High-order post-Newtonian expansion of the redshift invariant for eccentric-orbit nonspinning extreme-mass-ratio inspirals

We calculate the eccentricity dependence of the high-order post-Newtonian (PN) series for the generalized redshift invariant hut it for eccentric-orbit extreme-mass-ratio inspirals on a Schwarzschild background. These results are calculated within first-order black hole perturbation theory using ReggeWheeler-Zerilli (RWZ) gauge. Our Mathematica code is based on a familiar procedure, using PN expansion of the Mano-Suzuki-Takasugi analytic function formalism for l modes up to a certain maximum and then using a direct general-l PN expansion of the RWZ equation for arbitrarily high l. We calculate dual expansions in PN order and in powers of eccentricity, reaching 10PN relative order and e(20). Detailed knowledge of the eccentricity expansion at each PN order allows us to find within the eccentricity dependence numerous closed-form expressions and multiple infinite series with known coefficients. We find leading logarithm sequences in the PN expansion of the redshift invariant that reflect a similar behavior in the PN expansion of the energy flux to infinity. A set of flux terms and special functions that appear in the energy flux, like the Peters-Mathews flux itself, are shown to reappear in the redshift PN expansion.