Hydrogen atom as indicator of hidden symmetry of a ring-shaped potential

The idea of an analogy between a ring-shaped potential and a Coulomb potential is advanced. It is shown that the expansion of the parabolic basis with respect to the spherical basis in the problem of a ring-shaped potential is determined by the Clebsch-Gordan coefficients of the group SU(2) continued to the region of arbitrary real indices. The connection between these coefficients and the functions 3F2 is found, and it is shown that they have a symmetry property under substitution of the parabolic quantum numbers.