.net/20.500.14352/44632
%X 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.
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