Ring-Shaped Functions and Wigner $6j$-Symbols

We obtain an explicit expression for the ring-shaped matrix relating the ring-shaped functions corresponding to different values of an axiality parameter and find the relation between this matrix and the Wigner $6j$-symbols. We investigate the motion of a quantum particle in a ring-shaped model with a zero “bare” potential and find bases factored in spherical and cylindrical coordinates for this model. We derive a formula generalizing the Rayleigh expansion for a plane wave in terms of spherical waves in a ring-shaped model.