In this Physics paper the authors claim a relation between the hypergeometric function $_2 F_1$ and the spherical Bessel functions in one particular limit. More specifically, in equation (2.3) they define functions
$$\phi_{n\ell J}(t,\hat{x},\rho)=\dfrac{1}{N^\phi_{\Delta n \ell}}e^{iE_{n,\ell}t}Y_{\ell J}(\hat{x})\bigg[\sin^\ell \rho \cos^\Delta \rho {_2F_1}\left(-n,\Delta+\ell+n,\ell+\frac{d}{2},\sin^2\rho\right)\bigg]\tag{2.3}$$
where $E_{n,\ell}=\Delta+2n+\ell$ and $Y_{\ell J}(\hat{x})$ are the spherical harmonics in $S^{d-1}$. The $N^{\phi}_{\Delta n\ell}$ are normalization constants: $$N^\phi_{\Delta n\ell}=(-1)^n\sqrt{\dfrac{n!\Gamma^2(\ell+\frac{d}{2})\Gamma(\Delta+n-\frac{d-2}{2})}{\Gamma(n+\ell+\frac{d}{2})\Gamma(\Delta+n+\ell)}}.$$
In that case they claim that if $\rho \ll 1$ the following limit holds
$$\dfrac{1}{N^\phi_{\Delta n\ell}}\sin^\ell \rho \cos^\Delta \rho {_2F_1}\left(-n,\Delta+\ell+n,\ell+\frac{d}{2},\sin^2\rho\right)\to \dfrac{1}{\rho^{(d-2)/2}}J_{\ell+(d-2)/2}\big((E_{n,\ell}^2-\Delta^2)^{1/2}\rho\big)\tag{2.5}.$$
My question here is: where does this limit connecting the hypergeometric and the Bessel functions come from? What property of the hypergeometric function is being employed? This seems to be non-trivial since it involves these rather complicated constants $N^\phi_{\Delta n\ell}$.