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We calculate the eccentricity dependence of the high-order post-Newtonian(PN) series for the generalized redshift invariant $\langle u^t \rangle_\tau$for eccentric-orbit extreme-mass-ratio inspirals on a Schwarzschild background.These results are calculated within first-order black hole perturbation theory(BHPT) using Regge-Wheeler-Zerilli (RWZ) gauge. Our \textsc{Mathematica} codeis based on a familiar procedure, using PN expansion of theMano-Suzuki-Takasugi (MST) analytic function formalism for $l$ modes up to acertain maximum and then using a direct general-$l$ PN expansion of the RWZequation for arbitrarily high $l$. We calculate dual expansions in PN order andin powers of eccentricity, reaching 10PN relative order and $e^{20}$. Detailedknowledge of the eccentricity expansion at each PN order allows us to findwithin the eccentricity dependence numerous closed-form expressions andmultiple infinite series with known coefficients. We find leading logarithmsequences in the PN expansion of the redshift invariant that reflect a similarbehavior in the PN expansion of the energy flux to infinity. A set of fluxterms and special functions that appear in the energy flux, like thePeters-Mathews flux itself, are shown to reappear in the redshift PN expansion.

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