  In this work, we elaborate on the development of a general relativistic
formalism that allows one to analytically express the mass and spin parameters
of the Kerr black hole in terms of observational data: the total redshift and
blueshift of photons emitted by geodesic massive particles revolving the black
hole and their orbital parameters. Thus, we present concise closed formulas for
the mass and spin parameters of the Kerr black hole in terms of few directly
observed quantities in the case of equatorial circular orbits either when the
black hole is static or is moving with respect to a distant observer.
Furthermore, we incorporate the gravitational dragging effect generated by the
rotating nature of the Kerr black hole into the analysis and elucidate its
non-trivial contribution to the expression for the light bending parameter and
the frequency shifts of photons emitted by orbiting particles that renders
simple symmetric expressions for the kinematic redshift and blueshift. We also
incorporate the dependency of the frequency shift on the azimuthal angle, a
fact that allows one to express the total redshift/blueshift along any point of
the orbit of the revolving particle for the cases when the black hole is both
static or moving with respect to us. These formulas allow one to compute the
Kerr black hole parameters by applying this general relativistic formalism to
astrophysical systems like the megamaser accretion disks orbiting supermassive
black holes at the core of active galactic nuclei. Our results open a new
window to implement parameter estimation studies to constrain black hole
variables, and they can be generalized to black hole solutions beyond Einstein
gravity.
