Mixing of spherical and spheroidal modes in perturbed Kerr black holes

The angular dependence of the gravitational radiation emitted in compact binary mergers and gravitational collapse is usually separated using spin-weighted spherical harmonics ₛY_m of spin weight s, that reduce to the ordinary spherical harmonics Y_m when $s=0$. Teukolsky first showed that the perturbations of the Kerr black hole that may be produced as a result of these events are separable in terms of a different set of angular functions: the spin-weighted spheroidal harmonics ₛS_mn, where n denotes the ``overtone index'' of the corresponding Kerr quasinormal mode frequency ω_mn. In this paper we compute the complex-valued scalar products of the ₛS_mn's with the ₛY_m's (``spherical-spheroidal mixing coefficients'') and with themselves (``spheroidal-spheroidal mixing coefficients'') as functions of the dimensionless Kerr parameter j. Tables of these coefficients and analytical fits of their dependence on j are available online for use in gravitational-wave source modeling and in other applications of black-hole perturbation theory.