Analytical solutions of bound timelike geodesic orbits in effective-one-body frame

We derive the approximate analytical solutions of the bound timelike geodesic orbits in the effective-one-body frame with an extreme-mass ratio limit. The analytical solutions are expressed in terms of the elliptic integrals using Mino time $\ensuremath{\lambda}$ as the independent variable. Since Mino time decouples the $r$ and $\ensuremath{\theta}$ motion, we also give explicit expressions for three orbital frequencies ${\mathrm{\ensuremath{\Omega}}}_{r},{\mathrm{\ensuremath{\Omega}}}_{\ensuremath{\theta}},{\mathrm{\ensuremath{\Omega}}}_{\ensuremath{\phi}}$ using the Fourier series expansion. With these analytical expressions at hand, we can perform Fourier expansions in Mino time $\ensuremath{\lambda}$ for any function expressed in terms of the coordinates $(r,\ensuremath{\theta},\ensuremath{\phi})$. In particular, the observer's time $t$ is decomposed into Mino time $\ensuremath{\lambda}$, and the frequency-domain description is constructed from the $\ensuremath{\lambda}$-Fourier expansion and the expansion of $t$. These analytical expressions are quite simple to implement, and can be applicable for calculating gravitational waves from extreme mass-ratio inspirals with the frequency-domain Teukolsky equation.