\(q\)-Classical orthogonal polynomial: A general difference calculus approach

Abstract

It is well known that the classical families of orthogonal polynomials are characterized as eigenfunctions of a second order linear differential/difference operator with polynomial coefficients.

In this paper we present a study of the classical orthogonal polynomials sequences in a more general framework by using the differential (or difference) calculus and Operator Theory. The Hahn’s Theorem and a characterization theorem for the \(q\)-polynomials which belongs to the \(q\)-Askey and Hahn tableaux are proved. Finally, we illustrate our results applying them to some known families of orthogonal \(q\)-polynomials.

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BibTeX

@article {MR2653053,
AUTHOR={Costas-Santos, R. S. and Marcellan, F.},
TITLE={{$q$}-classical orthogonal polynomials: a general difference calculus approach},
JOURNAL={Acta Appl. Math.},
FJOURNAL={Acta Applicandae Mathematicae},
VOLUME={111},
YEAR={2010},
NUMBER={1},
PAGES={107--128},
ISSN={0167-8019},
MRCLASS={33C45 (33D45 39A13 42C05)},
MRNUMBER={2653053},
ZBL={1204.33011},
MRREVIEWER={Thomas Stoll},
DOI={10.1007/s10440-009-9536-z},
URL={https://doi.org/10.1007/s10440-009-9536-z},
}