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An extended class of orthogonal polynomials defined by a Sturm-Liouville problem

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Gómez-Ullate Otaiza, David and Kamran, Niky and Milson, Robert (2009) An extended class of orthogonal polynomials defined by a Sturm-Liouville problem. Journal of mathematical analysis and applications, 359 (1). pp. 352-369. ISSN 0022-247X

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Official URL: http://dx.doi.org/10.1016/j.jmaa.2009.05.052




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 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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© 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.

Uncontrolled Keywords:Differential-equation; Bochner; Theorem
Subjects:Sciences > Physics > Physics-Mathematical models
Sciences > Physics > Mathematical physics
ID Code:30809
Deposited On:15 Jun 2015 08:54
Last Modified:10 Dec 2018 15:09

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