$S^3$ hyperspherical harmonics in terms of spherical harmonics

Question

In a problem I might want to work one, I would find myself having to work explicitly with hyperspherical harmonics on a three-sphere and their tensor decomposition. It’s well known that $S^3$ can be expressed as the total space of a non-trivial $U(1)$ bundle over the sphere $S^2$. My question is whether someone has tried to extend the spherical harmonics $Y_{lm}$ on $S^2$ to hyperspherical harmonics by replacing them with suitable sections of a bundle over the $S^2$ base?