  We present an analytic computation of Detweiler's redshift invariant for a
point mass in a circular orbit around a Kerr black hole, giving results up to
8.5 post-Newtonian order while making no assumptions on the magnitude of the
spin of the black hole. Our calculation is based on the functional series
method of Mano, Suzuki and Takasugi, and employs a rigorous mode-sum
regularization prescription based on the Detweiler-Whiting singular-regular
decomposition. The approximations used in our approach are minimal; we use the
standard self-force expansion to linear order in the mass ratio, and the
standard post-Newtonian expansion in the separation of the binary. A key
advantage of this approach is that it produces expressions that include
contributions at all orders in the spin of the Kerr black hole. While this work
applies the method to the specific case of Detweiler's redshift invariant, it
can be readily extended to other gauge invariant quantities and to higher
post-Newtonian orders.
