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Gel’fand pair associated with the quantum group of motions of the plane and \(q\)-Bessel functions. (English) Zbl 0882.16030
Summary: We give an algebraic construction of double cosets of a pair of Hopf algebras and their epimorphism. In the case of Hopf \(C^*\)-algebras, we consider connections of this construction with generalized shift operators, hypercomplex systems and hypergroups. At last, we study the Gel’fand pair associated with the quantum group of motions of the plane and its connection with the Hahn-Exton \(q\)-Bessel functions.

MSC:
16W30 Hopf algebras (associative rings and algebras) (MSC2000)
33D80 Connections of basic hypergeometric functions with quantum groups, Chevalley groups, \(p\)-adic groups, Hecke algebras, and related topics
17B37 Quantum groups (quantized enveloping algebras) and related deformations
81R50 Quantum groups and related algebraic methods applied to problems in quantum theory
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