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

Costas-Santos, R. S.and Marcellan, F.**Acta Applicandae Mathematicae**111 (2010), 107 — 128

## 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}, }