of the modes and $\Omega_p = 2\pi^{(p+1)/2}/\Gamma[(p+1)/2]$ is the volume of the unit $p$-sphere. Here, $M^{(1)}_l$ and $M^{(2)}_l$ are the multiplicities of the scalar and divergence-free vector spherical harmonics, respectively, on the $p$-sphere. These multiplicities are given by (see, e.g., Refs. [@CM; @CH]) $$M^{(1)}_l = \frac{(2l+p-1)(l+p-2)!}{(p-1)!l!}\,,$$ and $$M^{(2)}_l = \frac{(2l+p-1)(l+p-1)!}{(l+1)(l+p-2)(p-2)!(l-1)!}\,.$$ Eq. (\[ABSCR\]) is derived in Appendix C.

For either physical modes I or II, we have for large $r (\approx r^*)$ that \[see Eqs. (\[varphi1\]) and (\[varphi2\])\] $$\left\{ \omega^2 + \frac{d^2\ }{dr^2} - \left[l(l+p-1)+\frac{p(p-2)}{4}\right]
\frac{1}{r^2}\right\}\varphi^{(\lambda\leftarrow)}_{\omega l}(r) \approx 0\,.$$ The asymptotic solution with the correct normalization, i.e. with the coefficient of $e^{-i\omega r}$ being one, is $$\varphi^{(\lambda\leftarrow)}_{\omega l}(r)
\approx \sqrt{2\pi\omega r}J_{l+(p-1)/2}(\omega r)\,\ \  (r_H \ll r)\,.$$ Thus, for small $\omega r$ $$\varphi^{(\lambda\leftarrow)}_{\omega l}(r)
\approx C_l \left( {r}/{r_H}\right)^{l+p/2} \,\ \ 
(r_H \ll r \ll \omega^{-1})\,, 
\label{Cl}$$ where $$C_l = \frac{\sqrt{4\pi}}{\Gamma\left( l+ (p+1)/2\right)}
\left(\frac{\omega r_H}{2}\right)^{l+p/2}\,.
\label{Clf}$$

Now, near the horizon, where $V_\lambda \ll \omega^2\;\;$ ($\lambda=1,2$), these solutions behave like \[see Eqs. (\[varphi1\]) and (\[varphi2\])\] $$\varphi^{(\lambda\leftarrow)}_{\omega l}(r)
\approx D_l^{(\lambda)}e^{-i\omega r^*}\,\ \ (r-r_H \ll \omega^2 r_H^3)\,,$$ where the constants $D_l^{(\lambda)}$ will be determined later. Hence, for $|\omega r^*| \ll 1$ we have in this region $\varphi^{(\lambda\leftarrow)}_{\omega l}(r) \approx D_l^{(\lambda)}$ and the absorption probability is given 