Paper: Extensions of discrete classical orthogonal polynomials beyond the orthogonality

Extensions of discrete classical orthogonal polynomials beyond the orthogonality

Abstract

It is well-known that the family of Hahn polynomials is orthogonal with respect to a certain weight function up to degree \(N\). In this paper we prove, by using the three-term recurrence relation which this family satisfies, that the Hahn polynomials can be characterized by a \(\Delta\)-Sobolev orthogonality for every \(n\) and present a factorization for Hahn polynomials for a degree higher than \(N\).

We also present analogous results for dual Hahn, Krawtchouk, and Racah polynomials and give the limit relations among them for all \(n\in \mathbb N\). Furthermore, in order to get these results for the Krawtchouk polynomials we will obtain a more general property of orthogonality for Meixner polynomials.

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BibTeX

@article {MR2494714,
AUTHOR={Costas-Santos, R. S.; Sanchez-Lara, J. F.},
TITLE={Extensions of discrete classical orthogonal polynomials beyond the orthogonality},
JOURNAL={J. Comput. Appl. Math.},
FJOURNAL={Journal of Computational and Applied Mathematics},
VOLUME={225},
NUMBER={2},
YEAR={2009},
PAGES={440--451},
ISSN={1081-3810},
MRCLASS={33C45 (33C47 42C05)},
MRNUMBER={MR2494714},
ZBL={1167.42008},
MRREVIEWER={Alicia Cachafeiro},
DOI={10.1016/j.cam.2008.07.055},
URL={https://doi.org/10.1016/j.cam.2008.07.055},
}