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SIGMA, 2012, Volume 8, 042, 30 pages (Mi sigma719)  

This article is cited in 2 scientific papers (total in 2 papers)

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

Renato Álvarez-Nodarsea, Rezan Sevinik Adigüzelb, Hasan Taşelib

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.

DOI: https://doi.org/10.3842/SIGMA.2012.042

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Full text: http://emis.mi.ras.ru/.../042
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Bibliographic databases:

ArXiv: 1107.2423
MSC: 33D45; 42C05
Received: July 29, 2011; in final form July 2, 2012; Published online July 11, 2012
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Citation: Renato Álvarez-Nodarse, Rezan Sevinik Adigüzel, Hasan Taşeli, “On the orthogonality of $q$-classical polynomials of the Hahn class”, SIGMA, 8 (2012), 042, 30 pp.

Citation in format AMSBIB
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\by Renato \'Alvarez-Nodarse, Rezan Sevinik Adig\"uzel, Hasan Ta{\c s}eli
\paper On the orthogonality of $q$-classical polynomials of the Hahn class
\jour SIGMA
\yr 2012
\vol 8
\papernumber 042
\totalpages 30
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\crossref{https://doi.org/10.3842/SIGMA.2012.042}
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. Verde-Star L., “Recurrence Coefficients and Difference Equations of Classical Discrete Orthogonal and Q-Orthogonal Polynomial Sequences”, Linear Alg. Appl., 440 (2014), 293–306  crossref  mathscinet  zmath  isi  scopus
    2. F. Soleyman, M. Masjed-Jamei, I. Area, “A finite class of q-orthogonal polynomials corresponding to inverse gamma distribution”, Anal. Math. Phys., 7:4 (2017), 479–492  crossref  mathscinet  zmath  isi  scopus
  • Symmetry, Integrability and Geometry: Methods and Applications
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