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We present new results on the analytic eccentricity dependence of severalsequences of gravitational wave flux terms at high post-Newtonian (PN) orderfor extreme-mass-ratio inspirals. These sequences are the leading logarithms,which appear at PN orders $x^{3k} \log^k(x)$ and $x^{3k+3/2} \log^k(x)$ forintegers $k\ge 0$ ($x$ a PN compactness parameter), and the subleadinglogarithms, which appear at orders $x^{3k} \log^{k-1}(x)$ and $x^{3k+3/2}\log^{k-1}(x)$ ($k\ge 1$), in both the energy and angular momentum radiated toinfinity. For the energy flux leading logarithms, we show that to arbitrarilyhigh PN order their eccentricity dependence is determined by particular sumsover the function $g(n,e_t)$, derived from the Newtonian mass quadrupolemoment, that normally gives the spectral content of the Peters-Mathews flux asa function of radial harmonic $n$. An analogous power spectrum$\tilde{g}(n,e_t)$ determines the leading logarithms of the angular momentumflux. For subleading logs, the quadrupole power spectra are again shown to playa role, providing a distinguishable part of the eccentricity dependence ofthese flux terms to high PN order. With the quadrupole contribution understood,the remaining analytic eccentricity dependence of the subleading logs can inprinciple be determined more easily using black hole perturbation theory. Weshow this procedure in action, deriving the complete analytic structure of the$x^6 \log(x)$ subleading-log term and an analytic expansion of the $x^{9/2}$subleading log to high order in a power series in eccentricity. We discuss howthese methods might be extended to other sequences of terms in the PN expansioninvolving logarithms.

Eccentric-orbit EMRI radiation: Analytic forms of leading-logarithm and subleading-logarithm flux terms at high PN orders