Second Order in Mass Ratio Radiative-Recoil Corrections to Hyperfine Splitting in Muonium

Radiative-recoil corrections to hyperfine splitting in muonium of orders $\alpha (Z\alpha )(m\mathrm{/}M{)}^2{E}_F$ and $(Z^2\alpha )(Z\alpha )(m\mathrm{/}M{)}^2{E}_F$ are calculated. These corrections are of the second order in the small electron-muon mass ratio. An analytic expression $(-6 \ln2- \frac{3}{4})\alpha (Z\alpha) - \frac{17}{12} (Z^2 \alpha) (Z\alpha)^2 E_F$ is obtained. Numerically the correction is equal to $-0.0351kHz$ and is of the same order of magnitude as the expected accuracy of the current Los Alamos experiment to measure the hyperfine splitting.