Hyperfine structure of muonic lithium ions

On the basis of perturbation theory in fine structure constant $\alpha$ and the ratio of electron to muon masses we calculate recoil corrections of order $\alpha^4 (M_e/M_\mu)$, $\alpha^4 (M_e/M_\mu)^2\ln(M_e/M_\mu)$, $\alpha^4 (M_e/M_\mu)^2$, $\alpha^5(m_e/m_\mu)\ln(m_e/m_\mu)$ to hyperfine splitting of the ground state in muonic lithium ions $(\mu\ e\ ^6_3\mathrm{Li})^+$ and $(\mu e\ ^7_3\mathrm{Li})^+$. We obtain total results for the ground state small hyperfine splittings in $(\mu\ e\ ^6_3\mathrm{Li})^+$ $\Delta\nu_1=14153.03$ MHz and $\Delta\nu_2=21571.26$ MHz and in $(\mu\ e\ ^7_3\mathrm{Li})^+$ $\Delta\nu_1=13991.97$ MHz and $\Delta\nu_2=21735.03$ MHz which can be considered as a reliable estimate for a comparison with future experimental data.