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 $_3F_2$ is found, and it is shown that they have a symmetry property under substitution of the parabolic quantum numbers.