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Symmetry, Integrability and Geometry: Methods and Applications, 2006, Volume 2, 034, 8 pp.
DOI: https://doi.org/10.3842/SIGMA.2006.034
(Mi sigma62)
 

On Orthogonality Relations for Dual Discrete $q$-Ultraspherical Polynomials

Valentyna A. Grozaa, Ivan I. Kachurykb

a National Aviation University, 1 Komarov Ave., Kyiv, 03058 Ukraine
b Khmel'nyts'kyi National University, Khmel'nyts'kyi, Ukraine
References:
Abstract: The dual discrete $q$-ultraspherical polynomials $D_n^{(s)}(\mu(x;s)|q)$ correspond to indeterminate moment problem and, therefore, have one-parameter family of extremal orthogonality relations. It is shown that special cases of dual discrete $q$-ultraspherical polynomials $D_n^{(s)}(\mu(x;s)|q)$, when $s=q^{-1}$ and $s=q$, are directly connected with $q^{-1}$-Hermite polynomials. These connections are given in an explicit form. Using these relations, all extremal orthogonality relations for these special cases of polynomials $D_n^{(s)}(\mu(x;s)|q)$ are found.
Keywords: $q$-orthogonal polynomials; dual discrete $q$-ultraspherical polynomials; $q^{-1}$-Hermite polynomials; orthogonality relation.
Received: February 14, 2006; in final form February 28, 2006; Published online March 16, 2006
Bibliographic databases:
Document Type: Article
MSC: 33D45; 81Q99
Language: English
Citation: Valentyna A. Groza, Ivan I. Kachuryk, “On Orthogonality Relations for Dual Discrete $q$-Ultraspherical Polynomials”, SIGMA, 2 (2006), 034, 8 pp.
Citation in format AMSBIB
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\by Valentyna A.~Groza, Ivan I.~Kachuryk
\paper On Orthogonality Relations for Dual Discrete $q$-Ultraspherical Polynomials
\jour SIGMA
\yr 2006
\vol 2
\papernumber 034
\totalpages 8
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\crossref{https://doi.org/10.3842/SIGMA.2006.034}
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