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


SIGMA 2 (2006), 034, 8 pages      math.CA/0603408      http://dx.doi.org/10.3842/SIGMA.2006.034

On Orthogonality Relations for Dual Discrete q-Ultraspherical Polynomials

Valentyna A. Groza a and Ivan I. Kachuryk b
a) National Aviation University, 1 Komarov Ave., Kyiv, 03058 Ukraine
b) Khmel'nyts'kyi National University, Khmel'nyts'kyi, Ukraine

Received February 14, 2006, in final form February 28, 2006; Published online March 16, 2006

Abstract
The dual discrete q-ultraspherical polynomials Dn(s)(μ(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 Dn(s)(μ(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 Dn(s)(μ(x;s)|q) are found.

Key words: q-orthogonal polynomials; dual discrete q-ultraspherical polynomials; q-1-Hermite polynomials; orthogonality relation.

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