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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.

Item Type: | Article |
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Additional Information: | © 2009 Elsevier Inc. All rights reserved. |

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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