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

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