Gelʹfand pair associated with the quantum group of motions of the plane and q-Bessel functions

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 Gelʹfand pair associated with the quantum group of motions of the plane and its connection with the Hahn-Exton q-Bessel functions.