  The angular dependence of the gravitational radiation emitted in compact
binary mergers and gravitational collapse is usually separated using
spin-weighted spherical harmonics ${}_sY_{\ell m}$ of spin weight $s$, that
reduce to the ordinary spherical harmonics $Y_{\ell m}$ 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 ${}_sS_{\ell m n}$,
where $n$ denotes the "overtone index" of the corresponding Kerr quasinormal
mode frequency $\omega_{\ell m n}$. In this paper we compute the complex-valued
scalar products of the ${}_sS_{\ell m n}$'s with the ${}_sY_{\ell m}$'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.
