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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
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
Citation:
Valentyna A. Groza, Ivan I. Kachuryk, “On Orthogonality Relations for Dual Discrete $q$-Ultraspherical Polynomials”, SIGMA, 2 (2006), 034, 8 pp.
Linking options:
https://www.mathnet.ru/eng/sigma62 https://www.mathnet.ru/eng/sigma/v2/p34
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Abstract page: | 161 | Full-text PDF : | 48 | References: | 41 |
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