Is there a simple way to compute the overlap integral of three hyperspherical harmonics on the three-sphere?
To be more precise, is there a closed form expression for $$ \int_{0}^{2\pi}\int_{0}^{\pi}\int_{0}^{\pi} \sin^2\theta \ \sin\chi\ Y_{\ell_1\ \kappa_1\ m_1}(\theta,\chi,\phi)\ Y_{\ell_2\ \kappa_2\ m_2}(\theta,\chi,\phi)\ Y^{\star}_{\ell_3\ \kappa_3\ m_3}(\theta,\chi,\phi)\ \mathrm{d}\theta \ \mathrm{d}\chi\ \mathrm{d}\phi\,, $$
where ${}^\star$ represents complex conjugation and we have $\ell_i=0,1,2,3\ldots$, $0\leq\kappa_i\leq \ell_i$, $|m_i|\leq \kappa_i$ and the metric on the unit radius round three sphere being given by
$$ \mathrm{d}s^2 = \mathrm{d}\theta^2+\sin^2\theta\left(\mathrm{d}\chi^2+\sin^2\chi\ \mathrm{d}\phi^2\right). $$
For spherical harmonic on the $S^2$, we know that such overlap integral exists in closed form and are the so called Gaunt coefficients. These coefficients can be readily expressed in terms of $3J-$symbols. I guess I would like to know if the same can be done for the three-sphere.
In these coordinates, the hyperspherical harmonics read $$ Y_{\ell\ \kappa \ m}(\theta,\chi,\phi)=N_{\ell\kappa m}\;\sin^\kappa\theta\; {}_2F_1\left(-\ell+\kappa;2+\ell+\kappa;,\frac{3}{2}+\kappa;\frac{1-\cos\theta}{2}\right)\;P_{\kappa}^m(\chi)\;e^{{\rm i}\ m\,\phi} $$ where ${}_2F_1(a;b;c;z)$ is an hypergeometric function, $P_{\kappa}^m(\theta)$ is an associated Legendre polynomial and $N_{\ell\kappa m}$ a normalisation factor so that $$ \int_{0}^{2\pi}\int_{0}^{\pi}\int_{0}^{\pi} \sin^2\theta \ \sin\chi\ Y_{\ell_1\ \kappa_1\ m_1}(\theta,\chi,\phi)\ Y^{\star}_{\ell_2\ \kappa_2\ m_2}(\theta,\chi,\phi)\ \mathrm{d}\theta \ \mathrm{d}\chi\ \mathrm{d}\phi=\delta_{\ell_1\ell_2}\delta_{\kappa_1\kappa_2}\delta_{m_1 m_2}\,. $$