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Representations of the quantum algebra \(\text{su}_q(1,1)\) and discrete \(q\)-ultraspherical polynomials. (English) Zbl 1128.17013

Summary: We derive orthogonality relations for discrete \(q\)-ultraspherical polynomials and their duals by means of operators of representations of the quantum algebra \(\text{su}_q(1,1)\). Spectra and eigenfunctions of these operators are found explicitly. These eigenfunctions, when normalized, form an orthonormal basis in the representation space.

MSC:

17B37 Quantum groups (quantized enveloping algebras) and related deformations
33D45 Basic orthogonal polynomials and functions (Askey-Wilson polynomials, etc.)
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