Renato Álvarez-Nodarse, Rezan Sevinik Adigüzel, Hasan Taşeli, “On the orthogonality of <nobr>$q$</nobr>-classical polynomials of the Hahn class”, SIGMA, 2012, Volume 8,042, 30 pp.

This article is cited in 2 papers

On the orthogonality of $q$-classical polynomials of the Hahn class

Renato Álvarez-Nodarse^a, Rezan Sevinik Adigüzel^b, Hasan Taşeli^b

^a IMUS & Departamento de Análisis Matemático, Universidad de Sevilla, Apdo. 1160, E-41080 Sevilla, Spain
^b Department of Mathematics, Middle East Technical University (METU), 06531, Ankara, Turkey

Abstract: The central idea behind this review article is to discuss in a unified sense the orthogonality of all possible polynomial solutions of the $q$-hypergeometric difference equation on a $q$-linear lattice by means of a qualitative analysis of the $q$-Pearson equation. To be more specific, a geometrical approach has been used by taking into account every possible rational form of the polynomial coefficients in the $q$-Pearson equation, together with various relative positions of their zeros, to describe a desired $q$-weight function supported on a suitable set of points. Therefore, our method differs from the standard ones which are based on the Favard theorem, the three-term recurrence relation and the difference equation of hypergeometric type. Our approach enables us to extend the orthogonality relations for some well-known $q$-polynomials of the Hahn class to a larger set of their parameters.

Keywords: $q$-polynomials, orthogonal polynomials on $q$-linear lattices, $q$-Hahn class.

MSC: 33D45; 42C05

Received: July 29, 2011; in final form July 2, 2012; Published online July 11, 2012

Language: English

DOI: 10.3842/SIGMA.2012.042