Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 8 (2012), 042, 30 pages      arXiv:1107.2423      http://dx.doi.org/10.3842/SIGMA.2012.042

On the Orthogonality of q-Classical Polynomials of the Hahn Class

Renato Álvarez-Nodarse a, Rezan Sevinik Adıgüzel b and 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

Received July 29, 2011, in final form July 02, 2012; Published online July 11, 2012

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.

Key words: q-polynomials; orthogonal polynomials on q-linear lattices; q-Hahn class.

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