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Clebsch-Gordan coefficients for the quantum algebra \(su_ q(2)\). (English. Russian original) Zbl 0820.17019
Theor. Math. Phys. 98, No. 1, 1-7 (1994); translation from Teor. Mat. Fiz. 98, No. 1, 3-11 (1994).
The authors derive the expression of Rodrigues type for the Clebsch–Gordan coefficients of the tensor products of irreducible finite- dimensional representations of the quantum algebra \(\text{su}_ q(2)\). The \(q\)-gamma and \(q\)-beta functions are used for the calculation of the normalization constant in this expression. Then the \(q\)-analogue of the Fock–Racah representation for these Clebsch-Gordan coefficients is obtained. Clebsch–Gordan coefficients are expressed in terms of the basic hypergeometric functions \({}_ 3 \varphi_ 2\) of the unit argument. A connection of the symmetry relations for Clebsch-Gordan coefficients to these of the function \({}_ 3 \varphi_ 2\) is indicated. The relation between Clebsch–Gordan coefficients and \(q\)-Hahn polynomials is given. Note that most of the results are already contained in, for example, Chapter 14 of the book “Representation of Lie groups and special functions. Vol 3: Classical and quantum groups and special functions” by N. Ya. Vilenkin and the reviewer (Kluwer, Dordrecht, 1992; Zbl 0778.22001).
MSC:
17B37 Quantum groups (quantized enveloping algebras) and related deformations
33D45 Basic orthogonal polynomials and functions (Askey-Wilson polynomials, etc.)
81R50 Quantum groups and related algebraic methods applied to problems in quantum theory
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