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We derive new terms in the post-Newtonian (PN) expansion of the generalized redshift invariant hu tiτ for a small body in eccentric, equatorial orbit about a massive Kerr black hole. The series is computed analytically using the Teukolsky formalism for first-order black hole perturbation theory, along with the Chrzanowski, Cohen, Kegeles method for metric reconstruction using the Hertz potential in ingoing radiation gauge. Modal contributions with small values of l are derived via the semianalytic solution of Mano-Suzuki-Takasugi, while the remaining values of l to infinity are determined via direct expansion of the Teukolsky equation. Each PN order is calculated as a series in eccentricity e but kept exact in the primary black hole’s spin parameter a. In total, the PN terms are expanded to e16 through 6PN relative order, and separately to e10 through 8PN relative order. Upon grouping eccentricity coefficients by spin dependence, we find that many resulting component terms can be simplified to closed-form functions of eccentricity, in close analogy to corresponding terms derived previously in the Schwarzschild limit. We use numerical calculations to compare convergence of the full series to its Schwarzschild counterpart and discuss implications for gravitational wave analysis. 

We calculate the high-order post-Newtonian (PN) expansion of the energy and angular momentum fluxes onto the horizon of a nonspinning black hole primary in eccentric-orbit extreme-mass-ratio inspirals. The first-order black hole perturbation theory calculation uses Mathematica and makes an analytic expansion of the Regge-Wheeler-Zerilli functions using the Mano-Suzuki-Takasugi formalism. The horizon absorption, or tidal heating and torquing, is calculated to 18PN relative to the leading horizon flux (i.e., 22PN order relative to the leading quadrupole flux at infinity). Each PN term is a function of eccentricity e and is calculated as a series to e10. A second expansion, to 10PN horizon-relative order (or 14PN relative to the flux at infinity), is computed deeper in eccentricity to e20. A number of resummed closed-form functions are found for the low PN terms in the series. Using a separate Teukolsky perturbation code, numerical comparisons are made to test how accurate the PN expansion is when extended to a close p =10 orbit. We find that the horizon absorption expansion is not as convergent as a previously computed infinity-side flux expansion. However, given that the horizon absorption is suppressed by 4PN, useful results can be obtained even with an orbit as tight as this for e ≲1 /2 . Combining the present results with our earlier expansion of the fluxes to infinity makes the knowledge of the total dissipation known to 19PN for eccentric-orbit nonspinning extreme-mass-ratio inspirals.